Pohl's Introduction to Physics Volume 1

Klaus Lüders Robert O. Pohl Editors Pohl’s Introduction to Physics Volume 1: Mechanics, Acoustics and Thermodynamics

With videos of 77 experiments Pohl’s Introduction to Physics Klaus Lüders Robert O. Pohl Editors

Pohl’s Introduction to Physics

Mechanics, Acoustics and Thermodynamics, Vol. 1 Editors Klaus Lüders Robert O. Pohl Fachbereich Physik Department of Physics Freie Universität Berlin Cornell University Berlin, Germany Ithaca, NY, USA

Translated by Prof. William D. Brewer, PhD, Fachbereich Physik, Freie Universität Berlin, Berlin, Germany

Additional material to this book can be downloaded from http://extra.springer.com.

ISBN 978-3-319-40044-0 ISBN 978-3-319-40046-4 (eBook) DOI 10.1007/978-3-319-40046-4

Library of Congress Control Number: 2017944730

© Springer International Publishing Switzerland 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are be- lieved to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institu- tional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the Second English Edition

The first English edition of Pohl’s “Physical Principles of Mechanics and Acoustics” appeared in 1932 (published by Blackie & Son, Ltd., London and Glasgow). It was based on the second edition of Pohl’s “Einführung in die Physik, Mechanik und Akustik” (Julius Springer, 1931). The present new, second English edition, based on the 21st edition of “Pohls Einführung in die Physik”, Vol. 1, (Mechanik, Akustik und Wärmelehre) (Springer Spektrum, 2017), has now been published after nearly 85 years! Following R.W. Pohl’s death in 1976 and the posthumous appearance of the 18th edition in 1983, since 2004 we have edited three new revised and updated editions, based mainly on the 16th edition (1964), the 13th edition (1955), and the 18th edition. The major change in this new series was the addition of 74 videos demonstrating many of the experiments that R.W. Pohl had developed and used. It is also augmented by comments in the margins when they appeared to be helpful as additional explanations or were needed to provide more modern information (see the Preface to the 19th edition). We have in addition included the collection of exercises which Pohl provided for the first English edition, and we have supplemented these (see the Preface to the 20th edition). The exercises were not included in any of the first 18 German editions. We have furthermore modified some mathematical formulations, symbols, and units so that they conform to the recommendations of the International System of Units (SI). We gratefully acknowledge the help of Professor W.D. Brewer of the Physics Department of the Free University of Berlin, not only for carrying out the translation of the text with great quality and speed, but also, and this is probably even more important, for his help with the identification and clarification of unclear parts in the text and in our comments. The English-language readers will appreciate the numerous links he added for further information. We also wish to thank Dr. T. Schneider and Ms. D. Mennecke-Buehler of the Springer-Verlag for making this edition possible, and for their generous help in carrying out its preparation and production. Berlin and Göttingen, March 2017 K. Lüders R. O. Pohl

v Preface to the Twenty-First German Edition

One of the most extensive changes in this new edition concerns its format, and is intended to make the books more readable. “Pohl” will now be published for the first time as an e-book, but also as a printed version with a new format. The numbering of the chapters, figures, equations etc. now conforms to the usual system in modern textbooks. The relevant exercises are given at the end of each chapter. We have also made major changes to the accompanying videos. In the e-book format, they are now more readily accessible and can be called up directly using the appropriate links in the text. All those videos which were produced in cooperation with the Institute for Scientific Films (IWF) in Göttingen are now available in their original quality and with a spoken text. The remaining videos were to some extent supplemented and in one case replaced by an improved version. Two new videos have been added: “Kepler Ellipses” (an excerpt from the opening show of the 2009 “High- lights der Physik” in Cologne), and “The Magdeburg Hemispheres” (an excerpt from a Lichtenberg Lecture given by Prof. G. Beuermann in Göttingen). At the same time, we have taken advantage of the opportunity to review all of the text critically. This has led to a number of clarifications, both in the text and in the figures, including the addition of several new figures, notably in Chapters 12 and 19. We owe special thanks to Prof. K. Samwer from the First Physics Institute of the University of Göttingen for his committed and helpful support of the preparation of this new edition in a variety of ways. We also wish to give particular thanks to Prof. G. Beuermann, J. Feist and C. Mahn from that Institute for their various and dedicated assistance. Furthermore, we wish especially to thank Dr. J. Kirstein from the Physics Didactics group at the Free University of Berlin for his professional and speedy editing of the videos. We are once again indebted to the Physics Department of the Free University and its administration for providing working facilities and for the helpful efforts of many of its members in solving technical problems, especially in connection with computer technology. Finally, we heartily thank the Springer-Verlag, and in particular Dr. V. Spillner, Ms. M. Maly and Ms. B. Saglio for their stimulating and agreeable cooperation.

Berlin and Göttingen, July 2015 K. Lüders R. O. Pohl From the Preface to the 20th Edition (2008)

The many positive comments from readers of the 19th edition of POHL’s Introduction to Mechanics, Acoustics and Thermodynamics have encouraged us to publish a new, revised edition. This also gave us the opportunity to include some supplementary material which we believe to be important. In addition to new or revised marginal comments and a few factual clarifications within the text, this new material consists in particular of additional videos and a set of exercises for the readers. Also, the sections on osmosis and diffusion from earlier editions have now been included here. This time, the videos were filmed under our own direction in the new lecture hall in Göttingen, in addition to several filmed in cooperation with the Physics Didactics group at the Free University in Berlin. In choosing the topics, we have again been guided on the one hand by our attempt to present ’lively’ illustrations of physics, and on the other by our intention to document typical demonstration experiments in the tradition of POHL, which in some cases are no longer being showneveninGöttingen. The major portion of the exercises originates with an earlier English-language edition (from 1932!); they are thus POHL’s original exercises. However, we found it desirable to add some more exercises which deal with questions that either relate directly to the videos or illustrations, or that complement the experiments, which are sometimes described rather briefly in the text due to lack of space. These exercises are thus not problem sets in the usual sense, but rather they are intended to help the reader achieve a better understanding of the sometimes difficult physical concepts described in this volume, and furthermore they provide additional information.

vii From the Preface to the 19th Edition (2004)

For over thirty years, from 1919 to 1952, R.W. POHL gave the introductory lectures in experimental physics at the University of Göttingen for students of a variety of major subjects. The three-volume set of textbooks based on those lectures pursued a double goal for many years: On the one hand, they were intended to arouse the readers’ interest in physics; and on the other, they served as textbooks for teaching basic physics to interested students. Even though in more recent decades, physics education at the university level has adjusted more and more to the needs of various professions and now includes many specialized courses, the goals of POHL’s works are still valid and topical. We are therefore convinced that these books still convey a fascination for the experimental investigation of physical phenomena and deserve a place on the bookshelves of modern-day students. That is the reason for the present new edition, initially covering the ﬁelds of mechanics, acoustics and thermodynamics. A second volume will present the most important topics from electrodynamics and optics.

For many readers, the most noticeable characteristic of POHL’s books is the large number of experiments which they illustrate and describe in detail; these demonstrate how one must ask questions of Nature in order to uncover her secrets. The presentation of demonstration experiments using shadow projections, which fix the attention of the observer on the essentials of the demonstration, is an integral part of this program. But in addition, we want to provide readers with the opportunity to experience the demonstrations just as they have been presented in the Göttingen lecture hall for more than 80 years. For this reason, we have complemented this edition with two CD-ROMs containing short videos. The first of these is an original film of a lecture delivered by R.W. POHL in 1952 (Video 1).1 We hope that our readers will enjoy watching these videos as much as we have enjoyed filming them.

In order to retain the liveliness of the “POHLs”, as the books are often called, it seemed important to us to maintain the manner of presentation of their original author as nearly as possible. Since, however, the ﬁrst volume alone was available in no fewer than fourteen different editions, we had to make choices. This book is based mainly on the 16th edition, which appeared in 1964. Occasionally, however, we refer to other editions, in particular the 13th (1955) and the 18th (1983).

1 Video 1: “R.W. POHL Lecturing” http://tiny.cc/fpqujy This ﬁlm, shot by FRITZ LUETY (now professor emeritus at the University of Utah in Salt Lake City) while he was a graduate student in 1952, for the summer celebration of the Göttingen physics institute, shows a lecture on oscillatory motion given by POHL, with several demonstration experiments that are described in Chap. 11 of this book. From the Preface to the 19th Edition (2004) ix

We have as far as possible avoided making changes to the text. Among the exceptions are our more frequent use of vectors and integrals, i.e. mathematical objects with which today’s readers are in general familiar. Furthermore, we have adapted the symbols and units to modern usage, in order to spare our readers the unnecessary annoyance of conversion. Our own attempts at enriching the text are limited to comments in the margins, which contain both direct explanations of material in the text and references to newer developments in the areas of physics treated. From the Preface to the First Edition (1930)

This book contains the first part of my lectures on experimental physics. An effort has been made to present them as simply as possible. This is intended to make the book accessible not only to students and teachers, but also to other readers with an interest in physics. Basic experiments occupy the most prominent place in the presentation. They serve in particular to clarify the concepts and to provide an overview of the magnitudes of the quantities involved. Quantitative details are not emphasized. A large collection of demonstration experiments occupies considerable space. In our lecture hall in Göttingen, we have a smooth-floored area of 12 5m2. The cumbersome accessory of earlier lecture halls, i.e. the heavy, stationary demonstration table, has long since been dispensed with. Instead, smaller tables are set up as needed, and they are no more anchored to the floor than is the furniture in a living room. The clarity of the experimental arrangement and the accessibility of the individual experimental setups are enhanced considerably by the use of these convenient tables. Most of them can be rotated around their vertical axis and they are readily adjustable in height. Thus, the annoying overlap of perspective between different setups can be avoided. The setup currently being demonstrated can be highlighted and made visible to every member of the audience by panning the tables. The apparatus used is as simple as possible and consists of a moderate number of devices. Many of the setups are described here for the first time. They can be obtained, as can other accessories for lecture demonstrations, from the Spindler & Hoyer company in Göttingen. The main portion of the illustrations in the book are based on photographs. Many of the images are presented as silhouettes. This method of presentation is especially suitable for reproduction in book form; in addition, it often provides some indication of the dimensions of the experimental setup. Finally, showing the experiments as silhouettes makes them visible even in large lecture halls, which demand clear-cut outlines, not interrupted by incidental details such as laboratory stands, frames etc.

Göttingen, March 1930 R. W. Pohl R.W. Pohl (1884–1976)

R.W. POHL (1884–1976) discussing color centers (F-centers), elementary crystal lattice defects which were discovered at his institute and investigated there for many years. He is shown during a visit to the Ansco Re- search Laboratory in Binghamton, NY in the year 1951. Details of POHL’s life and work can be found on the website http://rwpohl.mpiwg-berlin.mpg.de of the Max Planck Institute for the History of Science (MPIWG). There, one can ﬁnd links to other literature, scientiﬁc institutions and websites which offer information and doc- uments on the teaching and research of the famous physicist in Göttingen. In addition, the documentary video “Simplicity is the Mark of Truth” by Ekkehard SIEKER (Video 1 from Vol. 2) can be found on the MPIWG web site, together with all the other videos from both volumes and other audiovisual materials, available both for videostreaming or as downloads.

xi Contents

Part I Mechanics

1 Introduction; Distance and Time Measurements ...... 3 1.1 Introduction ...... 3 1.2 Distance and Length Measurements. Direct Distance Measurements 4 1.3 The Meter as a Unit of Length ...... 6 1.4 Indirect Length Measurements of Very Large Distances ...... 7 1.5 Angle Measurements ...... 9 1.6 Time Determinations. True Time Measurements ...... 10 1.7 Clocks and Graphical Registration ...... 11 1.8 Measurement of Periodic Sequences of Equal Times and Lengths .. 13 1.9 Indirect Time Measurements ...... 15 Exercises ...... 16

2 The Description of Motion: Kinematics ...... 17 2.1 Definition of Motion. Frames of Reference ...... 17 2.2 Definition of Velocity. Example of a Velocity Measurement ...... 18 2.3 Definition of Acceleration: The Two Limiting Cases ...... 20 2.4 Path Acceleration and Linear Motion ...... 22 2.5 Constant Radial Acceleration and Circular Orbits ...... 26 2.6 Distinguishing Physical Quantities and Their Numerical Values .... 28 2.7 Base Quantities and Derived Quantities ...... 29 Exercises ...... 30

3 Fundamentals of Dynamics ...... 31 3.1 Force and Mass ...... 31 3.2 Measurements of Force and Mass. NEWTON’s Fundamental Equation of Motion ...... 34

xiii xiv Contents

3.3 The Units of Force and Mass. Expressions Containing Physical Quantities ...... 38 3.4 Density and Speciﬁc Volume ...... 38 Exercises ...... 39

4 Applications of NEWTON’s Equation ...... 41 4.1 Constant Acceleration in a Straight Line ...... 41 4.2 Circular Motion and Radial Forces ...... 44 4.3 Sinusoidal Oscillations: The Gravity Pendulum as a Special Case ... 49 4.4 Motions Around a Central Point ...... 54 4.5 Elliptical Orbits and Elliptically Polarized Oscillations ...... 56

4.6 LISSAJOUS Orbits ...... 57

4.7 KEPLER’s Elliptical Orbits and the Law of Gravity ...... 58 4.8 The Gravitational Constant ...... 60 4.9 The Law of Gravity and Celestial Mechanics ...... 62 Exercises ...... 65

5 Three Useful Concepts: Work, Energy, and Momentum ...... 67 5.1 Preliminary Remarks ...... 67 5.2 Work and Power ...... 67 5.3 Energy and Its Conservation ...... 71 5.4 First Applications of the Conservation Law of Mechanical Energy .. 74 5.5 Impulse and Momentum ...... 75 5.6 Momentum Conservation ...... 76 5.7 First Applications of Momentum Conservation ...... 77 5.8 Momentum and Energy Conservation During Elastic Collisions of Objects ...... 79 5.9 Momentum Conservation in Inelastic Collisions and the Ballistic Pendulum ...... 80 5.10 Non-Central Collisions ...... 82 5.11 Motions Against Dissipative Forces ...... 82 5.12 The Production of Forces with and without Consuming Power ... 86 5.13 Closing Remarks ...... 87 Exercises ...... 88 Contents xv

6 Rotational Motion of Rigid Bodies ...... 91 6.1 Introductory Remarks ...... 91 6.2 Deﬁnition of Torque ...... 91 6.3 The Production of Known Torques, the Constant D*, and the Angular Velocity ! ...... 94 6.4 The Moment of Inertia, Equation of Motion for Rotations, Torsional Oscillations ...... 97 6.5 The Physical Pendulum and the Beam Balance ...... 102 6.6 Angular Momentum ...... 104 6.7 Free Axes ...... 108 6.8 Free Axes of Humans and Animals ...... 111 6.9 Deﬁnition of the Spinning Top and Its Three Axes ...... 112 6.10 The Nutation of a Force-Free Top and Its Fixed Spin Axis ...... 115 6.11 Tops Acted on by Torques. Precession of the Angular-Momentum Axis ...... 117 6.12 Precession Cone with Nutation ...... 121 6.13 A Top with Only Two Degrees of Freedom ...... 123 Exercises ...... 125

7 Accelerated Frames of Reference ...... 129 7.1 Preliminary Remarks. Inertial Forces ...... 129 7.2 Frames of Reference with Only Path Acceleration ...... 130 7.3 Frames of Reference with Radial Acceleration. Centrifugal and CORIOLIS Forces ...... 133 7.4 Vehicles as Accelerated Frames of Reference ...... 141 7.5 The Gravity Pendulum as a Plumb Bob in Accelerated Vehicles .... 144 7.6 Earth as an Accelerated Frame of Reference. Centrifugal Acceleration of Bodies at Rest ...... 146

7.7 Earth as an Accelerated Frame of Reference. CORIOLIS Force on Moving Bodies ...... 148 Exercises ...... 151

8 Some Properties of Solids ...... 153 8.1 Preliminary Remarks ...... 153 8.2 Elastic Deformation, Flow and Solidiﬁcation ...... 153

8.3 HOOKE’s Law and POISSON’s Relation ...... 155 8.4 Shear Stress ...... 156 xvi Contents

8.5 Normal, Shear and Principal Stress ...... 157 8.6 Bending and Twisting (Torsion) ...... 160 8.7 Time Dependence of Deformation. Elastic Aftereffects and Hysteresis ...... 165 8.8 Rupture Strength and Speciﬁc Surface Energy of Solids ...... 167 8.9 Sticking and Sliding Friction ...... 169 8.10 Rolling Friction ...... 172 Exercises ...... 173

9 Liquids and Gases at Rest ...... 175 9.1 The Free Displacements of Liquid Molecules ...... 175 9.2 Pressure in Liquids. Manometers ...... 178 9.3 The Isotropy of Pressure and Its Applications ...... 179 9.4 The Pressure Distribution in a Gravitational Field. Buoyancy ..... 182 9.5 Cohesion of Liquids: Tensile Strength, Speciﬁc Surface Energy, and Surface Tension ...... 185 9.6 Gases as Low-Density Liquids Without Surfaces. BOYLE’s Law ...... 193 9.7 A Model Gas. Pressure Due to Random Molecular Motions (Thermal Motion) ...... 195 9.8 The Fundamental Equation of the Kinetic Theory of Gases. Velocity of the Gas Molecules ...... 196 9.9 The Earth’s Atmosphere. Atmospheric Pressure in Demonstration Experiments ...... 198 9.10 The Pressure Distribution of Gases in the Gravitational Field. The Barometric Pressure Formula ...... 202 9.11 Static Buoyancy in Gases ...... 205 9.12 Gases and Liquids in Accelerated Frames of Reference ...... 206 Exercises ...... 208

10 Motions in Liquids and Gases ...... 211 10.1 Three Preliminary Remarks ...... 211 10.2 Internal Friction and Boundary Layers ...... 211 10.3 Laminar Flow: Fluid Motions Which Occur when Friction Plays a Decisive Role ...... 214

10.4 The REYNOLDS Number ...... 217

10.5 Frictionless Fluid Motion and BERNOULLI’s Equation ...... 220 Contents xvii

10.6 Flow Around Obstacles. Sources and Sinks. Irrotational or Potential Flows ...... 226 10.7 Rotations of Fluids and Their Measurement. The Irrotational Vortex Field ...... 228 10.8 Vortices and Separation Surfaces in Nearly Frictionless Fluids ..... 232 10.9 Flow Resistance and Streamline Proﬁles ...... 234 10.10 The Dynamic Transverse Force or Lift ...... 237 10.11 Applications of the Transverse Force ...... 241 Exercises ...... 244

Part II Vibrations and Waves

11 Vibrations ...... 247 11.1 Preliminary Remarks ...... 247 11.2 Producing Undamped Vibrations ...... 247 11.3 The Synthesis of Non-Sinusoidal Periodic Processes from Sine Curves 251 11.4 The Spectral Representation of Complex Oscillatory Processes .... 256 11.5 Elastic Transverse Vibrations of Linear Solid Bodies Under Tensile Stress ...... 259 11.6 Elastic Longitudinal and Torsional Vibrations of Stressed Linear Solid Bodies ...... 263 11.7 Elastic Vibrations in Columns of Liquids and Gases ...... 265 11.8 Normal Modes of Stiff Linear Bodies ...... 269 11.9 Normal Modes of 2-Dimensional and 3-Dimensional Bodies. Thermal Vibrations ...... 271 11.10 Forced Oscillations ...... 273 11.11 Energy Transfer Stimulated by Resonance ...... 278 11.12 The Importance of Resonance for the Detection of Pure Sinusoidal Oscillations. Spectral Apparatus ...... 279 11.13 The Importance of Forced Oscillations for Distortion-Free Recording of Non-Sinusoidal Oscillations ...... 281 11.14 The Ampliﬁcation of Oscillations ...... 282 11.15 Two Coupled Oscillators and Their Forced Oscillations ...... 283 11.16 Damped and Undamped Wobble Oscillations ...... 286 11.17 Relaxation (or Toggle) Oscillations ...... 288 Exercises ...... 289 xviii Contents

12 Travelling Waves and Radiation ...... 291 12.1 Travelling Waves ...... 291

12.2 The DOPPLER Effect ...... 293 12.3 Interference ...... 295 12.4 Interference with Two Slightly Different Source Frequencies ..... 296 12.5 Standing Waves ...... 297 12.6 The Propagation of Travelling Waves ...... 299 12.7 Reﬂection and Refraction ...... 302 12.8 Image Formation ...... 303 12.9 Total Reﬂection ...... 304 12.10 Shockwaves when the Wave Velocity Is Exceeded ...... 307

12.11 HUYGHENS’ Principle ...... 308 12.12 Model Experiments on Wave Propagation ...... 309 12.13 Quantitative Results for Diffraction by a Slit ...... 311

12.14 FRESNEL’s Zone Construction ...... 315 12.15 Narrowing of the Interference Fringes by a Lattice Arrangement of the Wave Sources ...... 317 12.16 Interference of Wave Trains of Limited Length ...... 321 12.17 The Production of Longitudinal Waves and Their Velocities ...... 321 12.18 High-Frequency Longitudinal Waves in Air. The Acoustic Replica Method ...... 323 12.19 The Radiation Pressure of Sound. Sound Radiometers ...... 326 12.20 Reﬂection, Refraction, Diffraction and Interference of 3-Dimensional Waves ...... 328 12.21 The Origin of Waves on Liquid Surfaces ...... 336 12.22 Dispersion and the Group Velocity ...... 341 12.23 The Excitation of Waves by Aperiodic Processes ...... 345 12.24 The Energy of a Sound Field. The Wave Resistance for Sound Waves 347 12.25 Sound Sources ...... 351 12.26 Aperiodic Sound Sources and Supersonic Velocities ...... 354 12.27 Sound Receivers ...... 355 12.28 The Sense of Hearing ...... 356 12.29 Phonometry ...... 360 12.30 The Human Ear ...... 362 Exercises ...... 365 Contents xix

Part III Thermodynamics

13 Fundamentals ...... 369 13.1 Preliminary Remarks. Definition of the Concept ‘Amount of Substance’ ...... 369 13.2 The Definition and Measurement of Temperature ...... 370 13.3 The Definitions of the Concepts of Heat and Heat Capacity ...... 374 13.4 Latent Heat ...... 377 Exercise ...... 380

14 The First Law and the Equation of State of Ideal Gases ...... 381 14.1 Work of Expansion and Technical Work ...... 381 14.2 Thermal State Variables ...... 384 14.3 The Internal Energy U and the First Law ...... 384 14.4 The State Function Enthalpy, H ...... 386

14.5 The Two Speciﬁc Heats, cp and cV ...... 388 14.6 The Thermal Equation of State of Ideal Gases. Absolute Temperatures ...... 390 14.7 Addition of Partial Pressures ...... 394 14.8 The Caloric Equations of State of Ideal Gases. GAY-LUSSAC’s Throttle Experiment ...... 395 14.9 Changes of State of Ideal Gases ...... 398 14.10 Applications of Polytropic and Adiabatic Changes of State. Measurements of ...... 403 14.11 Pneumatic Motors and Gas Compressors ...... 406 Exercises ...... 407

15 Real Gases ...... 409 15.1 Phase Changes of Real Gases ...... 409 15.2 Distinguishing the Gas from the Liquid ...... 411 15.3 The VAN DER WAALS Equation of State for Real Gases ...... 414 15.4 The JOULE-THOMSON Throttle Experiment ...... 416 15.5 The Production of Low Temperatures and Liquefaction of Gases .. 418 15.6 Technical Liquefaction Processes and the Separation of Gases .... 420 15.7 Vapor Pressure and Boiling Temperature. The Triple Point ...... 421 15.8 Hindrance of the Phase Transition Liquid ! Solid: Supercooled Liquids ...... 424 15.9 Hindrance of the Phase Transition Liquid $ Gas: The Tensile Strength of Liquids ...... 425 Exercise ...... 426 xx Contents

16 Heat as Random Motion ...... 429 16.1 Temperature on the Molecular Scale ...... 429 16.2 The Recoil of Gas Molecules Upon Reﬂection. The “Radiometer Force” ...... 433 16.3 The Velocity Distribution and the Mean Free Path of the Gas Molecules ...... 435 16.4 Molar Heat Capacities in a Molecular Picture. The Equipartition Principle ...... 437 16.5 Osmosis and Osmotic Pressure ...... 440 16.6 The Experimental Determination of BOLTZMANN’s Constant k from the Barometric Equation ...... 445 16.7 Statistical Fluctuations and the Particle Number ...... 447 16.8 The BOLTZMANN Distribution ...... 449

17 Transport Processes: Diffusion and Heat Conduction ...... 453 17.1 Preliminary Remarks ...... 453 17.2 Diffusion and Mixing ...... 453 17.3 FICK’s First Law and the Diffusion Constant ...... 454 17.4 Quasi-Stationary Diffusion ...... 457 17.5 Non-Stationary Diffusion ...... 459 17.6 General Considerations on Heat Conduction and Heat Transport .. 460 17.7 Stationary Heat Conduction ...... 463 17.8 Non-Stationary Heat Conduction ...... 464 17.9 Transport Processes in Gases and Their Lack of Pressure Dependence ...... 465 17.10 Determination of the Mean Free Path ...... 468 17.11 The Mutual Relations of Transport Processes in Gases ...... 470

18 The State Function Entropy, S ...... 475 18.1 Reversible Processes ...... 475 18.2 Irreversible Processes ...... 477 18.3 Measurement of the Irreversibility Using the State Function entropy ...... 479 18.4 Entropy in a Molecular Picture ...... 482 18.5 Examples of the Calculation of the Entropy ...... 483 18.6 Application of Entropy to Reversible Changes of State in Closed Systems ...... 487 18.7 The H-S or MOLLIER Diagram with Applications. Supersonic Gas Jets ...... 488 Exercise ...... 493 Contents xxi

19 Converting Heat into Work. The Second Law ...... 495 19.1 Heat Engines and the Second Law ...... 495

19.2 The CARNOT Cycle ...... 497

19.3 The STIRLING Engine ...... 498 19.4 Technical Heat Engines ...... 500 19.5 Heat Pumps (Refrigeration Devices) ...... 501 19.6 The Thermodynamic Deﬁnition of Temperature ...... 504 19.7 Pneumatic Motors. Free and Bound Energy ...... 504 19.8 Examples of Applications of the Free Energy ...... 506 19.9 The Human Body as an Isothermal Engine ...... 509 Exercise ...... 510

Table of Physical Constants ...... 511

Solutions to the Exercises ...... 512

Index ...... 518 List of Videos

All of the videos for this volume, and also those for Vol. 2, may be downloaded from http://extra. springer.com.

1R.W.POHL Lecturing http://tiny.cc/fpqujy ...... viii, 261, 264, 267, 274, 282, 283

Part I Mechanics 2.1 “Free fall” http://tiny.cc/jpqujy ...... 23, 24 3.1 “Action D Reaction” http://tiny.cc/ppqujy ...... 33

4.1 “MAXWELL’s wheel” http://tiny.cc/9pqujy ...... 43 4.2 “Dynamic stability of a bicycle chain” http://tiny.cc/cqqujy ...... 49 4.3 “Circular oscillations” http://tiny.cc/1pqujy ...... 57

4.4 “LISSAJOUS ﬁgures” http://tiny.cc/6pqujy ...... 58

4.5 “KEPLER’s elliptical orbits” http://tiny.cc/tpqujy ...... 59 5.1 “Dancing steel ball” http://tiny.cc/3qqujy ...... 75 5.2 “Conservation of linear momentum” http://tiny.cc/tqqujy ...... 77, 78 5.3 “Elastic collisions in slow motion” http://tiny.cc/xqqujy ...... 79 5.4 “Elastic collisions” http://tiny.cc/xrqujy ...... 80, 89 5.5 “Measuring the velocity of a bullet” http://tiny.cc/qrqujy ...... 81

xxiii xxiv List of Videos

5.6 “Non-central collisions” http://tiny.cc/jqqujy ...... 82 6.1 “Twisting a steel bar” http://tiny.cc/csqujy ...... 96, 163, 173 6.2 “Moments of inertia” http://tiny.cc/gsqujy ...... 102 6.3 “Conservation of angular momentum using a rotating chair” http://tiny.cc/4rqujy ...... 105 6.4 “Angular momentum as a vector” http://tiny.cc/ltqujy ...... 107 6.5 “The physics of swinging” http://tiny.cc/ssqujy ...... 108 6.6 “Rotation around free axes” http://tiny.cc/itqujy ...... 109 6.7 “Supple shaft as stable axis of rotation” http://tiny.cc/5squjy ...... 46, 109 6.8 “Free rotation of a rectangular parallelepiped” http://tiny.cc/1squjy ...... 110 6.9 “The three axes of a top” http://tiny.cc/2tqujy ...... 113 6.10 “Precession of a spinning top” http://tiny.cc/wtqujy ...... 119, 127 6.11 “Precession of a rotating wheel” http://tiny.cc/otqujy ...... 119, 122 6.12 “Physics of riding a bicycle with no hands” http://tiny.cc/vsqujy ...... 120 6.13 “Stabilization using a spinning top” http://tiny.cc/5tqujy ...... 125 7.1 “Freely falling frame of reference” http://tiny.cc/1uqujy ...... 133 7.2 “Simple pendulum in a rotating reference frame” http://tiny.cc/euqujy ...... 138, 139 7.3 “Gyrocompass” http://tiny.cc/ququjy ...... 140 7.4 “Torsional pendulum on a rotating table” http://tiny.cc/cuqujy ...... 142

7.5 “FOUCAULT’s pendulum” http://tiny.cc/luqujy ...... 54, 148, 149, 151

8.1 “Elastic deformation: HOOKE’s law” http://tiny.cc/cvqujy ...... 154, 173 List of Videos xxv

8.2 “Bending a rod” http://tiny.cc/4uqujy ...... 160, 162, 173 8.3 “Plastic deformation, rupture strength” http://tiny.cc/kvqujy ...... 167, 168 8.4 “Reducing sliding friction” http://tiny.cc/mvqujy ...... 172

9.1 “BROWNian motion” http://tiny.cc/4wqujy ...... 175 9.2 “The Compressibility of water” http://tiny.cc/6vqujy ...... 182 9.3 “Buoyancy in a model ﬂuid” http://tiny.cc/fwqujy ...... 184 9.4 “The tensile strength of water” http://tiny.cc/3vqujy ...... 169, 187, 426, 431 9.5 “Surface work” http://tiny.cc/tvqujy ...... 188 9.6 “Coalescence of Hg droplets” http://tiny.cc/mwqujy ...... 192, 209 9.7 “A Model Gas: Its barometric density distribution” http://tiny.cc/rwqujy ...... 196, 204 9.8 “Magdeburg hemispheres” (Otto von Guericke’s experiment) http://tiny.cc/vwqujy ...... 199

9.9 “BEHN’s tube” http://tiny.cc/0vqujy ...... 206 10.1 “Model experiments for streamlines” http://tiny.cc/8wqujy ...... 215, 217, 221, 225, 239 10.2 “Turbulence of ﬂowing water” http://tiny.cc/cxqujy ...... 217 10.3 “Fluid Flow around obstacles” http://tiny.cc/fcgvjy ...... 220, 226, 235, 237–239 10.4 “Smoke rings” http://tiny.cc/ocgvjy ...... 233

10.5 “MAGNUS effect” http://tiny.cc/dcgvjy ...... 240

Part II Vibrations and Waves 11.1 “Vibrations of a tuning fork” http://tiny.cc/idgvjy ...... 249 11.2 “Transverse normal modes of a stretched rubber band” http://tiny.cc/negvjy ...... 261 xxvi List of Videos

11.3 “Transverse vibrations of a string” http://tiny.cc/odgvjy ...... 262, 263 11.4 “Longitudinal vibrations of a helical spring” http://tiny.cc/ydgvjy ...... 270

11.5 “HELMHOLTZ resonators” http://tiny.cc/0cgvjy ...... 273

11.6 “Free and forced oscillations of a torsional pendulum (POHL’s pendulum)” http://tiny.cc/uegvjy ...... 274, 275, 290 11.7 “Forced oscillations with a pocket watch” http://tiny.cc/zegvjy ...... 279 11.8 “Coupled pendulums: Force coupling” http://tiny.cc/pfgvjy ...... 283, 285 11.9 “Acceleration coupling and chaotic oscillations” http://tiny.cc/8cgvjy ...... 284, 290 11.10 “Coupled oscillations with damping” http://tiny.cc/5egvjy ...... 285 11.11 “Antirolling tank” http://tiny.cc/jfgvjy ...... 286 11.12 “Wobble oscillations” http://tiny.cc/qcgvjy ...... 287 12.1 “Model of a travelling wave” http://tiny.cc/2fgvjy ...... 292 12.2 “Experiments with water waves” http://tiny.cc/tfgvjy ...... 293, 294, 295–299, 300–303, 305, 341, 347 12.3 “The acoustic radiometer” http://tiny.cc/gggvjy ...... 327 12.4 Fresnel’s zones http://tiny.cc/fggvjy ...... 331

Part III Thermodynamics 13.1 “Model experiment on thermal expansion and evaporation” http://tiny.cc/nggvjy ...... 370 14.1 “Adiabatic changes of state” http://tiny.cc/yggvjy ...... 400 14.2 “Compressed-air motor” http://tiny.cc/0ggvjy ...... 406 15.1 “Liquefaction of oxygen” http://tiny.cc/8ggvjy ...... 419 15.2 “Liquid and solid nitrogen” http://tiny.cc/ihgvjy ...... 422 List of Videos xxvii

15.3 “Solid carbon dioxide (dry ice)” http://tiny.cc/lhgvjy ...... 423 16.1 “Model experiments on diffusion and osmosis” http://tiny.cc/xhgvjy ...... 175, 196, 432, 443 16.2 “Crookes’ radiometer (light mill)” http://tiny.cc/qhgvjy ...... 433

19.1 “Operation of a STIRLING engine” http://tiny.cc/gigvjy ...... 498, 500, 502

19.2 “Technical Version of a STIRLING engine” http://tiny.cc/kigvjy ...... 500 19.3 “Heat pump/refrigerator” http://tiny.cc/1hgvjy ...... 502 Mechanics I 1

Part I hdwo h ador square, cardboard the of shadow n ieMeasurements Time and Distance Introduction; .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 1.1 Figure examples a) two offer We deceived. this: are readily of Facts be may but learned; inexperienced be science. observations, the must the Observing chance of observations. experimentally- sometimes carefully-planned progress usually on observations; historical based the from of is obtained change may course It interpretations the their remain, in facts science. The empirical facts. determined an is Physics Introduction 1.1 lcrclm stre n tilmntstewiewall white the the illuminates It only Initially, on. – turned is square. lamp cardboard electric a e.g. object, opaque arbitrary o h es hsclo betv hnet h area the to causes change This objective on. or switched physical is least lamp the electric the not burning, is lamp gas the o ese at seen be now api gie.Aan h alapaswie hstime this white, appears wall gas area the the marked only Again, Then the – wall. ignited. the is to pinned lamp paper of snippet a with ample adsetgslamp gas candescent es ooree h mg nege udmna hne At change. fundamental a undergoes image the eyes our to less, h ooe shadows colored The ooe shadows Colored S 2 S o ecm oteata xeiet While experiment: actual the to come we Now – . 1 lc hdwfo h ador qaecan square cardboard the from shadow black A . C1.1 .InFig. olsItouto oPhysics to Introduction Pohl’s n neetia nadsetlamp. incandescent electrical an and 1.1 S 1 h hdwi akd o ex- for marked, is shadow The . eseawiewall white a see we , O 10.1007/978-3-319-40046-4_1 DOI , S xetfor except 1 Neverthe- . W including ,anin- P san is the S 1 , agasﬂame(C.A by heated is ‘mantle’) Ce (the and Th of oxides with permeated net a lamp, gas incandescent the In C1.1. e o.2 et 28.6). Sect. 2, Vol. see 1 UER 1885, , 3

Part I are not Colors .Thatareaishowever 1 S . C1.2 . But still, only the light from the 2 , everyone sees a system of spirals . manner. Nevertheless, experiments S 1 S 1.2 due to the light from the electric lamp. shadow. It looks quite different from the halo qualitative .InFig. olive-green Spiral illusion Measurements. Direct Distance Measurements The spiral illusion This demonstration is instructive forphysical every properties, beginner: but rather results ofNot psychology and taking physiology! this factuseless effort into in the account past. has givenb) rise to a good deal of change in the color of the area surrounded by a bright The presence of this halo by itself gives rise to the very noticeable gas lamp can reach our eyes from the area Figure 1.2 knowledge, often knowledge ofcarried great out import, only even in when a they were cal observations of Nature!? Inwhich particular, the have most been general developed concepts sinceman experience, earliest such times as in space,suspect. the time, course forces of Physics etc., hu- must maysome be misinterpretations. yet considered have to deal with many a prejudice and 1.2 Distance and Length Without doubt, experiments and observations have yielded new These and many othersensory phenomena organs which function result seldomservers. from cause the difﬁculties But way for theyyet our practiced still unrecognized subjective ob- warn inﬂuences may us be to lurking be in our careful. physi- How many other as we see a lively (now reddish-brown) shadow at curving around a common midpoint.concentric circles. However, in One can fact verify itof this consists the immediately of circles by with following the one point of a pencil cshows 11.42 Introduction; Distance and Time Measurements 1 C1.2. Figure another, similar optical illusion. 4

Part I 1.2 Distance and Length Measurements. Direct Distance Measurements 5 and observations attain their full value only when all the quantities involved are determined in terms of precise numerical values with units. Measurements play an important role in physics. The art Part I of physical measurement is highly developed, the number of applicable techniques is large, and they are the subject of an extensive specialized literature and their own technical field (metrology). Among the manifold types of physical measurements, those involving lengths and times occur particularly often – sometimes alone, frequently paired with measurements of other quantities. It is therefore expedient for us to begin by discussing the measurement of lengths and of times, and to elucidate their fundamentals, while not concerning ourselves with the technical details of how they are carried out. We learn the usage of the words ‘length’ and ‘distance’ as children. Every direct measurement of a length is based on applying and comparing to a ruler or other length standard. One counts how many lengths of the ruler are contained within the length to be measured. This may seem trivial, but it is often not adequately taken into account. The procedure of measurement itself, i.e. comparison with a length standard, is not sufficient; in addition, a unit of length must be defined. Every definition of physical units is completely arbitrary. The most important requirement is always an international agreement, which must be as all-inclusive as possible. Furthermore, ready reproducibil- ity is desirable, along with convenient numerical values for the most frequently applied measurements in everyday life. Length or distance measurements are based on the unit of length, the meter. The meter was previously (before 1960) defined by the length of a metal bar (the “archival meter”), kept at the Bureau des Poids et Mésures in Sèvres; it is also called the “standard meter”. The modern definition of the meter will be given below. For calibration purposes, length standards are commercially available. They take the form of gauge blocks; these are rectangular steel blocks with planar, parallel, highly polished end surfaces. When struck together, they stick to each other (cf. Fig. 9.19). They can be used to reproduce lengths to a precision within 103 mm D 1 m (termed 1 micrometer, or 1 micron). For practical length measurements, one uses divided length scales or rulers, and various forms of measuring instruments. Rulers should 1 have division marks whose length is 2 2 times as great as their spacing; then fractional distances can be estimated most accurately. Measurement instruments for lengths facilitate reading off the fractional distances by means of mechanical or optical arrangements. The mechanical instruments make use of length conversions using some sort of arrangement of levers, or screws (“screw micrometers”), or gears (“dial indicators”), or spirals. Vernier calipers are also frequently used. m. m (i.e. the 30 ˙ mm). mm or 40 has been increased 2 b 1 20 50 to 10 a ˙ The microscope is among the : C1.3 . – One scale is attached to the fixed Mechanical methods can be used down to 1.4 C1.4 m. 0.1 , left side. Then the hair is removed and replaced ˙ 1.3 Length measurements using a microscope m. The naked eye is limited to around , right side. We can read off four scale divisions between the 1 1.3 diameter of a hair!). The error limits forduced length to measurements around with optical˙ methods can be re- arrows; the thickness of the hair was thus 4 on the microscope’s object stageplate by a (object small micrometer), scale showing etched100 e.g. into scale a one glass marks. millimeter dividedFig. The into field of view now shows the image seen in most important optical instruments forare length direct determinations. length These measurements. Asstrated an to example which a can large be audience, demon- man we hair. could measure the thickness ofWith a hu- a simple microscope,a screen. an The image thickness ofarrows, of cf. the the Fig. hair hair is is marked on projected the onto screen with two by the spacing of the division marks (in this example 1/10 mm). As In the zero positioncover the of gaps on the the other, instrument, soand that the the appears whole dark. overlap marks region is If on opaque to one one now the moves scale right, the the just caliper overlapEach with region new its appears darkening means periodically scale that bright slowly the and distance dark. For direct length measurements,extremely fine one divisions, which can are employ no longer aThis visible to is ruled the demonstrated naked scale eye. in with Fig. part and one toglass the platelets movable with part fine of divisionare a marks. one screw behind caliper. As the seen other, Bothdivision by and marks scales the overlap and are over viewer, the a transparent they gaps largewidth between region. (more them precisely, The have each the black same has a width of e.g. 1.3 The Meter as a Unit of Length Figure 1.3 Observations with a microscope ). m, i.e. 9 10 D https://www.ptb.de/cms/ Introduction; Distance and Time Measurements 1 news-ausgaben/ptb-news/ news07-2/high-precision- length-measurements.html reduce this limit even further (cf. en/presseaktuelles/journals- magazines/ptb-news/ptb- the requirements of man- ufacturing technology, efforts are being made to 1 nanometer). Because of C1.3. The optical microscope, referred to here, is discussedindetailinVol.2, Sects. 18.11 and 18.12. be measured with uncertainties down to about 20 nm (1 nm C1.4. Today, with optical methods, distances can 6

Part I 1.4 Indirect Length Measurements of Very Large Distances 7

Figure 1.4 An interference micrometer, enlarged for clarity Part I

a result, by counting the number of dark-light intervals of the invisibly fine scales, one can carry out a direct length measurement. This is, put succinctly, a length measurement using geometric interference. There is an optical analog to this length measurement by interference: In optics, one can replace the man-made scales by naturally given scales. One uses the light waves of a particular spectral line 86 emitted by excited atoms of the krypton isotope 36Kr. Their wavelength in vacuum (the “division mark spacing”) has been compared to the standard meter in Sèvres, and by international agreement, the 1 650 765.73-fold multiple of this wavelength has been defined as one meter C1.5. C1.5. The unit of length ‘meter’ was redefined in 1983: In this way, the hope is that the precise sense of the word ‘meter’ “The meter is the length of will be passed on to later generations more securely than by using the distance which light tra- the “archival meter” as a prototype for defining the unit. An archival verses during a time period of meter rod, in spite of all the care that may be taken, remains an imper- (1/299 792 458) seconds”. manent object. All rulers change their lengths in the course of long This fixes the numerical time periods. This is the result of internal material changes which value of the velocity of light occur within all solid bodies. in vacuum and relates the meter to the unit of time, the second (see e.g. http://en. 1.4 Indirect Length Measurements wikipedia.org/wiki/History_ of_the_metre,andhttp:// of Very Large Distances physics.nist.gov/cgi-bin/cuu/ Info/Units/meter.html). Baseline methods, stereogrammetry: Very long distances are often no longer accessible to direct measurement methods. Consider for example the distance between two mountain peaks, or the distance of a celestial object from the earth. One must then apply an indirect method of length measurement, e.g. the well-known baseline method as indicated in Fig. 1.5. The length BC of the baseline is found when possible by a direct length measurement. Then the angles ˇ and are determined. From the length of the baseline and these angles, the required distance x is obtained graphically or by calculation. of and one C B and B can be com- how the two A 1.6 . The images II of the object x and of the lenses, a calibration table I on the one hand, and the overall f ssumes without further consider- CR or It is the preferred method for surveying ter- BL . were determined using some sort of protrac- C1.6 C1.6 and .From ˇ CR are shifted at their centers (different viewing angles) by ngular measurements at the ends of the baseline by two or A and a given focal length length measurements, i.e. when applying a ruler and BL II on the other, the desired distance , the angles stereogrammetry Length measurements I 1.5 BC direct . This difficulty can however be avoided. One combines the two pho- length puted. This can bebaseline understood simply by applyingcan geometry. be compiled. For aThus given far, the methodrious offers nothing difficulty: notable. But Itexample now to would we determine be encounter the azig-zag time individual path se- consuming of segment a lengths and lightning in discharge often from the the impossible complicated, corresponding images for image field which appears 3-dimensional. We see in Fig. C tographic images in the well-known manner in a stereoscope to give the same object the distances In Fig. tor (e.g. a telescopereplaces with the divisions two a on acameras. circular Their scale). objectives are Stereogrammetry denoted as To end our briefgant discussion technical of variation of length the baseline measurements,so-called method we of length mention determination, an the rain, ele- especially in mountainous regions.things In for physics, determining it is complexof used 3-dimensional lightning among discharges. orbits other or paths, e.g. those Figure 1.5 using a baseline, and stereogram- metric length measurements the earth. They becometermining the important enormous only distances inNevertheless, which special the are cases, beginning relevant e.g. to student insuch astronomy. of potential de- physics difficulties; otherwise, should heat be or all aware she with may of see lengthial no measurements problems sort and of hold measurementsonly them in for to general. be This thecomparing opinion most lengths. however triv- is valid This method, well known fromfree of school fundamental mathematics, difficulties. is It howeveration a not that the light rayswith used the to straight measure lines the ofand angles Euclidian can its geometry. be admissibility That identified is can– a in Fortunately, presumption, we the need end nottal be concern uncertainties confirmed ourselves in only with the such empirically. case fundamen- of the usual physical measurements on ition the Introduction; Distance and Time Measurements 1 radio signals from satellites. methods use in add global positioning system (GPS), based on the arrival of dition to surveying terrain, also e.g. in architecture and in medicine (X-ray stere- ograms). Modern surveying employed using digital data processing technology, is applied in many fields; in ad- C1.6. The method of stereogrammetry, which is today 8

Part I 1.5 Angle Measurements 9 Part I

Figure 1.6 A stereoscope with a moveable marker. The images represent forked lightning discharges

individual photographs are combined in a stereoscope. And now comes the decisive trick: the use of a “movable marker”. This movable marker or cursor is obtained by employing two identical pointers 1 and 2. They can be moved over the image surface horizontally and vertically. The distances through which they have been moved can be read off the scales S1 and S2. In addition, the distance between the two pointers can be varied systematically in a measurable way (S3, with a scaled screw drive). Looking into the stereoscope, we see these two pointers superimposed as one, apparently ﬂoating freely in space. If we change their spacing (by turning the screw S3), then the cursor appears to move towards us or away from us in the ‘image space’. By using all three possible motions (S1, S2, S3), the cursor can be adjusted to indicate any given point in the image, thus to point to a mountain peak, to an arbitrary position along the crooked path of a lightning stroke, etc. This demonstration is very impressive. A calibration table then allows us to read off the distances (height, depth, width) which determine the location of the given point (i.e. its three spatial coordinates) from the values of the scale readings S1, S2,andS3.

1.5 Angle Measurements

Proceeding from length measurements, we can determine surface areas, volumes and angles. We note only a few aspects of angle measurements: Arc length b Planar angles (Fig. 1.7) are defined by the ratio ; solid Radius r Spherical surface sector A angles (Fig. 1.8) by the ratio .Thenall .Radius r/2 angles are determined as pure (dimensionless) numbers. ı is ı mo- (ab- (1.1) . 100 r steradian D radian ˛ on a sphere 180 A 1steradian.It of time, or a D D 1 : D 1 360 . The most important = r , is simply a numerical ˝ r D interval . 2 clock ˛/ on the sphere, i.e. the fraction 0175 D degree : 2 cos r 0 , so is every direct time measure- 75. D 3 : 1 : . 1 A 2 intersects a surface sector r 57 D ˛ 2 D D 57.3° simply expresses the identity A ::: 0175 radius : D 96 % of the surface area of a sphere of radius 0 : 7 1radian length standard 360 circumference D 01745 = ˝ : 100 1 0 , the corresponding solid angle is 4 has two meanings: either an ). If these names for the number 1 occur in some com- ı = D , representing the word 8 ); or in referring to a solid angle as the unit : D D 1 sr ı Radiation intensity can be referred to the solid angle subtended ˛ ı 32 The def- The definition D 1 time rad 2 Measurements D r ˛ 4 in time. Just as a length is limited by its two end points, a time ' = 2 For intersects a surface sector of area Example: by the beam of radiation. See Vol. 2, Chap. 19. A cone with the openingcentered at angle its apex, with r is the abbreviation for the number 3.1415. . . Correspondingly, identical to inition of the solid angle Figure 1.7 of the planar angle unit, defined by the equation 1.6 Time Determinations. True Time The word The symbol is an abbreviation for the number 0.01745. . . Thus e.g. Figure 1.8 breviated (abbreviated bination of units, one seesangle is immediately that included in the the determination implied of measurement an procedure. The equation: 1 radian ment associated with a comparison to a ment interval is limited byits the end. two moments Just ina as time comparison every to at direct a its length beginning measurement and is associated with clocks are based on counting uniformly repeated processes (usually this number 1 in referring to a planar angle as the unit The unit of all angles is the number 1. It is often expedient to denote Introduction; Distance and Time Measurements 1 10

Part I 1.7 Clocks and Graphical Registration 11 rotations or oscillations). Here, we cannot define “uniform” as a concept, but rather only experimentally: One compares many clocks of all possible different types with each other and with periodically re- Part I curring astronomical phenomena. This comparison leads to a “strug- gle for survival”: Clocks whose behavior deviates from that of the majority are eliminated, and the regularity of operation of the surviving clocks is declared to be “uniform”. Just like the definition of the unit of length, the definition of the unit of time is a matter for international agreement. The unit called the second was originally defined in terms of astronomical phenomena (initially by the rotation of the earth around its axis, and later by its annual orbit around the sun). These definitions, which in the final analysis are mechanical, have proved to be inadequate1. Thus, they have been replaced by an electrodynamic definition. It is based on 133 an electromagnetic wave emitted by 55Cs atoms under certain excitation conditions, with a wavelength in the range of 3 cm. Since 1967, the second has been defined as the time in which 9 192 631 770 oscillations (wave crest C wave trough) of these waves occur (and can be counted) (“atomic clock”).

1.7 Clocks and Graphical Registration

Clocks which can be used for practical time measurements are well known. They make use of mechanical oscillation phenomena. Ei- ther a hanging pendulum oscillates in the gravitational ﬁeld of the earth (e.g. as in wall clocks or upright clocks), or a balance wheel oscillates rotationally around a spiral spring (e.g. in a wristwatch or pocket watch)C1.7. We still have to show that the oscillations of such C1.7. Today, most small a pendulum or balance wheel can be referred to a uniform rotation. clocks and watches utilize the mechanical oscillations of The swinging of a pendulum, put concisely, is similar to a circular quartz crystals (Sect. 11.8). motion as seen from the side. Looking in the plane of the circular motion, we see an object on a circular path as though it were moving back and forth. Its motion over time is precisely the same as that of a swinging pendulum. An optical recording can show this in a particularly graphical way; it converts the temporal sequence into a spatial diagram and represents the motion in terms of a curve. In order to register this motion, we use the arrangement which is illustrated in Fig. 1.9: A slit S is imaged onto a screen P by the lens L. The light source which illuminates the slit (an arc lamp) is not shown.

1 This is due to the non-constancy of the earth’s rate of rotation. The frictional forces associated with the ocean (and surface) tides increase the rotational period of the earth (with a power dissipation of ca. 109 kilowatts!), adding about 1:5 103 s per century. As a result, each century lasts about 30 s longer than the preceding one. Furthermore, the rotational period varies within the course of a year; for not clearly understood reasons, it is ca. 2 103 s longer in May than in July. Finally, additional random fluctuations in the period of the earth’s rotation have been observed. See Exercise 1.1. . P lit. , there is a horizontal pin attached to the edge of a cylinder S , metronome pendulum). Its maximum deflection is A metal pin is moved by a slider in the direction of the arrow; this for the second The relation between circular motion and a sine curve. In front of L 1.10 ), and 1.9 1.9 in Fig. , we arrange one behind the other: a long time after beingS illuminated briefly. In front of the vertical1. slit a metal pin whichder moves with in a a circle horizontal around(Fig. the axis surface which of a is cylin- parallel2. to the plane a of(cf. wire the Fig. slit pointeradjusted which to be is the same fastened asmetal the pin to radius for of the the the first cylinder which experiment. side carries the of a pendulum This screen is coated with a phosphorescent powder, which glows for This arrangement replaces S experiment. connected to a metronome pendulum in front of a s Figure 1.10 Figure 1.9 The lens causes the image of the slit to move uniformly across the screen the vertical slit which can rotate around aaround horizontal this axis. axis The using cylinderslit. can a be flexible made shaft; to rotate the axis is parallel to the plane of the Introduction; Distance and Time Measurements 1 12

Part I 1.8 Measurement of Periodic Sequences of Equal Times and Lengths 13 Part I

Figure 1.11 A sine curve gives the dependence of the angular function sin ˛ on the angle ˛

In both cases, we obtain the same curve, deep black on a bright green glowing background: a sine curve, see Fig. 1.11. This close connection between circular motion, the oscillation of a pendulum and a sine curve plays an important role in many different areas of physics. We will come back to this topic in Sect. 4.3. Graphical registration is useful for many rapidly-occurring processes, and in some cases it is indispensableC1.8. For this purpose, e.g. oscil- C1.8. Images of this type are loscopes are useful; they can also be employed for measuring short often obtained using CCD times. They can be used to register periods of time down to 109 s. cameras (charge-coupled Today, in order to synchronize and calibrate clocks for science and devices), which also permit technology, time signals are broadcast; they are controlled by preci- the images to be recorded in sion standard clocks (Cs atomic clocks). color. Today, of course, there are also convenient computer programs which generate graphical representations 1.8 Measurement of Periodic Sequences from measured data. of Equal Times and Lengths

Let us assume that within a given time t, N identical processes occur in sequence, each one lasting a time T; e.g. oscillations or rotations. One then deﬁnes in general t=N D T as the period (1.2) and (from the Latin frequentia) N=t D 1=T D as the frequency (1.3) (the unit of frequency is 1/s D 1 Hertz (Hz)). Within a given length l, N identical forms occur in sequence, each one of length D.Thenwe deﬁne2 l=N D D as the period of length, (1.4) N=l D 1=D D as the frequency of length. (1.5)

2 Generally accepted terminology is lacking. For D, terms like ‘wavelength’ and ‘lattice constant’ are in use. In optics, 1=D is called the ‘wavenumber’.Thisterm is however a poor choice, as is the term ‘number of revolutions’ used in technology. An electric motor, for example, has a rotational frequency of D 3000/min D 50/s D 50 Hz. Reciprocal lengths and reciprocal times are not numbers, but rather dimensioned quantities with units. . x x D or In T . x 11.10 (1.6) (1.7) D )that: .The 11 cm or : x x 0 strobo- x T D 1.12 D D < ). The beats D . D T B ( :/ . This oscilla- B x or from C1.9 D T x x ˙ T < D ˙ , which can readily 3/cm), : 11.45 B T 8 ). The upper comb has T or or of identical times T D D ). In the figure, the length D 1.4 x B x x x is the frequency of the disk D x T ( D < B or a period D D D 83/cm). We find (for or x or : t 0 = odic sequence (by “interference”) 12 cm ( ; we cause it to oscillate at a very N is equal to 10 in the example shown in : B B D , defined later in the text, e.g. in Fig. F 0 T ) follow from this (for the general case of N D B = is obtained from the rotational frequency = D 1 1.7 , as shown in Fig. 1 .Then x x t ‘beats’ ˙ ˙ D . Quantitatively, for the measurement of or “stretched” times (where 3 2cm( T : D B = )and( B 1 = 1 D D 1 of a frequency D 1.6 D D B D ). x , which differ only slightly from x D x D ), and the lower comb has the period T T is the frequency of the light pulses. T . One can imagine two parallel combs whose teeth partially x / D = . One can use a stopwatch to count the number of revolutions x within the time 1 T D . the minus sign applies when = 1 D ? T ( . 1 shows a leaf spring N C Measurement of a periodic sequence of identical lengths x T 20 12.4 D N . D 1/cm) and and 1.13 : D x 9 x D D D of the disk and The frequency of illumination of the disk, ? The index B refers to the word Superposed on it is aor known second times periodic sequence of known lengths superposition produces a third peri of “enlarged” lengths ( high, unknown frequency and for the measurement of times tion is projected onto thesequence wall of by individual intermittent flashes. light Thisduced pulses, kind simply a of by uniform illumination using canis a be placed rotating pro- in disk the with, light for beam example, from 20 a suitable slits. source It of light. be counted or measured identical times N lengths, we find produced by interference have the period periods (or frequencies) are the figure). Equations ( D and in Sect. 3 Out of the many practical examplesscopic we measurement mention only one, the Figure the upper part of the figure, we see the periodic sequence of Figure 1.12 A periodic sequence of identicalcan be lengths measured D using the same scheme, as illustrated in Fig. overlap (an arrangement likethe that period shown in Fig. s. 15 Introduction; Distance and Time Measurements 1 C1.9. Modern high-speed stroboscopes attain time reso- lutions of 10 14

Part I 1.9 Indirect Time Measurements 15

Figure 1.13 A leaf spring F for demonstrating stroboscopic time measurement. The oscilla-

tions of this spring are shown in Fig. 11.45. Part I The driving force is provided by a ﬂexible shaft W and a holder which is tapped on one endbythepinA. More details are given in Sect. 11.10 under “forced oscillations”.

We start with a high light pulse frequency and reduce it gradually. The image of the oscillating leaf spring seems to move, and the frequency of its apparent oscillations becomes lower and lower as the light pulse frequency decreases (stroboscopic time dilation). When the image appears to move very slowly, one can ﬁnally determine B conveniently, for example B D 1.5 Hz. We insert B and into Eq. (1.7) and for the example, we ﬁnd x D 50 Hz. – In the limiting case B D 0, the image of the leaf spring is stationary and Eq. (1.7) gives x D .

1.9 Indirect Time Measurements

Instead of today’s usual “direct” or true time measurements, i.e. measurements based on counting periodic motions, in earlier times non-periodic phenomena were often used, for example in hour glasses and water clocks. These types of clocks played an important role in the early history of mechanics (e.g. in the work of GALILEI, Sect. 2.4). Today, they are found only in the puny form of “egg timers”. But a modern variant which makes use of radioactive decays for determining the age of historical objects is of considerable importanceC1.10. C1.10. This refers for example to the so-called C-14 One concluding remark: We have described only some measure- method. Here, the radioactive ment procedures for lengths and times, but have not attempted to carbon isotope 14 C (half-life define these two concepts in words or sentences. Both of them have 6 ca. 5700 years) is employed; developed over long times from extremely diverse experiences and its concentration in living observations. Physicists base their use on only a narrow selection of organisms has a roughly con- these. For ‘time’, for example, they might say the following: stant equilibrium value, while Every physical measurement requires at least two “readings”; for in no-longer living objects, it length measurements, the beginning and the end of the length must decreases due to the radioac- be “read out”; electrical measurement instruments show the differ- tive decay, so that the age ence between the zero point (ground potential or zero current) and of such objects (since their the actual reading, etc. Between the first and the second readout, our death) can be estimated (see heart beats or the clock ticks. All observations can be ordered into e.g. http://en.wikipedia.org/ one of two groups: In the first group, the results of a measurement wiki/Radiocarbon_dating.) ) 1.7 . C1.11 , it is pointed out that the coming 1.7 does the pendulum seem to be at rest to T ) 1.8 This by no means exhaustively encompasses the concept In the footnote in Sect. A swinging pendulum is observed through a rotating disk , but it is at least not just a meaningless phrase time of Exercises 1.1 depend upon how often our heartfirst beats and or the the clock second ticks readout. betweenof In the no the significance second for group, the by resultthat of contrast, the the that processes measurement. is in Then the wetime, first can and group say depend which on we athe measure quantity clock. that by we counting call heartbeats or the ticks of century will be aboutprecise number 30 s using the longer information given than in the the text. one (Sect. just past. Find the 1.2 with four slits, whichwhich is oscillation rotating period at five revolutions per second. At the observer? (Sect. , physi- ERGER Y ENE Introduction; Distance and Time Measurements 1 “This by no means exhaustively encompasses the concept of time, but itleast is not at just a meaningless phrase”. “The Meaning of Time”, whose subtitle is “A Theory of Nothing” (Perfect Paper- back Press, 2008). C1.11. There is a voluminous literature on the question raised here, which is onlydifficult too to answer; namely, “What is time?”. It haswritten been not just by cists, but also by scientists and scholars from many other disciplines. To name just one (arbitrarily chosen) example, we mention the paperback book by G 16

Part I Kinematics Motion: of Description The .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 2.1 Figure explicitly. into mentioned be rotation always earth’s will the of This deformation take the will earth. also We sometimes the and discussions, some reference. in of account frame our i.e. change deformable.) occasionally is will instead we but Later, rigid, really not also is earth The w carousel. reality, the (In of floor consideration. the of or out earth located. the are is descrip- chapter we the this where for of room reference rest the of in frame motion our of is tion which body solid rigid path, The Fig. wavelike a in follow sketched feet are observer the which motion him, cycloids An the or the of her namely picture for pedals. different feet; very bicyclist’s the the a with of sees sidewalk paths the circular and on This feet in standing her moving at reference”. down them time, looks of from sees bicyclist with this “frame A see object the example: can randomly-chosen We an frame, a essential. of rigid quite location is fixed, specification the a supplementary in from change seen a as to refers Motion Frames Motion. of Definition 2.1 r observer ary h ocpsof concepts the o h ecito falmtos localled also motions, all of description the For rotational treat we when motions. especially confusion, endless an have would fReference of h ahfloe ytepdl fabccea enb station- a by seen as bicycle a of pedals the by followed path The velocity and olsItouto oPhysics to Introduction Pohl’s acceleration elaetediyrtto fteearth the of rotation daily the leave We r rciigpyiso large a on physics practicing are e h tnpito u observations, our of standpoint the ewl ei ihthem. with begin will We . kinematics 2.1 O 10.1007/978-3-319-40046-4_2 DOI , . tews we Otherwise erequire we , oainlmotions”. rotational treat we when pecially es- confusion, endless an Ohriew ol have would we “Otherwise 2 17

Part I is (2.1) at the as the .This u l (2.2) u within the time l 0. The symbol l t ! t which is then measured, which res the measurement of rather t l m d d u D u Time interval Distance moved D m by taking the limit . u m C2.1 u in general if one successively decreases the distance of a bullet from a pistol. velocity along the direction of the distance is fixed by two thin cardboard disks; its length could be for . Then we define shows a suitable setup for this measurement. The distance Measurement of the velocity of a bullet from a pistol with a sim- t l of a Velocity Measurement changes 2.2 mean . However, the changes gradually decrease to below the precision l of the measurements. The value of quotient starting point. Mathematically, onelimiting thus value finds of the velocity as the depends only on the starting point, is denoted as the velocity conventionally replaced by d, giving for the definition of the velocity rotational frequency ple “time recorder”. On the right is the tachometer which registers the i.e. the differential quotienttime of interval. the distance travelled dividedThis by the definition inshort many times. cases requi Asmuzzle velocity an example, we considerFigure the measurement of the Figure 2.2 2.2 Definition of Velocity. Example Suppose that an object movesinterval through a distance interval example 22.5 cm. The timeforward measurement manner is by performed referring in to a the straight- basis of all time measurements, 5.9 )isused 2.1 )or( 2.2 ), which makes use 5.19 The Description of Motion: Kinematics 2 of momentum conservation. C2.1. For this velocity measurement, the defining equation ( one described in Sect. (Fig. of molecules in a molecular beam. For macroscopic objects, however, a much more elegant method is the directly. This principle is also applied e.g. to thesurement mea- of the velocities 18

Part I 2.2 Definition of Velocity. Example of a Velocity Measurement 19 a uniform rotation. The time markers are registered automatically. For this purpose, an electric motor causes the two disks on a common shaft to rotate at a uniform, rapid rate. Their rotational frequency , Part I i.e. the quotient of the number N of rotations/time t, is determined by a tachometer which measures rotational frequencies1, for example D 50 Hz. The bullet first passes through the left disk, and the bullet hole that it leaves is our first time marker. While it travels over the distance of 22.5 cm to the second cardboard disk, a certain time elapses, and the bullet hole or time marker in the second disk is shifted relative to the first marker by a certain angle, corresponding to the rotation of the shaft during this elapsed time. After stopping the rotation, we measure an angle of ca. 18° or 1/20 of the circumference of the disks.

By pushing a pin through the two bullet holes, we make the angular shift C2.2. The word ‘trajectory’ visible for distant observers in the silhouette of the apparatus. originally meant ‘flight path’. It is now used in a more gen- 1 1 3 The time delay t was thus 20 50 s D 10 s. From this, we find the eral sense to mean the path velocity (location vs. time or geomet- 22:5cm 0:225 m m u D D D 225 : rical form) followed by any 103 s 103 s s moving object. When this path is circular (like the path We then repeat the experiment with a smaller distance l between the disks of only 15 cm. The end result is the same. Thus, the distance of a satellite around a planet), it is called an ‘orbit’. We will chosen in the first experiment was already short enough. It permitted use the simpler term ‘path’ us to measure the true muzzle velocity and not the smaller mean value over a longer trajectoryC2.2. here in most cases.

Only in cases with a constant or uniform velocity can we choose the quan- “One should early on adopt tities l (measured distance) and t (elapsed time) freely to allow the the habit of always writing most convenient measurement. In such cases, one can write the velocity in the units after every nu- abbreviated form as u D l=t. merical value of a physical One should early on adopt the habit of always writing the units af- quantity. It belongs to good ter every numerical value of a physical quantity. It belongs to good physics practice!” physics practice! This spares the reader from having to deduce the C2.3. POHL indicates here units meant from the physical context, and spares oneself from com- the advantage of using mitting frequent computational errors. When different units are used, the numerical values of the measured quantities also change. Their physical-quantity equations, which he consistently recalculation is automatically reliable if the quantitative results are employs. In a preliminary always given as numerical values and unitsC2.3. remark on writing physical Example equations (included in the The velocity u D 225 m/s is to be recalculated in terms of kilometers per volume on ‘Mechanics’ since hour. We have 1 m D 103 km and 1 s D (1/3600) hour; then the 12th edition), he writes 103 km among other things, “For u D 225 D 810 km/h: .1=3600/ hour each symbol, we write both the numerical value and the unit. The choice of units 1 If no tachometer is available, one can make use of a simple reducing gear: a pulley of circumference d is mounted on the motor shaft. An endless belt made of is free. Those mentioned string with length L d drives another pulley a few meters away; its knot serves under some equations are as a marker. The number N0 of revolutions of the string in a time t can be counted simply examples” (see also N0 L using this marker. Then D t d is the rotational frequency. Sect. 2.6). C and .In 1 (the 1 u 3 u u or (e.g. the at the be- 2 of the two 2 u u direction of a velocity, one D of the time inter- 1 u u t magnitude in an arbitrary direction, u c refers to the geometrical At the end the time interval, the body is 2.3 . It is determined graphically in ed quantities, i.e. mathematically, 2 u (e.g. the velocity of an aircraft rela- 1 in Fig. During u 2 . Time interval t u . . Velocity increase indicates the velocity of a body C C2.4 1 1 . Magnitudes are denoted by vertical bars on D . j u – We will encounter this in many places in this u 2 2 m u u a b is described by the equation (represented graphically as an arrow). This can be jj 1 2.3 u The geometrical addition , the vector a, the high velocity The Two Limiting Cases vectors jDj of a time interval as the arrow 2 2.4 2.3 , the body has the velocity u 0. – Therefore, t 2.4 C D addition of two velocities . The resultant vector has a magnitude (length of the arrow) of 1 2 2 u u groundspeed of the aircraft). Vectors which point inexample, opposite Fig. directions differ in their signs; for u supposed to gain an additionalrepresented velocity by the short second arrow. local wind velocity), whichare generally points vectorially in combined a different to direction, yield the “resultant” velocity Fig. Then we deﬁne tive to the surrounding air) and the much lower velocity both sides of the symbol Figure 2.3 addition or combination of the two oppositely-directed vectors Motions with a constant velocityand the are direction rare. of the In velocity general, change along the theIn magnitude path Fig. of the motion. 2.3 Deﬁnition of Acceleration: Well-formulated units can often besurement instructions. seen as a compactbook. form of mea- In everyday life, we oftene.g. make 10 m/s. do In with physics, the however, thedetermining magnitude is quantities only for aphysics, velocity. velocities The are always second direct isthey its are most clearly seen in theof process, well known also to non-physicists, In Fig. of vectors, e.g. of two velocities ginning val j The Description of Motion: Kinematics 2 bars for simplicity. their symbols in boldface, while their magnitudes are represented by normal type, dispensing with the vertical in an intuitive manner using the example of combining velocities. Vector quantities are denoted by printing C2.4. The rules for vector addition are explained here 20

Part I 2.3 Deﬁnition of Acceleration: The Two Limiting Cases 21

Figure 2.4 The general deﬁnition of the acceleration Part I

Figure 2.5 The deﬁnition of the path acceleration

as the mean acceleration. The time interval t is chosen so that the quotient no longer changes measurably when t is further decreased. Mathematically, one carries out the limit t ! 0, replaces the symbol by d, and thus obtains for the acceleration:

du a D : (2.3) dt

Just like the velocity, the acceleration is also a vector. The direction of this vector is the same as that of the increase in the velocity u (Fig. 2.4). In Fig. 2.4, the angle ˛ between the increase of the velocity u and the initial velocity u1 was arbitrary. We now consider two limiting cases: 1. ˛ D 0or˛ D 180ı,Fig.2.5. The velocity increase lies along the same direction as the original velocity. Then only the magnitude, not the direction of the velocity changes. In this case the acceleration is referred to as a path acceleration with the magnitude

du d2l a D D : (2.4) dt dt2

2. ˛ D 90ı,Fig.2.6. The velocity increase points in a direction perpendicular to the original velocity u. Now, only the direction and not the magnitude of the velocity changes; within the time interval dt through the angle dˇ. In this case, one refers to du=dt as the transverse acceleration or radial acceleration ar.FromFig.2.6,we see immediately that the following relation2 holds:

du dˇ D or du D u dˇ: u

dˇ 4:5 0:0175 2 Example: dˇ D 4:5ı, ı D 0:0175, dt D 0:1s, ! D D D dt 0:1s 0:79=s. 1 u (2.5) (2.6) and transverse ac- a ; these velocities are t (positive or negative) and nguage leaves the direction / . That is, the course of the 1 u : acceleration changes only the u ! 2 u ! D . . and the radial acceleration becomes , t ˇ D d D C2.6 d must be used. The latter are measured r a a t u C2.5 D t , according to the above definitions, is used in = u acceleration in everyday la . Practically, this requirement usually means that registration procedure t are both present at the same time; along the path of , 1564–1642.) The path r acceleration a The definition of the Motion . Then we compute ALILEI 2 u The word physics in aall, quite in different everyday sensea life than higher an speed, in accelerated e.g.Secondly, everyday motion the the language. accelerated usually word circulation means ofcompletely a First a out of motion document of consideration. or at file. – , as we have already pointed out, must be chosen to be sufficiently t form the quotient with some sort of radial acceleration the motion, both the magnitudechanging during and the the motion. direction Nevertheless, we of willthe the limit moment ourselves velocity for to are theline limiting motion) cases or of pure pure radial path acceleration acceleration (circular (straight- orbit). is called the angular velocity Figure 2.6 procedure. For example, time marks can be imprinted on the moving and small. The resultther of decrease the of measurement shouldrather not short time change intervals on a fur- motion is first recordedated automatically after and the the motion record is complete. is then But evalu- there is also a much simpler For the majority ofcelerations motions, path accelerations 2.4 Path Acceleration and Linear (G. G The quotient magnitude, not the directionfollows of a the straight-line velocity. path; As its trajectory a is result, linear. A the path motion acceleration is in principlevelocity easy at to two measure: times with We determine the the time interval ) 2.6 ). In is 6 OHL accelera- u , the magnitude u ? ! ! The Description of Motion: Kinematics D 2 r of velocities and tions given here are tounderstood. be many textbooks. It is insense this that the detailed descriptions of measurements but rather he always gives a compact instruction for its measurement; an aspect which is not emphasized in concerned not only with the definition of a quantity, direction, we see the rotation as clockwise. Then Eq. ( C2.5. The angular velocity is also a vector. Itthe points direction in of the axis of rotation: if we look along this C2.6. Here, we see among other things that P with all cases considered here, (for the definition of the vector product, see Chap. a has the general form: equation is sufficient. 22

Part I 2.4 Path Acceleration and Linear Motion 23

Figure 2.7 Measurement of the acceleration of a freely-falling body (Video 2.1) Video 2.1:

“Free fall” Part I http://tiny.cc/jpqujy

object by a clock. Of course, the imprinting process must not disturb the motion itself. We give a practical example: The acceleration of a freely-falling wooden bar is to be determined. Fig. 2.7 shows a suitable arrangement. It can also be used for many other types of acceleration measurementsC2.7. C2.7. For example, with modern apparatus, the ac- The essential element is a fine ink jet which is rotating in the hor- celerated motion of a freely- izontal plane. The jet is sprayed out of a nozzle D on the side of falling body can be conve- a rotating ink container (Fig. 2.8) (electric motor with its shaft ver- niently recorded electron- tical). The frequency, e.g. D 50 Hz, is determined by a frequency ically using photoelectric meter. Here again, the time measurement is referred to a uniform triggers. Beams of light rotation. which are directed at pho- The wooden bar is wrapped in a jacket of white paper and hung at tocells are interrupted briefly the point a. A cable trigger lets it fall at the desired time. The bar by the passing object, and the then falls through the rotating ink jet and on to the floor. – Figure 2.9 resulting signals control an shows the result: a clean sequence of time marks at time intervals of electronic stopwatch. 1/50 of a second.

The object continues falling while the ink jet swishes past; this causes the curvature of the time marks.

Figure 2.8 TheinkjetusedinFig.2.7, at half its actual size .Amore . For the magni- is one of the rare through which the l are written beside the acceleration of gravity t free fall = l : 2 D 8m/s u : The constant acceleration a for free 9 (1/50) s, increases continuously. The . It is called the D 2 D a t =9.81m/s body with time marks and their evaluation, with the usual ) g constant or uniform path acceleration Falling Video 2.1 computed values of themarks; velocity the velocity increasesthe on the amount average 19.5 cm/s. withinindividual each Here, values (1/50) s (scatter). we by ignore Thisexamples the of case inevitable a of errors intude the of this constant acceleration, we ﬁnd precise value is Repetition of the experiment withterial, for an example object a made brass ofsame tube value a instead for of different the the ma- acceleration. woodenfall bar, is yields the the same for all falling bodies. It is denoted by g Figure 2.9 We can see with thespacing unaided of eye that the the timeobject motion marks, falls is accelerated: in i.e. the The the time distance experimental and readout errors.surement of This experiment a alsomatter. second-order ( shows differential that quotient the mea- is in general an awkward The Description of Motion: Kinematics 2 “Free fall” http://tiny.cc/jpqujy Video 2.1: 24

Part I 2.4 Path Acceleration and Linear Motion 25

Figure 2.10 The velocity u as a function of time for constant path

acceleration Part I

or apparent gravity3. These are incidental experimental facts at this point. Their great signiﬁcance will become apparent later. Our practical example of a measurement led us to the special case of a constant path acceleration. This is an important case. Constant acceleration means that the increase in the velocity u is the same in equal time intervals t. The velocity u increases as shown in Fig. 2.10 linearly with time t. In each time interval t, the object travels through the distance l. Therefore, we have l D ut. Here, u is the average value of the velocity within each time interval t.Such an interval is shown in Fig. 2.10 as a shaded area. The entire triangular area 0BC is the sum of all of the distance segments l traversed in the time t. Thus for the case of constant path acceleration, the overall distance l traversed in the total time t is given by the equationC2.8 C2.8. The formula explained graphically here (2.7) follows 1 2 ; l D 2 at (2.7) mathematically from the deﬁning equations u D dl=dt i.e. the distance increases as the square of the time during which the (2.2)anda D du=dt (2.3). acceleration acts. For a D const and from the rules of integral calculus, we If the body already had an initial velocity u0 before the beginning of the obtain: acceleration, then instead of Eq. (2.7), the equation R R l D Ru dt D at dt 1 2 D a t dt D 1 at2 . l D u0t C 2 at (2.8) 2 would apply.

The origin of the constant path acceleration is completely irrelevant. It could be for example an electrical force instead of a mechanical force. Usually, for verifying Eq. (2.7), one makes use of the constant acceleration a D g which acts on a freely falling body. As an example, we mention the well-known falling rope experiment.

This experiment consists of a thin rope, hung perpendicularly, with a series of lead weights attached to it; cf. Fig. 2.11. The lowest weight nearly

3 This numerical value holds near the surface of the earth; for most purposes, g can be considered to be constant. A more precise consideration shows that g depends weakly on the geographical latitude of the location (Sect. 7.6). Furthermore, it also depends on local variations in the surface features of the earth (e.g. deposits of heavy ores under the ground), and, although only weakly, on the altitude of the location where the observations are made. does not r a changes by the u al acceleration , but only its direction. Let u be constant and let us assume that no other r a , 1629–1695.) The radi ). A lead ball and a fluffy feather fall at the same speed in a vac- Falling rope . 5.20 and Circular Orbits C2.9 UYGHENS uum, and strike thefeather ground falls at much the more samea slowly, time. as relatively In small we room well surface air,(cf. know. area in Fig. are But contrast, heavy only the bodies weakly with slowed by air resistance touches the floor. Thespond to spacing the of squares theis of other released, whole weights the numbers. is weights arrangedhears When hit their to the the impacts upper corre- floor at end equal oneStrictly of time after speaking, intervals. the another. observations rope of Theuum. free experimenter fall Only should then be cana carried perturbations highly out due evacuated in to a glassspeed air vac- tube, resistance all be bodies eliminated. indeed In fall at exactly the same change the magnitude of a velocity accelerations are present. Then the direction of the radial acceleration Figure 2.11 2.5 Constant Radial Acceleration (C. H itions of weight- The Description of Motion: Kinematics 2 high (cf. Comment C7.2). under cond lessness, e.g. in the evacuated drop tower at University of Bremen, which is over 120 m C2.9. Free fall can befor used carrying out experiments 26

Part I 2.5 Constant Radial Acceleration and Circular Orbits 27 same angular increment dˇ in equal time intervals dt. The resulting orbit is a circle. It is traversed with the constant angular velocity

! D dˇ=dt. Part I The time required for a complete revolution was termed the (rotational) period T in Sect. 1.8, and its inverse was called the rotational (or mechanical) frequency ; thus D 1=T. Then for the orbital velocity u,wehave

Circumference 2r u D D D 2r (2.9) Period T and the angular velocity is

2 Angle ! D D 2 (2.10) Period so thatC2.10 C2.10. In general, Eq. (2.11) u D !r : (2.11) is written as a vector equation: When a circular orbit is traversed with a constant velocity, the an- u D ! r. gular frequency ! is thus a factor of 2 greater than the mechanical frequency ,thatis! D 2; therefore, ! is often called the circular frequency (unit usually 1/s). This definition and these relations hold quite generally for periodic processes (e.g. the rotation of an electric motor shaft). We combine Eqns. (2.6)and(2.11) and obtain the resultC2.11 C2.11. Using the vector formulations of Eqns. (2.6)and (2.11), we obtain for the first u2 part of Eq. (2.12) likewise a D !2r D : (2.12) r r a vectorial form: ar D ! .! r/ or, for !?r: !2 This radial acceleration ar must be present so that a body can tra- ar D r. verse a circular orbit of radius r with the constant angular velocity The radius vector r points (circular frequency) ! or the constant orbital velocity u. from the center of the circular orbit outwards, while the ac- Intuitively, the constant radial acceleration required for a circular or- celeration vector points in the bit has the following interpretation (Fig. 2.12): opposite direction towards Suppose that a body traverses a circular segment ac within the time the center. interval t. Imagine this orbit to be composed of two steps that occur one after the other, namely: 1. A motion along a (tangential) path perpendicular to the radius with the constant velocity u, ad D ut; 2. an accelerated motion along a (radial) path (anti-parallel to the 1 . /2 radius), l D 2 ar t . The thin horizontal lines (time markers) in the figure permit us to see that the motion along l is accelerated, so that we can apply Eq. (2.7). A numerical example can be useful. The earth’s moon moves during the time t D 1 s along the direction ad,thatisperpendicular to , , . 7 u D l D C2.13 u (e.g. dis- , velocity t numerical values : definitions, such as physical quantities quantities , time l . A velocity of 100 unit D . For example, a hat might incorrect . unit with a 2 100 cent 5 km/h) with their 10 cent , but rather the quotient of distance/ and a D 10 u etc. are measured as 70 mm/s : D 2 D distance r a numerical value 10 $ the price of every item is a “quantity”, i.e. the 10 cent , 35 mm. Thus, the net effect is that the radius remains 7 m/s. The confusion of physical : , frequency C2.12 numerical value 1 a The explanation of radial D 5 km and velocity D u and Their Numerical Values 2 D / t l . r a 1 2 for example “the velocity is the distance traversed in a unit time” is meaningless. It becomesexample meaningful only when expressed as for cost 10 $ andprices a to be pencil the 10 same. cents. The ratio No of the one two would prices consider is rather these two The same principle holdsacceleration in physics: Distance tance (in the example, the value oflocity the is distance is 5) 5 gives and rise the to value widespread of but the ve- i.e. as products of a acceleration product of a Figure 2.12 The velocity is not a 2.6 Distinguishing Physical Quantities In commerce its orbital radius, by 1earth km, slightly. thereby “increasing” At its the distanceorbital same from radius time, the in it “approaches” an the accelerated earth motion, along traversing the the distance unchanged, and the orbitmoon is is found circular. to be The radial acceleration of the time. –tions Or in still one worse: second”. “The First of frequency all, is a the frequency is number not of a oscilla- number, but OHL ound in The Description of Motion: Kinematics 2 tions” can still be f many textbooks! consistently treated in most textbooks even today. C2.13. Such “false defini- with examinations. They deal with simple and indeed self-evident topics, which are however to some extent not portant concepts, from the 10th edition on; possibly because of his experience rectly to the content of the rest of this chapter. P however added them here, after introducing the first im- C2.12. The following two sections do not belong di- 28

Part I 2.7 Base Quantities and Derived Quantities 29 rather the quotient number/time; e.g. the pulse frequency of a person is ca. 70/minute. Secondly, a physical quantity which is supposed to be generally applicable cannot be deﬁned in terms of a particular unit Part I such as the second.

2.7 Base Quantities and Derived Quantities

Some few physical quantities are termed base quantities and are measured in units defined especially for those quantities, the base units; for example, time with the unit second,ortemperature with the unit kelvin. If one wishes to introduce a base quantity, it can be defined only in terms of axioms which are founded on extensive experiments or observations, and not on equations. Most physical quantities are defined as derived quantities.This means that they and their units can be defined not in terms of axioms, but instead by means of equations which contain other quantities and their units. We recall the example of the velocity, u D dl=dt and its units, chosen for example to be meters/second or kilometers/hour, etc. The possibility of defining quantities and their units in terms of equations is the only point in which derived quantities differ from the base quantities employed. No physical quantity is in its essence a base quantity; one could introduce many different quantities as base quantities. The number and type of the base quantities should be chosen insofar as possible so that no two derived quantities have the same defining equation. – In distinguishing between base quantities and derived quantities, one should in no case envision a hierarchy; base quantities are not im- bued with a special aura, nor should limiting them to a certain number (e.g. three) be raised to the status of a dogma. The currently agreed-upon base quantities in the international unit system (SI) and their base units areC2.14: C2.14. In these books (Vol. 1 and Vol. 2), we use mainly Length (or distance), unit meter (m), (but not exclusively) the units Time, unit second (s), defined within the SI. See Mass, unit kilogram (kg), for example http://physics. (Thermodynamic) temperature, unit kelvin (K), nist.gov/cuu/index.html or Electric current, unit ampere (A), http://www.bipm.org/en/ Luminous intensity, unit candela (cd); and measurement-units/. Amount of substance, unit mole (mol).

Note that the names of units and their abbreviations are printed in Romantype(m,s,kg,...),while physical quantities are printed in italics (distance = l or d, mass = M or m, etc.). Avoid confusing . , 2 2 81 m/s : 9 D did he jump? after jumping g h 2 and moves along take in each case? of a person who is P O r 81 m/s : a to 9 P is also 300 m upriver from . position for (a) constant D B vs g otion back and forth through the , a distance of 3 m away. There, 20 kg increases its velocity from The online version of this chapter (doi: acceleration and the corresponding O which is traversed during the fourth D l m ) 0. The acceleration of gravity is 3.2 ) . D ) t are points opposite each other on either side of the and 2.5 C2.15 B 4.3 2.4 ) was he in the air, and from what height t and and ) ) ) A . How long does the motion from How large is the radial acceleration A particle begins at rest at the point Calculate the distance An object of mass A parachute jumper initially falls freely over a distance of A person wants to row her boat along a straight-line path from . 2.4 2.2 2.4 2.4 O B to . The person rows at the same velocity as the upriver ﬂow rate 2.6 (Sect. 2.5 a straight-line path to the point its velocity is 6 m/s. Plot its velocity acceleration, and (b) a sinusoidalpoint m mouth of a river, which is 600 m wide; A of the tide which(Sect. is coming in. In which direction should she row? 2.2 second by an object in freesition fall at which the began time falling from a resting po- (we may neglect airhis friction). fall (i.e. Then it produces his a parachute negative opens, acceleration, delaying upwards) by 2 m/s (Sect. 2.4 (Sect. so that he ﬁnally landslong on (time the ground with a velocity of 3 m/s. How 2.3 50 m with the acceleration of gravity 15 m/s to 18 m/s,is covering the a magnitude distance of of its 20 m constant in the process. What force? (Sect. at 51.5° north latitude6378 (e.g. km. in (Sect. London)? The radius of the earth is Electronic supplementary material 10.1007/978-3-319-40046-4_2) contains supplementary material,able which to is authorized users. avail- Exercises 2.1 A them; there are not enough lettersto in allow the us Roman and to Greekthat alphabets use we different require symbols for all the quantities and units tity http:// . physical-quan The Description of Motion: Kinematics 2 C2.15. Recommendations for writing expressions and equations and for naming or abbreviat- ing units are given at physics.nist.gov/cuu/pdf/ checklist.pdf 30

Part I ure the For bodies. t egh(a 4m;i csa n‘pia ee’ estdown set We – lever’. ‘optical an point as the acts at it weight to metal m); owing 24 a “light- motions This (ca. small arrows. length to illu- small sensitivity its the great an by provides source, shown pointer” light direction as beam the the them of in between image mirrors passes the an rays casts light It slit of minated sketch. beam the a in and shown table, this on set oc sdtrie ulttvl ytocaatrsis tcan It characteristics: two by form qualitatively determined is force A n ntebok aldits called a block, that the says on one technology, ing and physics In deformed. be u uclrfre ial,w elc h lc ihaln rod long a (Fig. with rod namely block the block, the down the replace hand on our we more. acting slide Finally, bend is and to force force. top second muscular table a our the Now causing say: finger, could little We our with block downwards the accelerated being from block udmnaso Dynamics of Fundamentals .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 3.1 Figure of concept The as defined mean- be physics. of to of variety need a first have language, everyday which in terms, ings two These “ac- “mass”. the and “force” and “velocity” in the are celeration”; quantities relevant the kinematics, In Mass and Force 3.1 ocs ..afigrpesn ona h point the at down pressing finger a e.g. forces, 3.1 oi oiswihaehl xd n tcan it and fixed, held are which bodies solid hw noktbewt hc frame thick a with table oak an shows deformation pia eeto ftedfraino al o ysmall by top table a of deformation the of detection Optical S notewl.Aybnigo h al o iltilt will top table the of bending Any wall. the onto , dynamics force fasldbd egv la xml:Fig- example: clear a give we body solid a of sbsdo h es foronmuscles. own our of sense the on based is olsItouto oPhysics to Introduction Pohl’s erqiei diintecnet of concepts the addition in require we , weight A ..a1k lc.Tetbetpwill top table The block. kg 1 a e.g. , ; h eomdtbetpkesthe keeps top table deformed the 3.2 A hnw rs onon down press we Then . .Aan h al o is top table the Again, ). C3.1 Z accelerate O 10.1007/978-3-319-40046-4_3 DOI , w irr are mirrors Two . ehia terms technical force movable sact- is de- hw nFig. in shown as deformation torsional a also (Compare light. of laser beam a using today out carried readily be can ment experi- striking This C3.1. 3 6.7 .) 31

Part I ˛ into G F counter and tan D ˛ and . The vector F F ponents. A roller ,orforce , and is produced here 1 reactio D . 10.2 shows an example of this decom- . If the ramp is very steep, actio ˛ must be large. 3.3 F tan external friction = G F D , are held in equilibrium by the elastic force of the F II . They can be decomposed into their components , pull the roller downwards and upwards, respectively. In ˛ and sin I ’s terms, this is called F vectors The decomposition of vectors into their com The force due to external friction acts on the rod, and its counter force acts and ˛ EWTON denotes the weight of the roller. We decompose both cos is being held ﬁxed on a steep ramp by a horizontal force G G For internal friction (viscosity), cf. Sect. force. We offer three examples: are quite small, and thus the force slightly deformed surface of theF ramp. The components parallel to the ramp, components parallel to the rampcomponents and components which perpendicular are to perpendicular it.by to The the two the surface arrows of the ramp, represented by the slipping motion of our hand slidingForces down are the rod. Figure 3.2 A F 1 Figure 3.3 deformed and we can sayto that its a own force weight; is it acting is on called the rod, in addition downwards in an upwards direction onindicates the the direction hand. in The which arrow thealong hand the slides rod, and of its frictional force. in different directions. Figure position. Forces always act in pairs:places, The but they two are forces oppositely mayIn directed N act and at have two the different same strength. equilibrium, we have Fundamentals of Dynamics 3 32

Part I 3.1 Force and Mass 33 Part I

Figure 3.4 Deformation of a circular spring. In the center of the spring is a guide rod. This simple apparatus will be used later for demonstration experiments as an uncalibrated force meter.

Figure 3.5 Force D counter force, actio D reactioC3.2 (Video 3.1) C3.2. At the left in the picture is master mechanic W. S PERBER, and the author 1. In Fig. 3.4, at the left a stretched circular spring is shown being is on the right. pulled by two hands. A force acts on each hand. If the spring is held by only one hand, no force and no deformation occurs; cf. Fig. 3.4 Video 3.1: on the right. “Action D Reaction” http://tiny.cc/ppqujy 2. In Fig. 3.5, we see two flat, nearly frictionless carts on a smooth, flat floor which supports their weight without noticeable deformation. The arrangement is completely symmetric, the carts and the men on each side have the same size, shape and mass. – Both of the men could pull at the same time, i.e. they could work together as a “motor”; or only the man on the left or the man on the right could pull. In each case, the two carts meet in the middle. Thus, always two forces occur simultaneously. They are oppositely directed and of equal strength. This is indicated by the arrows. 3. In the case of the force that we call “weight”, the counter force would seem to be lacking. But that is simply due to our choice of the frame of reference. Figure 3.6 shows on the left the earth and a large stone. As seen from the sun or the moon, this picture would have to be drawn with two arrows. The earth pulls on the stone, and the stone pulls on the earth. Both bodies are accelerated towards each other. In Fig. 3.6 on the right, their approach is prevented by inserting a spring between them. In this case, two new forces occur, denoted by FD and FD. Now, equal but opposite forces act on each of the two bodies. Their sum is zero, and therefore the two bodies remain motionless relative to one another. The concept of mass is even more ambiguous in everyday language than the word ‘weight’. For example: dough is a mass that can be kneaded; the news media appeal to the broad mass of people, they consume a mass of paper, etc. . set of (e.g. a beam : One defines the 1 metric ton (t)). scales 3.7 D is defined by a primary kg which act on the weights 3 forces ; it is internationally denoted as the unit of mass 2 grams (g), 10 3 The 10 . D ’s Fundamental Equation C3.3 , 1643–1727). For the measurement of masses and , upper part) and some sort of (1 kg counter force 3.7 C3.4 D EWTON Measurement of On the pairwise occurrence of EWTON (kg) N of Motion (Fig. N ) and a beam balance. The . http://www.bipm.org/en/publications/mises-en-pratique/kilogram.html SAAC It consists of 90 % Pt and 10 % Ir (wt.-%) and is kept at Sèvres, near Paris. See above a certain (gravitational) force. The Like all massive bodies, the weights are attracted to the earth with standard made of noble metals kilogram a mass using a set( of weights masses of two objects arewhen the both same have the sameat weight the same place, i.e.tracted they to are the at- earth with theforces same balance or a spring balance). The measurement of massmasses is of two explained bodies as in equal Fig. whenon they can a be scale used interchangeably or balance of forces, oneweights can use the same basic apparatus, namely a forces, force Figure 3.6 2 also Figure 3.7 (I In physics, however, the concept ofof mass refers every to object, two namely characteristics “gravitational”tational” means and that “inertial” every massive mass. objecta is “Gravi- attracted force to which the we earth with callcan its change its “weight”. velocity – (magnitude and/or “Inertial”change direction!) means in by velocity that itself; requires no every the object action of a force. 3.2 Measurements of Force and Mass. http:// ition_of_ http:// and ). ) or with an oscillat- ). 5.8 4.3 Fundamentals of Dynamics 3 SI_base_units www.nist.gov/pml/newsletter/ siredef.cfm made to redefine the unit of mass (see e.g. en.wikipedia.org/wiki/ Proposed_redefin “prototype”, and it seems to be showing signs of aging. Efforts are therefore being (Sect. C3.4. This is the onlywhich unit is still defined by a e.g. in collision experiments (Sect. ing spring-and-mass system Two masses are equal when they have the same weight. In principle, the “inertial” property can also be used, C3.3. In this case, massmeasured is by making use of its “gravitational” property. 34

Part I 3.2 Measurements of Force and Mass.NEWTON’s Fundamental Equation of Motion 35 Part I

Figure 3.8 Measurement of a force using a set of weights and a beam balance. The force exerted by a spring which has been stretched by an amount x is compared to the force acting on a metal block from the earth’s gravity, for short the weight of the block. (The weight of the relaxed spring is cancelled out by the weight of the small metal block.) For practical applications, a spring balance is more convenient to use than a beam balance; compare Fig. 3.9. are called simply their weights3. These weights, i.e. forces, can also be used to measure forces in general. This is made clear by Fig. 3.8. There, the force of a spring is compared with another force, namely the weight of a kilogram. The unit of force is the newton (N) (cf. Eq. (3.4)). The weight of a kilogram block at a place where the acceleration of gravity is 9.81 m/s2 (rounded off) is equal to 9.81 newton. Compare Fig. 3.9.

Why can we use the same apparatus, namely a set of weights and some sort of balance or scales, to measure both masses and forces? The answer is indicated in Fig. 3.7: The masses of two objects are defined to be equal when they have the same weights at the same place, i.e. when they are attracted with the same forces by the earth. The weight of an object depends first of all on a property of that object, called its mass, and secondly on the earth: the forces that we call weights are always directed vertically towards the center of the earth. The influence of the earth is the same on both sides of the balance and therefore cancels (more about the force that acts between masses, called the gravitational force, in Sect. 4.7).

The relation between force and mass is derived experimentally by varying the two quantities independently of one another and measuring themC3.5. The simplest experimental arrangement is shown in C3.5. Now, having intro- Fig. 3.10. As the known force, we make use of the force F D FG duced forces in terms of the which acts on the small block A and which we call the weight of the “gravitational” mass, we will block. By means of a cord, this force is used to accelerate a loaded derive the relation between cart. The overall mass of the objects that are accelerated together, force and “inertial” mass. that of the cart and its load as well as that of the block A, is taken to be m. The acceleration is constant and therefore can be easily computed from the distance travelled, s, and the corresponding time t;it is a D 2s=t2 from Eq. (2.7). In this way, one ﬁnds that the acceleration a is proportional to the force F and inversely proportional to the

3 The word ‘weight’ inevitably reminds those who are not steeped in physics of a property of the object instead of a force acting on it. This makes it more difﬁcult to realize that these forces come about through the effect of the earth (or e.g. of the moon if the object were located on the lunar surface). (3.3) (3.2) indepen- )atall 7.6 If one sets the proportion- : ; const. (3.1) F F m m ) takes on the simple and convenient D D F m a a 0.3 % (cf. Sect. 3.1 D ˙ a 90 newton. : 4 1, then Eq. ( D D 2 Calibration of a spring balance as a force me- ,thatis of each other; or expressed positively, one makes use of the 81 kg m/s : 9 1 2 This ﬁnding forms the basis ofality mechanics. constant form one quantity for the measurement of the other. One thus can dispensedently with measuring the mass and force or, more generally using vector notation, Figure 3.9 points on the surface of the earth and in this case is equal to ter. The known force is inforce this that example acts the on gravitational asame metal within block the error of limits mass of 1/2 kg. It is the mass m Fundamentals of Dynamics 3 36

Part I 3.2 Measurements of Force and Mass.NEWTON’s Fundamental Equation of Motion 37 Part I

Figure 3.10 On the experimental derivation of the fundamental equation of motion. The force which produces the acceleration is the weight FG of the small block A. The weight of the cart and its load is compensated by an unmeasurably small deformation of the horizontal surface (plate glass ﬂoor). – Friction and rotational inertia of the wheels reduce the acceleration by 5 to 10 %. To eliminate these sources of error, it is sufﬁcient to incline the surface on which the cart moves by a small amount (a few tenths of a degree). This inclination is adjusted so that the cart, given a small push without the cord, moves along its path with practically constant velocityC3.6. C3.6. This basic experiment can be demonstrated today with a cart that moves almost Physics makes use of the mass for the measurement of forces.The without friction on an air- resulting unit for force, kg m/s2,isthederived unit of force which is track. POHL mentioned such called the newton, that is airtracks in the later editions of his book, but did not de- 1newton.N/ D 1kgm/s2: (3.4) scribe any actual experiments with them.

Eq. (3.3) contains the great discovery of ISAAC NEWTON, the relation between force and acceleration. It will be found in later sections to withstand the most rigorous experimental tests. This is rather surprising: Eq. (3.3) contains the common property of all massive bodies, their inertia (Sect. 3.1) and the forces which are therefore necessary to accelerate them. However, the masses m of the objects were measured using the other property which all massive bodies have in common, namely their “gravitational mass”; and this in a state with the bodies at rest! – This fundamental fact is often described by the brief but frequently misunderstood statement: “gravitational mass D inertial mass” (equivalence principle). Important applications of the equation a D F=m will be treated in the next chapters. At the end of the present chapter, we first clarify some of the other concepts that will often be needed for what follows. ex- (3.5) this force and the mass 2 only (free fall!). : 2 : ) which will help to . Assume a body of 2 ,when V s m 1kgm/s 2 : m 3.2 81 N (previously called : 2 D 81 9 m s : 81 m/s 9 : 1 density i.e. we replace each symbol 9 D 1N Mass D D , Vo lu m e F D D 2 2 a D C3.7 1 kg. The result is: F % D 1 kg is acted upon only by its weight, 1kg . Then we define m 1kg 4 its unit. 1kgm/s V 81 kg m/s : 9 D and a D a Mass density or more briefly, simply and the mass is known from the dimensions of the body. In such ), we insert the force ) ), we insert the force V 3.2 C3.8 3.2 Expressions Containing Physical Quantities to have the volume m 1 kg. The result is: mass density D The use of the word mass in the place of ‘body’ or ‘object’ is apparently inerad- m then it experiences an acceleration is acting on the objectof (the the counter object). force is the inertial reaction2. force In Eq. ( In words: If a body of mass with a numerical value Or in words: 1a newton mass is of the 1 force kilogram which an can acceleration give of to 1 an m/s object with This quantity, for fixed external conditions (pressure,a temperature), constant is that characterizes theposed. material of which the body is com- icable. Over and over,than one a finds body, for i.e. example instead‘weight’ a of ‘mass’ is the hanging used object, on ina a one a body string, of similar with rather its a way: properties! certain weight. a In This ‘weight’ English, is, is the however, word hanging officially tolerated on by a the string, SI. rather than cases, one can compute its mass byof making use of the helpful concept 4 pressions with physical quantities 3.4 Density and Specific Volume In the cases ofmass very can large often not and bethe clumsy measured volume or directly of with a very scale small or bodies, balance, the but clarify the units of force and mass. To this end, we make use of 3.3 The Units of Force and Mass. We begin by examining two results of Eq. ( mass 1. In Eq. ( 1 kilopond him- tities is OHL physics, the .P ’s book to aid in newton ’s books! OHL Fundamentals of Dynamics 3 OHL making the distinction between force and mass units, following a suggestion by the German standards laboratories, the Physikalisch- Technische Reichsanstalt (today the PTB). It hasbeen now suppressed in favor of the unit self wrote (from the 17th edition on): “In kilopond as a unit ofalongside force the newton is just as dispensable or superfluous as is the energy unitalongside ‘calorie’ the watt second.” C3.8. The unit of forcepond”, “kilo- which is no longer used, was initially included in P C3.7. The consistent use of expressions and equations with physical quan a particular advantage of P 38

Part I Exercises 39

Sometimes, instead of the mass density %, one uses its reciprocal, the

Vo lu m e V Part I Specific volume V D : (3.6) s Mass m Specific in general denotes a physical quantity when the quantity itself is not meant, but rather a quotient of the ‘specific’ quantity and another quantity. This makes it unnecessary to introduce a new name for the quotient. The ‘specific volume’ is thus the quotient of volume and mass, or the volume referred to the mass.

Exercises

3.1 An object sitting on a frictionless, horizontal plane on the earth’s surface is subject to the following forces: F1 D 50 N in the direction of the azimuthal angle ˛ D 155° (geoscientiﬁc notation, i.e. north: ˛ D 0; east: ˛ D 90° etc.); F2 D 30 N in the direction ˛ D 230°; and F3 D 20 N to the north. Which additional force to the northeast (˛ D 45°) would be required to make the resultant total force Ftot lie along the east-west axis? What is the magnitude of this total force and in what direction does it act? (Sect. 3.1)

3.2 An object with the weight FG is on a rough inclined plane with an inclination angle of ˛ D 40°. The coefﬁcient of sticking friction is h D 0:3. The object is to be just kept from sliding down the incline by a force F which makes an angle to the vertical. a) Find the expression relating the force F to the angle . b) By variation of , ﬁnd the direction of the smallest force that will still hold the object from sliding. (Sects. 3.1 and 8.9)

3.3 A body of mass m1 D 0:5 kg is sitting on a planar table at a distance of 1 m from its edge. A cord attached to the body is hanging over the edge of the table and carries a weight of mass m2 D 20 g (see Fig. 3.10). The system begins to move starting from this conﬁg- uration. How large are its velocity u and its kinetic energy Ekin when the body arrives at the edge of the table? What is the magnitude of the force along the cord, FF? (Friction and the mass of the cord are negligible.) (Sect. 3.2 and 5.3) For Sect. 3.2 see also Exercise 2.4.

Electronic supplementary material The online version of this chapter (doi: 10.1007/978-3-319-40046-4_3) contains supplementary material, which is available to authorized users. Equation knee-bends plctoso N of Applications .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 4.1 Figure form force acceleration the the the take of We cause convention: the useful be a introducing by start We Straight a in Acceleration Constant 4.1 ecl the call we t antd a era f h cl ftefremeter. force the of and scale upwards, the directed off is read be force can This magnitude its (spring). meter force a pressing hs xeiet,w hudke ai ati id o each for mind: in all its fact In only basic specify a direction. can keep vertical one the force, should ( in we Eq. body experiments, of a these application of first acceleration our the on concerns variations several are There role no play they effect sometimes useful. concepts and familiar, even the and are speaking, convenient are In ‘effect’ and arbitrary: ‘cause’ completely is convention Our e sasm httofre c nabd mr rcsl,te act force, they first precisely, The (more gravity body of a center on its act at forces two that assume us Let (compres- deformation spring”. the a to of related tension) force sion, “the only means spring a direction u ee its never but , Line h onad acceleration downwards The weight F C4.1 G . stefreo rvt,drce onad,which downwards, directed gravity, of force the is , oee,i h qain fpyis as and cause physics, of equations the in However, ftebd;tesecond, the body; the of S f Sect. cf. ; on forigin of point olsItouto oPhysics to Introduction Pohl’s a on faction of point a D n rt h qaino oinin motion of equation the write and 5.6 m F a and : famndoing man a of o xml,tefreof force the example, For . 6.2 F n opr Fig. compare and , 1 EWTON its , spoue ycom- by produced is , O 10.1007/978-3-319-40046-4_4 DOI , magnitude 3.3 n its and ’s ( 4.1 F 3.3 ;it ); to ): ) enta ewl derive will we that mean not does title chapter the in “Applications” C4.1. on nohrtextbooks. other in found always not are and instructive are These examples. selected with validity its will demonstrate we that rather but tion”), mo- of “equation the as known (also equation ton’s New- from motions specific 4 41

Part I in- up- 1 is the F G ,wenow F 4.1 downwards have differing upwards mo- G F is the oppositely- 1 and the force has the larger magni- F and G accelerated G . altitude, or is flying up- F 1 accelerated F F or 1 F constant velocity, the scale reading includes downwards The weight ). of the man, and constant C4.2 weight 70 kg b D ). At the upper end of the force meter is a body of The direction of the acceleration (upwards or down- 4.2 The origin of the ‘elevator feeling’ . The answer: When thewards fly or maintains downwards a with a motion (“the fly is falling freely”),too the small balance indicates – a case weight 2. which is When the fly carries out an the weight of the fly –to case a 1. somewhat (The oversized air fly molecule.) can then During be an considered to be similar tion (“flying up faster and faster”),too the high balance – indicates a case 3. weight which is weight. The resultantwards force, and the acceleration,A are variation directed on thisa experiment trick is question: often Instead encountereduse of in a the sensitive the spring large form balance;containing spring of a on as live its fly. in weighing Does Fig. pan the balance is indicate a the closed weight bottle of the fly? dicated by the springresultant balance force is are zero. equal and opposite,2. so The that man their acceleration, does the an upward accelerated forcethan read knee-bend. his off downwards-directed the During weight. spring his Asand balance downward the a is acceleration, result, smaller are the directed resultant force, 3. The man acceleratestion, the back upward up force to read his off the full spring height. balance is During greater this than his mo- tude. The test body infigure; the the first force experiment is meter a is man, a as sketched common in spring the balance. magnitudes. wards) depends on whether the force downwards-directed directed (upwards) force of thetions, spring. in logical We order: can make three observa- 1. The man(e.g. 687 stands newton at rest. The spring balance shows his weight Another variant: Ana experimenter circular spring, holds which we the havevertical already force (Fig. encountered, with meter its made spring of Figure 4.2 We observe accelerations only so long as ’s Equation EWTON with g = G . A body 2 F s = D m m 81 : 9 Applications of N D 4 g weight: the unit ‘kilogram’, i.e. they indicate the mass which corresponds to the measured the ‘newton’, has not become common in everyday use. Commercially available scales continue to use C4.2. The unit of force, of mass 1 kg has aof weight 9.81 N. 42

Part I 4.1 Constant Acceleration in a Straight Line 43 mass M, acted upon by its weight FG. If the hand holding the force meter is moving with a constant velocity upwards or downwards, the indicator reading (the compression) of the circular spring is the same Part I as when the hand is at rest. When, however, the hand is accelerating upwards or downwards, the spring will be more or less strongly compressed, i.e. the upwardly-directed force F1 is larger or smaller than the downwards-directed weight FG, respectively.

A similar experimental setup often plays an unpleasant role in our daily lives. The hand can be thought of as the floor of an elevator cabin, and we can think of the circular spring, in a drastically simplified version of Video 4.1: anatomy, as our gut, while the body M is our stomach. When the elevator “MAXWELL’s wheel” is accelerating downwards, the spring is stretched relative to its normal http://tiny.cc/9pqujy resting position. This relaxation is the physical basis for the much-disliked We can even follow the mo- “elevator feeling”, and when repeated periodically, it causes seasickness. tion without intervention when the wheel reaches its Finally, we consider the same experiment in a quantitative version. upper reversal point and First, the principle: We hang a block of mass m on a beam balance moves downwards again. (Fig. 4.3) and measure the force FG, i.e. the weight of the block. Then The essential observation, an invisible mechanism causes the block to move downwards with namely that a body which is a small acceleration. The balance deflects in a clockwise direction, being accelerated downwards its center of gravity is moved somewhat to the left and upwards. In is lighter, no matter which this position, the force pulling the block upwards, F1, is smaller than direction its velocity vector the weights on the right balance pan. One would thus have to remove is pointing, is somewhat ob- several small weights from the pan to return the balance to its zero scured in the first part of the position during the downwards acceleration of the block. Then, the experiment by two effects: force pulling the block upwards, F1, has the same magnitude as the When the wheel initially be- remaining weights on the right pan. During the downwards accelera- comes lighter, the damped > tion, we thus have jFGj jF1j, i.e. the resultant force F D FG CF1 is oscillations of the balance directed downwards; its magnitude is jFGjjF1j. This downwards- make it hard to read its deflection accurately. The same problem occurs again when Figure 4.3 A beam balance holds the balance stops moving ablockofmassm, at rest. If the briefly at the lower reversal block is accelerated downwards, point of the wheel. In the sec- the indicator of the balance shows a deflection (the right pan sinks) ond part of the experiment (after about 75 s), when some weight has been added to the pan on the side of the balance where the MAXWELL wheel is hanging, it is in equilib- Figure 4.4 Continuation of Fig. 4.3. rium during the downwards The object which experiences a constant motion. Now, when the im- downwards acceleration takes the form pulse at the lower reversal of a flywheel (“MAXWELL’s wheel”). point is absorbed by the The balance is damped by an oil column (not shown). At the lowest point of the balance, it remains in equi- wheel, it changes its direction and moves librium during the following upwards again. This causes a downward upwards- and downwards jerk (an impulse; cf. Sect. 5.5). This im- motions, without the disturb- pulse is absorbed by briefly holding the ing oscillations. Note that pointer of the balance. (Video 4.1) Fig. 4.4 shows the balance from the opposite side as in the video. ) m 2.12 .The ( 83 m : N. 0 2 ,‘towards required to D points from 10 s t r 6 : 2 accelerated This experiment of- jD 1 during the accelerated F : jj centripetal force / , computed from G j 2 1 F : j ’s equation of motion only to . It is the surface of the earth, F r smaller 2 (the radius vector , this motion is 2.1 ! m ): The force meter is a simple acceleration (along a straight-line ! jj and finding the time 048 m/s : 2.5 G 0 2 4.4 EWTON F D j at D r path . 1 2 a a 262 N, so that : D . The strings are again wound up, in the 5 D a s D (it is also called the 288 N, 1 : 5 F with a stopwatch. directed towards the center of the orbit; we will inertia s is supposed to move along a circular orbit of radius D F G m acceleration. F 9 s. Here, , since it is moving upwards more and more slowly, or 0g, : : 5 radial force radial Never draw any conclusions about circular or rotational mo- D 539 t D and Numerical example m with a constant angular velocity call it the travel the distance ten surprises even experienced physicists. requires a force From the equation of motion, this acceleration of a body of mass kitchen scale. The object is atwo flywheel strings on a which thin are shaft. woundaccelerated It around is downwards hung the by from shaft. itsby weight. When using released, The it the acceleration is equation is measured Practical demonstration (Fig. tion before agreeing with yourthe dialog textbook!) partner (perhaps on the thereference author was of frame agreed of upon reference in Sect. to be used. Our frame of or the floor of the lecture hall. Up to now, we have applied N the limiting case of a purely path). Now wea do purely the same for the other limiting case, that is for motion than at rest.downwards The accelerationwith of a the “braked” motion wheel (negative is acceleration). still directed opposite sense, and the wheel climbsto back repeat up. the One observation should incation not this forget of direction the of force motion. meter Again, (balance) the indi- is 4.2 Circular Motion and(Observer Radial at Forces rest!) Firstadvice: of all, as a preliminary remark, some good After the strings have rolledrotating off to due their to ends, the its flywheel continues directed force causesblock: the observed downwards acceleration of the radial acceleration, directed towardsorbit, the is center point of the circular the center point of the orbitour towards kinematic the treatment moving in body). Sect. According to r A body of mass ’s Equation EWTON good advice: Applications of N 4 “This experiment often Never draw any conclusions about circular or rotational motion before agreeing with your dialog partner on the frame of reference to be used.” “...some surprises even experienced physicists.” 44

Part I 4.2 Circular Motion and Radial Forces 45 Part I

Figure 4.5 A ball on a rotating table, held by a leaf spring at the left of a. The vertical force which acts on the ball, its weight, is compensated by an invisibly small deformation of the tabletop. the center’). Quantitatively, from the equation of motion, we must have F !2r D (4.1) m

(Circular frequency or angular velocity ! D 2; is the mechanical frequency, i.e. number of rotations/time (e.g. rpm).)

For an experimental test of Eq. (4.1), we replace the angular velocity ! by the mechanical frequency and obtain F 422r D : (4.2) m

The radial force F could be produced by deformation of a spring; i.e. in brief, by an elastic force. We offer three examples: 1. A leaf spring produces the radial force on a ball at the outer rim of a carousel or rotating table (Fig. 4.5). It pulls in the direction of the axis of rotation (towards the center of the table) and can produce a maximum force of Fmax. For this experiment, the spring is mounted below so that it can rotate, and its upper end is ﬁxed by the holding pin a. To determine Fmax, a counter force is applied via a cord and weight. If the maximum force Fmax is exceeded, the spring snaps out of its holder. This spring can be used only up to a certain maximum frequency max;thiscritical frequency can be computed from Eq. (4.2), giving r 1 F D max : (4.3) max 2 mr

Numerical example Fmax D 1:77 N; m D 0:27 kg; r D 0:22 m; then s 1 1:77 kg m s2 D D 0:87 s1 max 2 0:27 kg 0:22 m 1 Tmin D D 1:15 s : max , F 4.7 (4.4) (4.5) ,away escape while the o on a circular ) yields for the Dr outwards tangential D 4.2 F of the weights by tapping /: r : D m r D r ). phenomenon are the sparks flying is maintained, the body follows an 1 4.6 2 D spring constant F D path with constant velocity. Unfortu- D D . The force produced by a helical spring, is then the distance of the centers of gravity of the . When the radial acceleration is lost, the ball o frequency C4.3 r a body moves with a single frequency straight-line critical A grinding tangentially allows its extension to be measured even during the rotation. ), is directed towards the center point of the circular orbit The length of the orbital radius is totally unimportant. As F 4.7 Linear force law. The helical spring mustapparatus be is pre-tensioned at to rest; theweights force from their rest positions. (Fig. and its magnitude is proportional to the radius of the orbit, that is arbitrary circular orbit once it has beenThe established. linear force law can bethe implemented body in is various split ways. symmetrically Inpossible and Fig. on mounted two with as guidewe little rods. call friction the as These weights of rods thespring compensate two bodies. the The forces arrangement that of the helical frequency from the center of rotation (Fig. 2. orbit. long as this This means that wheel, spraying sparks The experiment verifies the predictions.rectly adjusted, When we the can frequency change is the spacing cor- Inserting this condition into the general equation ( If this limitingcarousel value is exceeded,flies the off ball on flies a off. It leaves the Figure 4.6 nately, its weightThe generally weight disturbs converts the our originallyfalling straight-line observation path curve; into of a but this parabolic velocities. this path. A disturbance good example is of lessoff this a important grinding wheel. at They highpaths. exhibit very orbital The clearly the glowing metal particles by no means fly ’s Equation ...” EWTON ), we can Applications of N 4 http://tiny.cc/5squjy perpendicular to the corresponding radii”. the contact point of thewith tires the road is the centerrotation. of All the mud flies off spraying grinding wheel, i.e. tangential spraying in all directions. For the pedestrian behind the car, in contrast, in the moving car, the mud- throwing tires exhibit the same picture as the spark- hit by the mud they(as throw long off as the wheelsslipping!). are not The explanation is simple: For the observer a car which is throwing off mud. One can cross aroad smooth right behind a cardirty with wheels without being “The spark-spraying grinding wheel seems to contradict our observations of the wheel of sparks tangentially. Up to the 11th edition, the following striking example could be found here: Video 6.7, “Supple shaft ( see a grinding wheel spraying C4.3. At the end of 46

Part I 4.2 Circular Motion and Radial Forces 47

Figure 4.7 Circular motion with a linear force law.

This ﬁgure also illustrates the Part I scheme of an “astatic” frequency controller for all types of motors. When deviations from the critical rotation frequency occur, the two bodies move either all the way outwards or all the way towards the center. The disk S can then activate a control element of the machine and restore the critical rotation frequency.

the disk-shaped endplate S of the spring with a ﬁnger, increasing or decreasing it at will. The bodies follow their circular orbit for any radius. At this critical frequency , the weights are in an indifferent neutral equilibrium, similar to that of a ball which is at rest on a ﬂat, horizontal table.

3. Nonlinear force law. Let the magnitude of a force produced by a spring and directed towards the center of the circular orbit be, for example, proportional to r2, with a new force constant D0;thatis r F DD0r2 (4.6) r .r=r D is a unit vector in the direction of r/:

Inserting this condition into the general Eq. (4.2) for the radial force yields the frequency r 1 D0 D r : (4.7) 2 m The frequency is now dependent on the radius r. For each frequency, there is only one possible orbital radius r. In this orbit, the body is in a stable equilibrium, similar to a ball resting at the bottom of a bowl. Experimentally, such a nonlinear force law can be implemented for example by using a circular spring as in Fig. 4.8. During rotation, the system can easily be perturbed, e.g. by tapping on the disk S. Following the perturbation, the correct value of r is immediately re- established. Up to now, our demonstration experiments for radial acceleration by means of a radial force involved rotating bodies with a very simple shape. They were “small” balls or blocks. We could neglect their diameters relative to the orbital radius r without making a significant error. They could be considered, put briefly, to be pointlike (“point masses”). Our final example illustrates the rotation of a more complex body, namely a chain ring. r (4.8) ; it thus r = on each link, as derived ), we obtain the magni- decreases as 1 ): F : r F 4.1 ). Each section of this chain 2 ! r D mu 4.11 u ). Within these segments, the forces D tly-fitting chain is stretched around d 4.4 0 r F directed towards their momentary centers F D accelerates each link of the chain towards ility F F ). If the individual links of the chain were not can be S 4.9 , which lead to a radial force 0 F . This force Circular motion A chain on a flywheel, for ). This force varies along the chain. (using the orbital velocity F 4.8 4.10 . The corresponding r produce the radial force 0 of curvature (midpointsEq. of ( the circles), as required for circular motion: A variation on this experimenthung is on even a more rotating instructive. sprocket A wheel long (Fig. chain is can be closely approximated by a(the circular radius segment with of a curvature, variable radius seeF Sects. the center of the circular orbit.tude of From Eq. ( At a high rotational frequency ofthe chain the by flywheel, tapping one it can from the thenbut side. rather push It rolls off does like not a sink stiffover slackly circular together, obstacles ring across along the the table.a good way. It example even of jumps In “dynamic stability”. this form, the experiment offers demonstrating dynamic stab in Fig. In a preliminary experiment, a tigh a flywheel (Fig. attached to one another, they would flyfrom off a tangentially like grinding the wheel, sparks once the flywheel wasLinked set in together motion. as asense, namely chain, by however, an theyduces expansion forces of all the behave chain. in This the deformation same pro- Figure 4.9 Figure 4.8 position of the disk read off a scale with theof aid a pointer. with a nonlinear force law. This figure also illustrates the scheme of a rotational frequency meter or tachometer. Each frequency corresponds to a particular value of the radius ’s Equation EWTON Applications of N 4 48

Part I 4.3 Sinusoidal Oscillations: The Gravity Pendulum as a Special Case 49 Part I

Figure 4.10 The origin of the radial force in a chain ring under tension. – We can imagine that a chain is stretched around a resting circular disk; the chain consists of balls with a spacing d connected by helical springs under tension. In the drawing, only 3 balls and 2 springs are shown. The long arrows begin at the center of gravity of the middle ball and represent the forces that act on it from the two springs, F0. Vector addition yields the net force F which is directed towards the center of the circular orbit. The quantitative relation between F0 and F is found from the similarity of the isosceles triangles which have the same sides and an opening angle ˛: F=F0 D d=r.

Figure 4.11 Oval shape of a bicycle chain before being thrown off the sprocket wheel (Video 4.2) Video 4.2: “Dynamic stability of a bicycle chain” http://tiny.cc/cqqujy In order to demonstrate that “the chain loop should be able to rotate in stable equilibrium not only as a circular ring, but also in any other arbitrary shape”, its rotation is shown in slow motion in the second part of the video. It is helpful to look at the still images one after another; becomes smaller when the chain is more stretched out. Therefore, the chain loop should be able to rotate in stable equilibrium not only as a circular one can then see how a bulge ring, but also in any other arbitrary shape! The results of the experiment in the chain caused by col- verify this expectation. A bicycle chain can be conveniently used for this liding with an obstacle is experiment. At a sufﬁciently high rotation frequency, it is thrown off the maintained during continued sprocket wheel. rotation. In earlier times, this experiment was sometimes demonstrated involuntarily For other experiments involv- when a drive chain was accidentally thrown off its pulleys in a factory, etc. ing chains, see e.g. https:// en.wikipedia.org/wiki/Self- siphoning_beads. 4.3 Sinusoidal Oscillations: The Gravity Pendulum as a Special Case

In Chap. 2, we limited our considerations of kinematics to the simplest orbits, and likewise for the dynamics treated in the present chapter: We have treated only motion in a straight line and circular orbits. , D T ;it t (4.9) is its D (4.11) (4.10) ,then ). Os- t t 0 ,andin x ). phase or oscilla- 4.12 -fold of the can be rep- 2.5 2 t which can be graphically as . The signifi- ,butalsothe ,and t d or the time .–Intheos- x 2 t = ! ; ˛ / : x sinusoidal (the displacement of the oscillation. period 2 . We start by de- =! Á 4.12 d a D C 2 (cf. Sect. t : t . ˛ D ,i.e.the2 or simply the ! C .! is the . a t 4.12 ! T sin 2 sin cshows amplitude 0 deflection x x 0 D D can readily be seen: at 2 x point masses sin T 4.12 ! phase angle to time, one can plot ! 0 N D C4.4 x = = D t ˛ ! 2 phase angle t ). For arbitrary phase angles, we have curve of the oscillation ! D D sin is convenient when one wishes to distin- circular frequency and t oscillations occur within a time 1.5 ! sin ! 0 A x 0 N , while Fig. x harmonic oscillator 2 cos D . , but instead the and the accelerations t will deal with linear pendulum oscillations and =! 0 ! is often called the x ˛ t x u :If 0 . Thus, d is called the t ! x = 4.9 frequency (cf. Sect. 1.8 x D ! – ˛ is called the D d x 2 D 2 t t x 2 , the symbol ! 4.3 D 0 d d d d . T bshows x is the D u D D ı = , whose meaning can be read off Fig. 2 u a 4.12 ˛ 360 D t . Both methods are illustrated in Fig. t D D is the deflection (momentary value) at the time = Figure functions of the time maximum value. x resented by sine curves.obtains By differentiating one and two times, one some motions around abodies central point. to a We treat good the approximation massive as moving scribing these motions kinematically andproduced by then forces discuss (dynamics). how they are The simplest of alla periodically-repeated straight-line motions path takes (linearits place oscillation). time along evolution A yields graphic representation the of guish several different amplitudes by using indices.) In sinusoidal oscillations, not only the deflections velocities ˛ For linear motion, therea is circular orbit only with a constant rotational linearation. velocity, (path) only Sects. a acceleration, radial and acceler- for described mathematically by a single sine function (Fig. ˛ (Instead of N increases proportionally to the time, so that cance of the proportionality factor cillation curve, tion time of theangle motion. – The abscissa of the sine function is an cillations of this type, and their curves, are termed Owing to the proportionality ofthe the oscillation curve either against the phase angle ˛=! The ordinate of theangular function graph sin of an oscillation curve is not simply the of the massive body of theproportional to pendulum from a its sine rest function, position), i.e. which is We recall Sect. general one speaks of a frequency ’s Equation direc- EWTON .The x velocity, force component of direction’. and x x Applications of N 4 sign however must be taken into account; a minus sign means ‘directed opposite to the positive tion occurs in the equations in this section, it suffices to use only the the vector quantities acceleration C4.4. Since only the 50

Part I 4.3 Sinusoidal Oscillations: The Gravity Pendulum as a Special Case 51 Part I

Figure 4.12 The time evolution of the deﬂection (position of the massive body of the pendulum), the velocity, and the acceleration for a sinusoidal oscillation. The period T is in general the time between two identical deﬂec- tions.

The sinusoidal oscillation curve of the velocity precedes the curve of the deﬂection by a “phase shift” of =2 D 90ı; i.e. the positive, upwardly-directed values begin one-fourth of a period (T=4) earlier than those of x. At the time t D 0, t D T=2, t D T etc., the oscillating body passes through its rest position (zero deﬂection). Then in Eq. (4.10), the sine function D 1, and the velocity has its maximum value u0 D !x0: (4.12)

The sinusoidal oscillation curve of the acceleration has a phase shift of D 180ı relative to the deﬂection x. This means, in words: the direction of the acceleration is at every moment opposite to the direction of the deﬂection. As a result, Eqns. (4.9)and(4.11) can be combined to yield a D!2x : (4.13)

This concludes the kinematic description. In order to treat the dy- namical explanation of a sinusoidal oscillation, we have to consider also the equation of motion a D F=m; then we obtain

F Dm!2x lin- (4.16) (4.14) (4.15) a). In the is held be- constant of m C4.6 4.14 are among the D ) belongs among the ndulum” and 4.16 m : ; : D m 2 : A body of mass ! Dx r ). m p can take many forms. A massive 4.13 dal oscillation, one requires a 1 s (comparing wooden and iron balls m 2 D , we find the frequency D , the proportionality factor between the D F D D 2 /mass D D ! for different values of . ) with Implementation of a “linearly polarized” sinusoidal oscillation . 4.14 . The weight, which is constant in magnitude and direction ). There, we found that the frequency is independent of the D C4.5 4.2 force law. The force which accelerates the massive body must of the frequency most popular and usefulcourses. exercises in – beginning The practical experimental setu object laboratory can simply be hungrest position, from the a quotient of helical weight/spring extension spring yieldsconstant (Fig. the spring most important formulas in physics. For that reason, measurements This equation is notmotion new to on us. circular We(Sect. orbits have in already encountered the itorbital special for radius; case here, of itlations. a is independent The linear of frequency force the isquotient law amplitude determined (force constant in of both the cases oscil- uniquely byEven with the qualitative experiment of the same diameter), oneof can the see oscillating the body decisive influence onlation of the period). the frequency The mass (or effect on ofcreasing its a the larger inverse, spring mass the constant, can oscil- etc. be Equation compensated ( by in- be proportional todirected the magnitude of the deflection and oppositely In words: Toear produce a sinusoi with a simple elastic oscillator or “ball and spring pe Figure 4.13 A linear force law can bemake obtained in use various of ways. the The force simplestof is due to a to spring the (i.e. deformation an (tension,the compression) “elastic” arrangement force). sketched in Thus, Fig. one arrives for example at or, with the abbreviation tween two helical springs. force of the springs and thewe deflection, have is the already “spring encountered. constant” which Inthe restoring general, force it is called the From Eq. ( ’s Equation , ma EWTON )). )), with D , into the F 4.9 4.16 Dx (Eq. ( D (Eq. ( t D ! D m F q sin 1 0 2 x Applications of N D D 4 x we obtain as the solution of the resulting differential equation a sinusoidal motion, equation of motion equation of motion: Insert- ing the linear elastic force of a spring, sketched here can be reversed and then leads to theing follow- direct application of the C4.5. The line of reasoning track; the air cushion would then take up the weight ofrider. the downwards, in a horizontal arrangement we could replace the ball by africtionless nearly rider on an air- C4.6. In order to compensate the weight of the pendulum body (ball), which is directed the frequency 52

Part I 4.3 Sinusoidal Oscillations: The Gravity Pendulum as a Special Case 53

Figure 4.14 a A vertically oscillating spring pendulum for testing

Eq. (4.16), b a gravity pendulum. Part I Its angular displacement ˛ is negative in this ﬁgure

and is thus simply an additional constant “background” force, has no inﬂuence on the frequency.

The linear force law is only a special case. Nevertheless, it is of great importance; for in every oscillatory system, no matter how complicated, one can replace the true force law by a linear force law. In that case, the motion must be limited to sufﬁciently small amplitudes (at very small amplitudes, every force law is approximately linear). Mathematically, this means that every force law F Df .x/ can be expanded as a series:

2 f .x/ D D0 C D1x C D2x C :::

The constant D0 must be zero, since the force must vanish for x D 0. With sufﬁciently small values of x (small amplitudes), the series can be terminated after the ﬁrst term, so that one obtains F DD1x.

Another example of this type of oscillator is the well-known gravity pendulum (also known as a simple pendulum). At small amplitudes, the construction sketched in Fig. 4.14b applies. It illustrates how the force acting on the pendulum ball, its weight FG, can be decomposed into two components. One of them, FT D FG cos ˛, serves only to maintain tension on the cord. The other, FR D FG sin ˛ (the “restoring force”), accelerates the ball along its (circular) orbit. At small angular displacements ˛, the orbit can be approximated as linear; furthermore, one can set sin ˛ x=l. Then for displacement angles of less than 4:5ı, the relative errors of these approximations are less than 103. (Note that ˛ is negative in the figure. The restoring force FR is always opposed to the momentary displacement x of the pendulum). We thus find FR DFG sin ˛ FGx=l.Thatis, . ı T or 15 mg a) or D (4.17) D 0 . ,which ˛ G 4.15 C4.7 F . The pro- x 8s : 6 ). The necessary T attained within the , and the accelera- of motion makes it maydifferindirec- 2.4 by 0.19 %, for G u of a pendulum very ı F u T ). The path acceleration 10 , but the path of motion 1 of the restoring force. – 4mand is small. One finds values of : : D and (Sect. radius of curvature l T 0 g D 4.16 11 u acceleration was present. In ˛ c). Then both path and radial s D ) yields l e gravity pendulum accordingly into the general equation for the is the (Fig. 2 path 4.15 mathematical gravity pendulum of the orbit. Its magnitude, from g D % 4.16 , either increasing it (Fig. a D tions of a spring pendulum, the ac- u are related by the equation T on the period periodic repetition (Fig. 2 0 D ˛ ˛ ) is suitable for the task of obtaining reli- of the orbital velocity. The radial acceler- no longer constant . Here, 1 is the constant by 0.76 %. b). Only a l curvature =% ı 81 m/s = 4.17 . Inserting 2 : l G 20 u 9 = F 4.15 acceleration of gravity D D D mg 0 006 s, i.e. half of an oscillation cycle in 1 s. This is the so- : remained always in the direction of the velocity at the ˛ % g 2 ) is important for metrology (the science and technol- magnitude total acceleration t a D of the pendulum ball, its weight D D T m 4.16 which are too large by 0.048 %, for ), is produces the ı 5 % 1m; a D 2.12 D 0 called “seconds pendulum”. Thelecture longest hall gravity in Göttingen pendulum has in a the length physics of Numerical example l ˛ changes the The influence of the amplitude precondition is the besting possible from a approach “massless” to cord a (i.e. a “point mass” hang- simple pendulum). tion of gravity Therefore, for beginning of the time interval, decreasing it (Fig. time interval d able values of the the general case, however, the vectors d by 0.43 % and for celeration was seen to be 1 4.4 Motions Around a CentralIn Point the case of sinusoidal oscilla Frequency and period of theof gravity the pendulum mass are of thus the independent pendulum ball. Th the restoring force is proportional to the displacement occupies a special position. Itshould should not be treated be as the a firstoscillations. special example case introduced and in the study of sinusoidal Equation ( ogy of measurements). The portionality factor precisely. Thus, Eq. ( was still a straight line. The additional velocity d possible to determine the oscillation period ation a frequency of oscillatory motion ( The mass Eq. ( tionbyanarbitraryangle (transverse) accelerations are present at theponents same of time. the Both are com- ’s Equation , , and EWTON 7.21 7.7 ). OUCAULT ’s pendulum” ,asinFig. Applications of N OUCAULT 4 “F http://tiny.cc/luqujy Video 7.5 pendulum, which hung from the roof ridgebeam above the hall (Sect. C4.7. The longest gravity pendulum used in the old physics lecture hall in Göt- tingen was the F 54

Part I 4.4 Motions Around a Central Point 55

Figure 4.15 The deﬁnition of the total acceleration Part I

Figure 4.16 Decomposition of a central acceleration ag into two components a and a%

points from the momentary center of curvature. The latter is the center point of the circle which most closely approximates the short segment of the orbit that one is considering at a particular time. From the enormous variety of possible motions (just think of the possible motions of our ﬁngers, arms and legs!), we choose a single group for closer examination, that of the motions around a central point. A central motion is the motion of a body (a “point mass”) on an arbitrary planar orbit in which an acceleration of varying magnitude and direction always points towards a single point, the center of acceleration. The line connecting the moving body with the center of acceleration is called the “radius vector”. From this deﬁnition, motion around a circular orbit as well as linearly polarized pendulum oscillations are both limiting cases of central motion. In the case of a circular orbit with constant rotational frequency, there is no orbital acceleration; in the case of linearly-polarized pendulum oscillations, there is no radial acceleration. For a generalized central motion, two simple propositions hold. First: The motion is played out in a plane. Second: The radius vector sweeps out equal areas in equal times (the “area law”). – Both propositions belong to kinematics. They are geometric consequences of the premise that the acceleration is arbitrary but always directed towards the same central point.

The area law can be seen from Fig. 4.17. This ﬁgure is based on Fig. 2.12. Three curved segments of the orbit of a central motion, corresponding to three successive time intervals, are approximated by the three arrows xa, ac,andce. The central acceleration increases in going from left to right. The thin arrows ab and cd show the continuation of the motion from the previous time interval with the same velocity along the tangent to the orbit. ). Oe , 4.7 ; Oc cd , D Oa . The angular ac orbits. We can r elliptical . e 2 0 of the ellipse (Sect. (“polarized” refers to the c r : t = paths traversed in the same time D . All the arrows are in the plane of since the altitudes of the triangles are cd since by construction area law. O of the ellipse, at the intersection of ; ; The center of acceleration of the orbit- focal points accelerated Oce Ocd Oac midpoint are the 0 D D D cc and 0 Ocd Oac Oce towards the central point elliptical orbits. aa t Thearealaw ’s increases proportionally to 1 8 ˆ ˆ ˆ < ˆ ˆ ˆ : Polarized Oscillations ! D Elliptically-polarized oscillations EPLER intervals Area The arrows the page, so that the orbits remain planar. The radius vectors etc. sweep out equal areas in equal times shape of the ellipse is determined by the ratio of the amplitudes of the ing body is at one of the two In this section, weically, treat they elliptically-polarized are oscillations. produced Kinemat- linearly-polarized, by sinusoidal the oscillations of superposition the of same frequency. two The perpendicular velocity Central motions need not followa closed spiral orbits; orbit. think But fortions one example is group of particularly of important: closed these orbitsdistinguish are among two the the cases: central mo- 1. its two principal axes. 2. K Demonstration experiment: The cordwhich of is a moving on catapulttube a holding in circular a the orbit stone left passeson through hand the a of cord short, the to smooth shorten experimenter. the The length of right the hand radius is vector pulling 4.5 Elliptical Orbits and Elliptically Figure 4.17 “shape” of thebiting oscillations). body lies The at center the of acceleration of the or- ’s Equation EWTON Applications of N 4 56

Part I 4.6 LISSAJOUS Orbits 57

Figure 4.18 The envelope of elliptical os- Video 4.3: cillation orbits for equal amplitudes of the “Circular oscillations”

two perpendicular component oscillations Part I http://tiny.cc/1pqujy (Video 4.3) With a simple experiment which is described in detail in Vol. 2, Chap. 9, last section, the elliptical orbits in Figs. 4.18 and 4.19 can be demonstrated. Two leaf Figure 4.19 The envelope of springs mounted perpendic- elliptical oscillation orbits for ular to each other carry disks unequal amplitudes of the two with slits which can allow perpendicular component oscilla- a light beam to pass in or- tions (Video 4.3) der to follow their motions by projection onto a screen. After demonstrating the excitation of the individual two component oscillations and by their phase difference '.The oscillations, both springs are phase difference plays the more important role. excited simultaneously but with varying phase differ- The set of all possible orbits has a square envelope (Fig. 4.18). ences. When the two amplitudes are unequal, it degenerates into a rectangle (Fig. 4.19)C4.8. C4.8. For the experimental demonstration of elliptical We summarize: For the kinematic demonstration of an elliptically- oscillations, POHL describes polarized oscillation of arbitrary shape, two perpendicular linearly- some impressive mechanical polarized sinusoidal oscillations of the same frequency but with ad- arrangements. For reasons of justable phase difference are sufficient. When their phase difference space, they are not included is 0ı or 180ı, the ellipse degenerates into a line. At a phase difference in this edition. However, of 90ı or 270ı, a circularly polarized oscillation, i.e. a circular orbit, we mention the convenient can result. In that case, the two individual amplitudes must be equal. possibilities of an electrical demonstration using frequency generators and an 4.6 LISSAJOUS Orbits oscilloscope. A simple mechanical experiment is shown in Video 4.3. The results and the methods of the preceding section can also be applied without difficulties to treat the most general case of elastic oscillations. We limit ourselves here to giving a summary overview. When the frequency difference between the two component oscillations becomes large, the change in phase difference is already noticeable after a single orbital passage. The ellipse becomes deformed. We see the characteristic picture of a planar LISSAJOUS figure. Fig- ures 4.20 and 4.21 show several examples of LISSAJOUS orbits. Their shape depends on two factors: 1. the ratio of the frequencies of the two component oscillations; 2. the phase difference of the two component oscillations at their rest point at the beginning of the experiment. For orbits is shown in ’s model of the atom. ISSAJOUS C4.9 OHR re- R ISSAJOUS , 1822–1880) illations. The vertical oscillation has the higher illations. The vertical oscillation has the higher figures for a frequency ratio of 3:2 for the two per- figures for a frequency ratio of 2:1 for the two per- ’s Elliptical Orbits and the image sequence is the well-known one shown in ISSAJOUS ISSAJOUS ISSAJOUS L L The production of L EPLER . The frequencies of the horizontal and the vertical com- Law of Gravity . ’s elliptical orbits have played a fundamental role twice in ISSAJOUS (Video 4.4) 4.22 4.21 EPLER Fig. ponent oscillations areobserve in them the after ratio giving a–TheL of horizontal about or a 2:3. vertical excitation One ‘kick’. can readily demonstration experiments, they are truly tortuous. They cannot Fig. chanics, and a second time in B the history of physics: Once in the development of celestial me- pendicular component osc frequency. pendicular component osc frequency. (J.A. L A mechanical method of producing L 4.7 K K Figure 4.21 Figure 4.22 orbits using the bending oscillations of a rod of rectangular crosswide, section 3 mm (2 mm high). Theflects small the mirror image of aonto pointlike light the source screen which isthe perpendicular rod. to Figure 4.20 ’s Equation EWTON figures , we can illations are 4.21 figures” ISSAJOUS Applications of N ISSAJOUS , pp. 93 and 814 (1855). 4 of roughly 3:2 to thathorizontal of oscillation. the see that the frequency ofvertical the oscillation has a ratio Video 4.4: “L shown in Fig. tortuous”. “For demonstration experiments, they are truly de l’Académie des Sciences 41 C4.9. J.A. Lissajous, Comptes Rendus des Séances shown individually. Then both are excited simultaneously. From the comparison with the L First, the horizontally and the vertically polarized, damped component osc http://tiny.cc/6pqujy 58

Part I 4.7 KEPLER’s Elliptical Orbits and the Law of Gravity 59 be demonstrated with simple and quickly understandable means (cf. Video 4.5). Video 4.5:

“KEPLER’s elliptical or- Part I In a KEPLER orbit, the center of acceleration is located at one of the bits” two focal points of the ellipse. A KEPLER orbit results when the http://tiny.cc/tpqujy object has an initial velocity which does not point towards the center. On a planar horizontal alu- The acceleration at every point is inversely proportional to the square minum plate, an object can of the distance (the length of the radius vector r), so that move with minimal friction. const It is filled with liquid nitro- a D : (4.18) gen, which evaporates and r2 escapes as gas through a hole The derivation of this equation can be found in every textbook on in the bottom of the object, theoretical physics. so that it can glide suspended on the gas film. A cord at- How can we implement the acceleration required by Eq. (4.18) phys- tached to the object leads at ically? The answer was first found on the basis of astronomical a distance r to a small elec- observations, by ISAAC NEWTON. tric motor, which produces The moon orbits around the earth. Its orbit is very nearly circular. Its a constant torque M.The radius is – take note of this number – equal to 60 earth radii. Kine- cord winds around a spiral matically, we have already described the moon’s orbit in Sect. 2.5: spindle of variable radius The moon has an orbital velocity of 1 km/s and experiences a radial on the motor shaft, so that 2 3 2 the force F transmitted by acceleration of ar D 2:7 mm/s D 2:7 10 m/s . Therefore, the ratio is the cord depends on the momentary radius ,whichis : 2 adjusted so that r2,and Acceleration of gravity g 9 8m/s 2 D D 3600 D 60 : = = 2 Moon’s radial acceleration 2:7 103 m/s2 thus also F D M 1 r . After a catapult launch, the From these facts, NEWTON drew the conclusion that the same force object therefore follows an acts on the moon as on every stone near the earth’s surface. This elliptical KEPLER orbit. force is directed towards the center of the earth and is called ‘weight’. However, the weight of a body is, counter to all our everyday preju- dices, not a constant force acting on a body. Instead, it varies depending on the distance r of the body from the center of the earth, and is proportional to r2. – Therefore, NEWTON set the weight of the moon not equal to F D mg, but rather to m F D const : (4.19) r2 And now the final conclusion was nearly inescapable: If the earth attracts the moon, then the converse must also be true; the moon must attract the earth. For an observer on the moon (a different frame of reference!), the earth has a weight. An observer supposed to be on the sun could apply the law of ‘actio D reactio’ (another change of frame of reference!). For this observer, both forces or weights must be identical but oppositely directed. Then in general, in place of the weight, we must consider the mutual attraction of two bodies by the force mM F D G : (4.20) r2 (4.21) shows the . 4.24 must be large r as dark circles into is the distance between of the smaller ball and 4.23 r a . The proportionality factor : from the equation 2 r M G G on. As the “earth”, a large lead ball D We employ a symmetrical arrangement a . law of gravity In the case of homogeneous spheres, this law 2 t universal gravitational constant . = ) made of some arbitrary material. The large s , several kg), and as the “moon” or a “stone”, . For bodies of arbitrary shape, r 2 m C4.10 ENRY M D with roughly 1600-fold linear magnification. We a s ’s famous in the law of gravity cannot be determined from as- G are the masses of the two bodies and Measurement of the , chemist, 1798) ). The two small balls are attached to the ends of a rod M EWTON and 4.23 m their centers of gravity ( holds for all valuescompared of to the dimensions of the bodies.) in this law is called the AVENDISH along a distance observe it for aroundacceleration from a minute with a stopwatch and compute the their final position (shownthis roating as the shaded large balls, circles). themove. small A balls mirror are – and accelerated a Immediately and long begin light after to pointer allow us to follow this motion G Carrying out the experiment: (Fig. (dumbbell) which is hungtorsion at band, its rotation center axis), using allowingsilhouette it a of to fine a rotate. metal well-tested Fig. band apparatuslarge (the (a balls. torsion To balance) carry withoutfrom out their the initial the position measurement, as indicated we in rotate Fig. the large balls Figure 4.23 C 4.8 The Gravitational Constant The constant tronomical observations. ItThe must principle be of the measured measurement in ismodel as the of follows: the laboratory. We construct astronomical a situati – small is employed (of mass a smaller ball (of mass ball is fixed, andpossible. the smaller one One is measurescomputes allowed the the to gravitational move constant acceleration freely as far as This is N gravitational constant (H ’s Equation EWTON . 5.6 Applications of N 4 of the center of gravity, see Sect. C4.10. For the definition 60

Part I 4.8 The Gravitational Constant 61 Part I

Figure 4.24 Silhouette of a torsion balance. The large balls have been removed from their movable support beam h. The dumbbell rod holding the small balls is held in a metal block closed at front and back by glass plates (good heat exchange). The screws s are used to arrest the torsional motion; they press four semicircular metal disks against the small balls. The torsion band has been highlighted in the figure so that it can be seen clearly. The mirror is mounted at one end of the band. The oscillation period T of the torsion balance is around 9 minutes. The lower part of the figure on the right shows the damped ringing-down of the oscillations. The upper-right inset of the figure illustrates the measurement of the acceleration a. The position of the light pointer after each 15 s was photographed at 108-fold magnification.

At the start of the experiment (immediately after rotating the large balls), the distance r of the centers of the balls and the twisting of the torsion band are practically unchanged and therefore, the acceleration is nearly constant. However, the torsion band was already twisted up to a maximum value at the beginning of the experiment: In the rest position, the attractive forces between the balls were in equilibrium with the torsional force due to the twisting of the band. As a result, after rotating the large balls, the resulting acceleration a of the small balls is exactly twice as large is if we had brought the large balls from far away in to the distance r.

Numerical Example M D 1:5 kg; r D 4:75 cm; the length of the dumbbell rod is l D 10 cm. The light pointer has a length of L D 40 m, giving a linear magnification of the distance of V D 2L=.l=2/. The factor 2 is justified by the following argument: A rotation of the mirror by the angle ˛ rotates the reflected light beam by the angle 2˛. On the screen, the measured acceleration of the light pointer is a D 1:28 102 cm/s2. From this we obtain G D 6 1011 m3/(kg s2 ). G D into 3 yield m (4.22) kg G 21 24 kg. 10 10 .Asare- 24 1 6 C4.11 : 3 1 10 m 21 D 6 10 D 2 1 2 : : m s 2 2 . Therefore, in the inte- 3 3 / 3 6 m m kg s 1 10 (planetary ‘years’) are in the 11 T 4 kg : 6 10 11 . 2 10 6738 m from the center of the earth. At the : 6 67 6 81 m s : : . 6 9 10 3 D (1571–1630) in the form of three laws. These D G 4 . Inserting these two values together with : 2 6 M 5g/cm ’s law of gravity not only correctly predicts the mo- : D 5 EPLER 81 m/s D K : 9 3 ’s laws” state that: ), we obtain Mechanics EWTON D g 6400 km 4.21 Earth’s mass EPLER D OHANNES The volume of the earth amounts to roughly 1 The experimental determinationrepresented of a great theearth step gravitational could forward: be constant using computed.r its – value, The the earth’s mass surface of is the at a distance of a value of Eq. ( sult, the average density of the earth is found to be 5500 kg/m earth’s surface, the accelerationvalue due to the force of gravity has the rior of the earth, weIndications are have that to the assume earth’s core materials has with a a high higher iron content. density. “K 1. Each planet moves on awith plane the around sun the at sun. one Its of orbit the is two2. an focal ellipse points. The radiusorbital vector plane in of equal each times. planet sweeps3. out The equal squares areas of the on orbital the periods ratios of the cubes of the semimajor axes of their orbits. tion of the moon; it iscelestial the dominant mechanics, principle throughout which the describes wholecomets, of the binary motions stars, of etc. the planets, the The observations of the motionsJ of the planets were summarized by This is of coursethe only earth’s crust a is mean on value. the average The 2.5 density g/cm of the stones in 4.9 The Law of Gravity and Celestial The discovery of a generalies attractive force is between rightly all massive countedmind. bod- among N the great achievements of the human Precision measurements of the gravitational constant ’s Equation , ). EWTON bodies (see balance, but AVENDISH https://royalsociety.org/ http://en.wikipedia.org/ Applications of N 4 events/2014/gravitation/ and wiki/Gravitational_constant# History_of_measurement to some extent also in experiments with falling e.g. C4.11. Compared with other physical constants, the universal gravitational constant is the least precisely known. Only four places after the decimal point are considered to be certain, mostly measured using the principle of the C 62

Part I 4.9 The Law of Gravity and Celestial Mechanics 63

The deviations of the elliptical orbits from circular orbits for the eight planets in the Solar System are rather small. If one for example draws the orbit

of Mars on a sheet of paper, giving its semimajor axis a length of 20 cm, the Part I deviation of the orbital shape from a circle is everywhere less than 1 mm. Given these numbers, we can appreciate KEPLER’s achievement in verifying the elliptical orbit of this planet.

The three laws formulated by his great predecessor could be explained in a uniﬁed way by NEWTON using his law of gravity2: 1. Every elliptical orbit requires a central (centripetal) acceleration. In the elliptical orbits observed by KEPLER, one focus was distinguished over the other. According to Sect. 4.7, the accelerations should be proportional to 1=r2. This is precisely the case from Eq. (4.20) for mutually attractive (gravitational) forces. 2. KEPLER’s second law is just the law of areas, which holds for every central motion (cf. Sect. 4.4). 3. KEPLER’s third law likewise follows from Eq. (4.20). This can be seen by considering a special case; we allow the KEPLER ellipse to become a circle. For a circular orbit, we have (cf. Eq. (4.2)):

42mr F D42m2r D : (4.23) T2 For the magnitude of F, we insert the value obtained from the law of gravity (Eq. (4.20)). Then we obtain

m 42mr const D ; T2 D const r3 : (4.24) r2 T2

In contrast to the planets, comets often have extremely long elliptical orbits. The semimajor axis of the ellipse can be as much as 100 times its semiminor axis. But even for the general case of these arbitrarily elongated elliptical orbits, KEPLER’s third law can be derived as aresultofNEWTON’s law of gravity. That, to be sure, requires more extensive computations. A simple example to conclude this section can serve to imprint the most important aspects of celestial mechanics in the readers’ memo- ries: We imagine a projectile to be ﬁred in a horizontal direction near the earth’s surface. The atmosphere (and with it the resistance of the air) are assumed not to exist. How great must the velocity u of the projectile be so that it becomes a little moon, circling the earth at a constant distance from the surface? A circular orbit with the orbital velocity u requires a radial acceleration of a D u2=r according to Eq. (2.12). This radial acceleration

2 KEPLER himself never progressed beyond qualitative efforts at explaining his laws. For example, in 1605, he wrote, “If we were to suppose that the earth is at rest in some particular place, and were to set another, larger earth nearby, it would be attracted by the earth just as stones are attracted to its surface”. . to 4.26 , right- closer 8km/s,the 4.25 > u . towards the cen- 3 11.2 km/s, known 2 of the earth, about > r : ; 81 m/s : m 9 6 8km/s D 2 10 u g from the cannon (the attraction . For velocities D 4 : 6 on the left: the projectile circles the 4.25 , the more elongated is the elliptical D u further 2 4.25 s m 8000 m/s 8 : D 9 u , the elliptical orbit degenerates into a hyper- 8 km/s, there is also an ellipse, cf. Fig. < u The parabolic trajectory of an Elliptical orbits m. We thus obtain 6 escape velocity 10 4 For the sun, the corresponding escape velocity is 618 km/s, and for earth’s moon, : projectile will orbit theThe earth center like of a the earth planet is or at a that focus comet, of on the an ellipse which ellipse. is earth close to theby surface artificial like earth a little satellitesof around satellite. which 400 km, are This moving placed in caseradius. a is in direction realized orbit perpendicular to at the earth’s anIf altitude this initial velocity istype exceeded, shown we in obtain the an center elliptical of orbit Fig. of the as the the cannon that fired the projectile. For velocities At a muzzle velocity of 8have km/s the in case a shown horizontal in direction, Fig. we therefore For velocities bola. The projectile leaves the earth, never to return ter of the earth.the earth On from the its6 other center hand, is the equal distance to of the the radius surface of for different initial velocities around the center of the earth hand side. But only that partbe of realized it which in is not practice. dottedfocus in In of the figure the this can ellipse case,of which the the is center earth’s of gravity acts thebody as earth around its though is midpoint, the at while earth maintaining the had the same shrunkThe mass). to lower a the small initial velocity 3 it is 2.3 km/s. Figure 4.26 Figure 4.25 is provided by thegives force the of projectile an the acceleration weight of of the projectile. The weight object thrown horizontally orbit. Finally, one arrives at the limiting case shown in Fig. ’s Equation EWTON Applications of N 4 64

Part I Exercises 65

The center of acceleration, the midpoint of the earth, appears to be practically infinitely distant. The radius vectors which point there are nearly parallel. One finds that the portion of the elliptical Part I orbit which remains above the earth’s surface is to a good approximation a parabola. It is the well-known parabolic trajectory of a horizontally-thrown projectile. – These considerations are useful, although the air resistance in reality makes their practical verification impossible. Even at normal velocities of several 100 m/s, the braking action of air resistance is considerable. The parabola can be seen only as a very rough approximation to the true trajectory, the so-called ballistic curve.

Exercises

4.1 Find the amplitudes of the velocity u0 and the acceleration a0 of a body that is undergoing sinusoidal oscillations with a maximum displacement (displacement amplitude) of x0 D 10 cm and a period of T D 1 s. (Sect. 4.3)

4.2 The acceleration of gravity on the moon is 1/10 of its value g on the earth. How long is the period TM of a simple pendulum on the moon, if its period on the earth is TE D 1 s? What is the mass mM of the moon, given its radius rM? (Sects. 4.3 and 4.7)

4.3 A stone of mass m attached to a rope of length l is being whirled around on a circular horizontal orbit at the angular velocity !. How does its angular velocity change if the length of the rope is suddenly decreased by l1? Treat the stone as a “point mass”. (Sects. 4.4 and 6.6)

4.4 The heat content of anthracite coal is about 3 104 kJ/kg (cf. Sect. 19.8). At what depth z under the earth’s surface could the coal lie if the work of raising it to the surface were just equal to its heat content? (See Sect. 13.3). (Sects. 4.8 and 5.2)

(In order to solve this exercise, one needs to know that the weight FG of an object of mass m decreases in the interior of the earth! At the center of the earth, its weight would be zero. It then increases linearly – assuming that the matter of the earth has a uniform, constant density – with increasing distance r from the center, ﬁnally reaching the value mg at the surface of the earth (r D RE). As an equation: FG D mg.r=RE/,whereRE D 6378 km.)

4.5 In Göttingen, there is a “Planetary Way”: A sphere representing the sun is in the Goetheallee, then towards the center of town follow Mercury, Venus, Earth, etc.; all the linear dimensions are reduced by a factor of 2 109 compared to the real distances. If this ) 4.9 The online version of this chapter (doi: m.) (Sect. 9 10 150 D R see also Exercise 2.5. 4.3 For Sect. Electronic supplementary material 10.1007/978-3-319-40046-4_4) contains supplementary material,able which to is authorized users. avail- model were set up in anitational otherwise forces, could empty the space, planets free circle ofwould the external the sun? grav- “year” And of if the so, modelof how Earth the long be? earth (The is radius of the real orbit ’s Equation EWTON Applications of N 4 66

Part I direction. .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © section. this of end the at introduced be will Momentum and Energy, Work, Concepts: Useful Three 1 displacement’ the of direction the as the defined be in will ‘force displaced’ ‘distance product The 1. outset: the at upon agreed are things Three Power and Work 5.2 N of use Making Remarks Preliminary 5.1 2. 3. directions; tn.I hsmr eea ae ednt h opnnsi the in components the denote we con- it case, the is general of nor more displacement, direction the this to In parallel not stant. is force the general, In aengetdu onw u ahrb osdrn h qainof equation we the considering which by facts closely. rather experimental more but motion of now, basis to up the neglected on have introduced be not will ue osdrbyb sn oeadtoa,ceel formulated cleverly are re- additional, these be – some often can concepts using effort by This requires considerably effort. motion computational duced of of equation the amount on great based a description a cases, motions such motions In the the or of parts only machine think of need one complex; extremely however are = n edfietework the define we and n hudaodsyn Tefrewrs swrig.Tecnetof concept The working”. is works, force “The saying avoid should One reactio C Fx Fx en:Teforce The means: en:Teforce The means: ,oecnteteeymto uniaiey aymotions Many quantitatively. motion every treat can one ”, h oc performs F force The oki performed is work F 1 x m 1 EWTON C t ipaeetelement displacement -th F ok energy, work, ebgnwt h ocp f‘work’. of concept the with begin We 2 W . olsItouto oPhysics to Introduction Pohl’s i seuto fmto n h a f“ of law the and motion of equation ’s x F D stesum the as 2 F n h displacement the and C 1 n h displacement the and against ; ::: 1 2 work. forbde n hi extremities. their and bodies our of ; C 3 and ;:::; F h oc F force the m momentum m x work m / D . x P . by O 10.1007/978-3-319-40046-4_5 DOI , x . F x on nopposite in point i F r ntesame the in are hs concepts These 1 x ; i F 2 ;:::; times power actio F m ; working”. is works, force “The saying, avoid should One work”. Teforce “The Fperforms 5 67

Part I (5.2) (5.1) : performs work second (W s) F second (W s) 1watt D ; : 2 . The force : x watt s F d h = 6 x 2 d F 10 F Z 6 D : 3 1 Joule (J) 1kgm D W , a muscular force lifts a massive block d D D D W 5.2 with the force In Fig. : h hour (kWh) meter (N m) C5.1 very slowly shows a graphical representation of a path integral over The definition of the work The definition of lifting work 5.1 1newton 1 kilowatt Lifting work. We will calculate the work for three different1. cases: straight up, as a path integral over the force Figure 5.1 Figure 5.2 or in the limit along the path d Figure the force. With this definition ofmust the consist work, of its the units productWe are call of also a determined; unit these of force and a unit of distance. ’ are F .The x F acts. Mi- and F F x d does not change. F W R Three Useful Concepts: Work, Energy, and Momentum D is positive, i.e. work is ’. This also fixes the sign 5 x sign errors can easily occur. used, whose direction is included in their magnitudes, components of Since often only the parallel sign of forces (reaction forces) are employed instead of which the force nus signs occur only when the corresponding counter- of the work. In allples the discussed exam- in this section, W performed on the object on and ‘displacement element d vector quantities ‘force of work is given by theintegral path W over the scalar product of the C5.1. The general definition 68

Part I 5.2 Work and Power 69 Part I

Figure 5.3 Lifting work up a ramp. The work is not performed directly against the weight FG of the object, but rather only against its component in the direction parallel to the surface of the ramp, FG cos ˛.However, the displacement x is greater than the perpendicular height h;itisx D h= cos ˛. The lifting work along the whole length of the ramp is thus W DFG cos ˛ h= cos ˛ DFGh. – Similar considerations apply to any ramp, no matter what its shape and curvature, or to other lifting machines, such as a block-and-tackle.

In a very slow lifting action, the velocity of the block lifted remains practically zero. As a result, to a very good approximation, F D FG. Then we ﬁnd dW DFGdh : (5.3)

This work is performed against the weight of the block. The weight FG is practically constant for all heights h in the neighborhood of the earth’s surface. So the shaded region indicated in Fig. 5.1 is a rectangle with the area FGh. We then obtain for the lifting work up to the C5.2 height h against the force of weight FG : C5.2. Lifting work is also positive. Since FG and h lifting work DFGh: (5.4) are directed opposite to one another, the sign of the corresponding scalar product All types of lifting machines, e.g. the simple ramp in Fig. 5.3, can do must be taken into account: nothing to change the magnitude of the product F h.Itisalways G W DF h D mgh. the perpendicular height h which determines the amount of work. G

Numerical Example A person with a mass of 70 kg climbs to the top of a 7000 m(!) mountain in one day. The force of his or her muscles perform lifting work of 70 kg 9.81 m/s2 7000 m 5106 N m or around 1.5 kWh. This “day’s work” has a market value of about 2 cents!C5.3 – When jumping, one need consider C5.3. This corresponds in only the height h by which the center of gravity of the jumper is raised. today’s currency to around When a person is standing, his/her center of gravity is about 1 m above the 30 Eurocents per kWh for ground. In jumping over a crossbar at a height of 1.7 m (cf. Fig. 5.4), the center of gravity is raised to a height of around 2 m. The height lifted is private users in Germany, thus only 2 m 1mD 1 m. So the muscular force of the jumper performs and around 11 Ct. in the a lifting work of 70 kg 9.81 m/s2 1 m, or around 700 N m. U.S. (Exercise 4.4).

2. Work against a spring. In Fig. 5.5, we see an object which is held by a spring. Muscular force is used to extend the spring very slowly in the direction x. The muscular force F performs work along the displacement element dx:

dW D Fdx : (5.5) ) ) D 5.6 D (5.8) (5.7) (5.6) 4.15 :The trian- F ( 5.5 200 N through D Dx D : 2 F : Dx : : x 1 2 x . Thus we find d d D Dx D F xDx Dx is a continuation of Fig. 1 2 the elastic force of the spring. The ), we obtain D D D 5.7 D 5.6 W F W d d , the path integral of the force is the against x .Fig. Elastic work 40 N m. D ) into Eq. ( , whose area is 4m area of the triangle : 4.15 0 D Proficient The calculation of work The definition C5.4 the sum of the shaded rect- 200 N D 5 : ,and Numerical example A bow is pulled backa with distance a of muscular 0.4 force m. of 0 The muscular force has to perform work amounting to Work of acceleration W d F Inserting Eq. ( of work against a spring, or “elastic work” Figure 5.4 This work is performed 3. object has just been released by the hand; the spring relaxes back When the spring isremains stretched practically very zero. slowly, As the athe velocity result, elastic of to force a the resulting very object from good approximation, the extension of the spring is Figure 5.5 Figure 5.6 high-jumpers roll over the crossbar against aR spring (elastic work): angular areas COB Along the displacement gular region COB elastic force is given by the linear force law (Hooke’s law), (Fig. OSBURY Three Useful Concepts: Work, Energy, and Momentum 5 medal with his backwards roll – the ‘Fosbury flop’ (sneered at by the skeptics). sible. In 1968, the American high-jumper R. F earned an Olympic gold C5.4. High-jump techniques have been developed in order to keep the center of gravity of the jumper as low as pos- 70

Part I 5.3 Energy and Its Conservation 71

Figure 5.7 The deﬁnition of work of acceleration Part I to its resting position by pulling together. It thereby accelerates the object, which was previously at rest, to the left, and the elastic force FD of the spring performs work of acceleration:

dW D FDdx : (5.9) According to the equation of motion, Eq. (3.3), we have du F D m (5.10) D dt and, from the deﬁnition of the velocity,

dx D udt : (5.11)

Equations (5.9) through (5.11) together yield

dW D mudu : (5.12)

The integration (analogous to Fig. 5.6)givesthe 1 2 : Work of acceleration D 2 mu (5.13)

Numerical examples A railroad train (locomotive C 8 cars), mass D 5:1 105 kg, velocity D 20 m/s, work of acceleration 108 Nm 28 kWh; a bullet from a pistol (cf. Sect. 2.2), mass D 3.26 g, velocity D 225 m/s, work of acceleration 82 N m.

The quotient work/timeC5.5 or – equivalently – the product of force C5.5. More precisely, times velocity are called the power P.Theunit of power is the differential quotient dW=dt D WP . 1watt(W)D 1 N m/s . (5.14)

When working continuously for several hours (turning a crank, walking on a treadmill etc.), a person can deliver a steady power output of around 0.1 kilowatt (kW). For a few seconds, the power output of a human can be greater than 1 kW; for example, a person can jump up a 6 m high staircase in 3 s. The power required is 70 kg9:81 m/s2 6m=3s 1:4 kW. More will be said about this topic in Sect. 5.11.

5.3 Energy and Its Conservation R In Sect. 5.2, we deﬁned the integral Fdx and called it ‘work’. We calculated the work for three different cases and gave numerical examples. abil- is the R / / was in- 8 4 : : 5 5 . . into the in equilibrium with F ). This work which 5.7 converted of the spring. ability to perform work pot E is here the constant lever arm, while r the potential of the lifted body energy 2 ) or accelerate it (e.g. Fig. Dx 1 2 mgh 5.9 at each moment; G A spring which has been compressed or extended can lift an A body which has been raised up can per- and F 5.8 to perform work: A body lifted higher or a compressed spring lifting work elastic work object and therebyA perform continuously lifting variable leverage work, keepsthe i.e. the weight lifting without force work of acceleration. form work: it could liftto another the body same height, of the without same accelerating mass it. Figure 5.8 In each ofcreased; these or, three expressed differently, cases, workity was the can themselves perform work.(Figs. They can forhas been example converted lift into an ‘the ability object to perform work’ is called Figure 5.9 lever arm which varies during the rotation. Three Useful Concepts: Work, Energy, and Momentum 5 72

Part I 5.3 Energy and Its Conservation 73

Similarly, an object which has been accelerated has not only a velocity, but also the ability to perform work; it can for example deform another body in a collision and thereby perform elastic work. In this Part I case, the work of acceleration which has been converted into the ability to perform work is called:

work of acceleration 1 mu2fthe kinetic energy E g 2 kin (5.13) of the accelerated body:

In the above examples, the sum of the two forms of energy (potential and kinetic) is a constant quantity; thus

Epot C Ekin D const: (5.15)

This is a fundamental law of mechanics, the conservation of energy (brieﬂy: energy conservation). Explanation: In Fig. 5.7, assume that the spring relaxes through a displacement of dx. Its elastic force FD performs the work dW in the process. This work can be described in two ways: First of all, as work of acceleration which increases the kinetic energy Ekin of the object, thus dW DCdEkin : (5.16) Secondly, it can be considered to reduce the potential energy stored in the spring, so that dW DdEpot : (5.17) Equations. (5.16)and(5.17) together yield

dEpot C dEkin D 0 or Epot C Ekin D const : (5.15)

Similarly, in the case of the free fall of a body: Its weight FG performs work of acceleration dW along the displacement dhC5.6, dW D C5.6. dh is directed down- CFGdh.ThisisDCdEkin and DdEpot. Thus, here also, dEpot C wards in this case. dEkin D 0andEpot C Ekin D const. We have thus far treated energy conservation in mechanics for only two types of forces, for elastic forces and for weight (the force of gravity). Such forces are called conservative. In their case, energy Non-conservative forces: is “conserved”. Frictional and muscular forces are non-conservative. “They will be included For non-conservative forces, the mechanical energy conservation law, later by means of a bril- i.e. Eq. (5.15), does not apply. They will be included later by means liant extension of the energy of a brilliant extension of the energy conservation law. conservation law”. ) ), )and 4.12 (5.19) ( (5.21) (5.20) ). 5.10 force law. amplitude : A good ex- 11.16 is ,or 0 h x of an object performs non-linear : : . Thus, the object’s final mg kin pot dancing ball D E E const . (5.18) : mgh : G D D 0 ) consist of a periodic conver- D F trongly dependent on their am- ), together with Eq. (4.5), leads gh D x ition, all of the oscillation energy 2 2 ! h 4.3 p G 5.20 mu const const D F 1 2 D 0 . D D u D u 0 C . Deformations are called elastic when (Sect. 2 0 2 0 2 2 . The dancing ball never quite returns to its Dx mu Dx mu 1 2 1 2 1 2 1 2 heat elastic ) equal to Eq. ( , it holds that x 5.19 The energy of a sinusoidal oscillation is proportional to of the Conservation Law of Mechanical Energy . During free fall, the weight Sinusoidal oscillations Oscillations whose frequency is s At the points of reversal (maximum displacement plitudes 2. The deflection does notincreases depend strongly sinusoidally with on decreasing time; amplitudetions’, (as its which in are frequency related ‘wobble to oscilla- the dancing ball; see Sect. In words: the square of its amplitude x all of the energy is potential, and we find sion of the twodisplacement mechanical energy forms into each other. For every velocity after falling from the perpendicular height work of acceleration, has been converted to kinetic energy; here we have During passage through the rest pos thereby convert its kinetic energyconverted back into to potential kinetic energy. energy Thisies. by will relaxation of be The the object deformedThis bod- again is rises, the gaining origin potential of energy, and the so motion forth. of the ample of an oscillatory phenomenon with a With the corresponding kinetic energy, theform object may itself elastically and/or de- the surface on which it impacts (cf. Fig. they fulfill the mechanical energycan conservation be law. realized Practically, onlymechanical this energy as is a continually converted limiting intoular case. the motions, i.e. energy into of A molec- fraction of the apparent 3. The definition of previous height. Setting Eq. ( 5.4 First Applications 1. to an important equation, which we have already encountered: Three Useful Concepts: Work, Energy, and Momentum 5 74

Part I 5.5 Impulse and Momentum 75

Figure 5.10 The conservation of energy. A steel ball is dancing on a steel

plate. The latter could also be a glass Part I plate covered with soot; then one could readily discern the ﬂattening of the ball on impact. Below: The time evolution of the dancing ball’s motion. (Video 5.1) Video 5.1: “Dancing steel ball” http://tiny.cc/3qqujy The steel ball is released magnetically, so that it falls straight down without rotation. 5.5 Impulse and Momentum R The path integral of the force, that is the work F dx,ledusto a fundamentally importantR concept, namely energy. The corresponding time integral F dt of the force does the sameC5.7: It is called C5.7. In contrast to the path the impulse and leads to the concept of momentum. integralR of the force, W D Many motions occur with jerks or impacts; forces which change F dx,theR time integral of quickly in magnitude and direction are at work. Figure 5.11 is in- the force, F dt, and thus the tended to illustrate the time variation of such a force. – Starting with momentum,isavector. processes of this kind, one arrives at the concept of ‘impulse’:

Z Impulse D F dt : (5.22)

The unit of impulse2 is for example the newton second (N s). Work performed upon an object increases its energy. What is the result of an impulse? The answer is provided by applying the equation of motion. Before the impulse, let the object have a velocity u1.Dur- ing a time interval dti, the acceleration is given by ai D Fi=m. Within the time interval dti, it gives rise to an increase in the velocity of 1 du D a dt D F dt (5.23) i i i m i i

Figure 5.11 The time integral of the force, or ‘impulse’

R 2 In electrodynamics, the corresponding quantity is the currentR impulse Idt, mea- suredinampere second (A s), and the voltage impulse Udt, measured in volt second (V s). m of 5.12 (5.27) (5.26) (5.25) (5.24) t changes : d p F R , was called by : to a ﬁnal value u 0 1 m u D : t 2 : : d t u u d m F An impulse I F , and we too must adopt this that is i m t C Z d 1 Z i D u D F p which are at rest. The overall mo- D 1 M p / can be combined with the empirical D m ) means: 1 . In recent times, this useful term has i u I u 2 5.5 2 momentum d and 5.24 u p m 2 m M u D . p m Momentum D ”: Forces always occur in pairs; they act on bod- 1 u M ) becomes quantity of motion reactio The conservation of momentum. Two carts with the masses 2 D 5.24 the travel distances which are in the ratio 1:2 in a given time. Therefore, actio . m 2 EWTON u the spring. The two cartsmagnitude are but each in acted opposite on directions. by Asmentum, an a also impulse result, equal of each but the receives opposite. same a As mo- by: a formula, we can represent this ies with the same magnitudesshows but the in simplest opposite example: directions. A Figure two compressed small spring carts is of placed masses between mentum of this “system” is zero. A trigger mechanism then releases the momentum of a body from its initial value m usage. Then, in words, Eq. ( rule “ and, after integration over time, their velocities are in the same ratio, 1:2. and m been supplanted by the word The symbol used for the momentum in most textbooks is N Figure 5.12 5.6 Momentum Conservation The deﬁnitions given in Sect. or Then Eq. ( The product of mass times velocity, quantity ,andwe the . In more recent momentum EWTON Three Useful Concepts: Work, Energy, and Momentum 5 “The product of mass word too must adopt this usage”. of motion times, this useful term has been supplanted by the times velocity was called by N 76

Part I 5.7 First Applications of Momentum Conservation 77

Figure 5.13 The deﬁnition of the center of gravity S Part I

The sum of the two momenta remains zero. That is, in a descriptive generalization: Without the effects of “external” forces, the sum of all the momenta in any system of arbitrarily moving bodies remains constant. This is the law of conservation of momentum. This momentum “This momentum conserva- conservation law is no less important than the law of conservation of tionlawisnolessimportant energy. than the law of energy conservation.” Momentum conservation is often called the “law of conservation of the center of gravity”. The reason for this can be seen in Fig. 5.12. The distances travelled during a given time interval obey Ms1 D ms2. The same equation defines the center of gravity of two bodies at rest (Fig. 5.13).C5.8 C5.8. The center of gravity is defined here using momentum conservation, that is by means of the inertial mass. 5.7 First Applications of Momentum Its name however is derived from the more frequently ap- Conservation plied definition based on the gravitational mass: it then As with energy conservation, we illustrate momentum conservation corresponds to the point at by means of a few simple examples. which a body can be hung so that the sum of all the 1. Let us start with a flat cart, about 2 m long. A man is standing torques resulting from gravity on its right end (Fig. 5.14). The cart and the man form a system. is zero (i.e. the body hangs The man begins to walk towards the left; he thus gains a momentum at rest without rotating; see directed to the left. At the same time, the cart rolls to the right. Ac- Sect. 6.2.) cording to momentum conservation, it has gained a momentum of the same magnitude but opposite direction to that of the man. – The man continues walking and walks off the left end of the cart. He takes his momentum with him. The cart now rolls with a constant velocity to the right, since it has the same momentum as the man, apart from its opposite sign. 2. To verify this quantitative statement, we let the empty, rolling cart meet up with a second man who is walking to the left (Fig. 5.15). The mass and velocity of this second man are chosen to be the same as

Figure 5.14 Momentum conservation. A man accelerates himself to the left on a cart and imparts a momentum in the opposite direction to the cart. (Video 5.2) Video 5.2: “Conservation of linear momentum” http://tiny.cc/tqqujy ˛ ,the cos ı p 90 , the cart ı D 60 ˛ D ˛ 5); for : towards the long axis 0 ˛ D ı ). The cart remains standing 5.16 nge relative to its initial value of 0). D ı C5.9 : p Mo- The 5.14 reacts with only half the velocity (cos 60 cart remains at rest (cos 90 Figure 5.15 mentum conservation. The walking man has keptthesamemomen- tum during and after walking over the cart. (Video 5.2) momentum of the cart in Fig. is equal to the momentum of the man momentum of the cartzero. cannot cha 4. The flatresists cart being has shoved. rubber However, it tires. canlong easily axis. At roll Therefore, right in it the angles can directionof to be of the its used its momentum to long demonstrate axis, the it vector nature Assume the man toof be the walking cart at when anof he angle his steps momentum onto is it. along the Then cart’s long only axis. the For component those of the first man.walking. The second Immediately, man the steps carttransferred onto also from the the stops cart and man rolling. stops opposite to direction The the to momentum cart the momentum was of of the the rolling3. cart. same magnitude The but flatwalking rapidly cart with a is constant standing velocity.right He at and steps onto rest. leaves the it cart onat on From the rest. the the The left man right, (Fig. brought hisnot a total change momentum man along it comes with noticeably him and while did he was on the cart. As a result, the Figure 5.16 of momentum. Three Useful Concepts: Work, Energy, and Momentum 5 “Conservation of linear momentum” http://tiny.cc/tqqujy Video 5.2: vector nature vantage compared to experiments on an airtrack, which are frequently used today, of being able to demonstrate the C5.9. These impressive demonstrations of momentum conservation have the ad- 78

Part I 5.8 Momentum and Energy Conservation During Elastic Collisions of Objects 79 5.8 Momentum and Energy

Conservation During Elastic Part I Collisions of Objects

Figure 5.17 shows two loaded carts with spring bumpers. They each have the same mass. The left-hand cart is moving towards the other with the velocity u. When they collide, they exchange their velocities; the right-hand cart moves away with the velocity u, while the left- hand cart is now at rest. It stops at the moment when the spring bumpers are completely relaxed. – to explain this experiment, one requires both momentum conservation and energy conservation. We will illustrate this using an example where the masses are not equal. Momentum conservation requires

Left-hand cart Left-hand cart Right-hand cart

mu D mu1 C Mu2 before the collision after the collision or m.u u1/ D Mu2 : (5.28) Energy conservation requires

1 2 1 2 1 2 2 mu D 2 mu1 C 2 Mu2 or . /. / 2 : m u C u1 u u1 D Mu2 (5.29)

Equations (5.28)and(5.29) together yield Video 5.3: . /: “Elastic collisions in slow u2 D u C u1 (5.30) motion” http://tiny.cc/xqqujy The major advantage of this arrangement is that the collision takes place slowly. Forces and energy transfers can be clearly observed. The Figure 5.17 Demonstration of a slow collision. The helical springs F are attached firmly to the cart at a and to the bumper at b. When the two bumpers collision process is made meet, the springs will be extended; an elastic force to the left thus acts on the slower by increasing the left-hand cart, and an equal force to the right acts on the right-hand cart. masses of the carts by adding These two forces have the same magnitudes at every moment during the the experimenters, while extension of the springs. They brake the motion of the left-hand cart and keeping the overall mass on accelerate the right-hand cart. (Video 5.3) each side equal. ). with after .For 5.28 (5.32) (5.31) 2 u u 5.17 D : ; from Eq. ( collision. It can m m 2 M m C u 2 C M or (target cart) m M 1 u 2 can be carried out with u u u inelastic D D 0; 2 u 1 5.17 u D this proceeds in sequence through , it thus follows for the velocity after / m M D / m M of mass . (colliding cart) 1 u of mass . ), one can eliminate either hold. But momentum conservation alone allows us to for the target object becomes negative, i.e. it is directed oppositely to the . The leftmost ball is pulled out and released, so that it The demonstration of elastic collisions not 5.30 1 (Video 5.4) (the cart is ‘reflected’). u in Inelastic Collisions and the Ballistic Pendulum for the colliding object u , 5.18 m > be demonstrated by replacing the elastic bumpers in Fig. “stick together”. In thisvation does case, the mechanical law of energy conser- a collision, one speaks of a (completely) lead or some other easily deformable material, so that the two bodies M In the special case thatthe collision that The resulting velocity is Figure 5.18 velocity between two objects of thehung same by mass. two threads The balls (“bifilarcan are suspension”), move freely so from that left they tobackward). right (but not forward or 5.9 Momentum Conservation When both bodies are moving with the common velocity Using Eq. ( original leftmost ball just before the first collision. a larger numberroad of switchyard. carts. In Oneuses the can lecture a hall, occasionally series instead seecf. of of Fig. this steel carts in balls onecollides a generally which with rail- are its neighboring hung ball.and as That takes simple ball on receives pendulums; theneighbor the momentum becomes role the target, of and the colliding whole line ball of (projectile),then balls flies within while up a and its very out next short to the time. right with The nearly rightmost the ball same velocity as the The demonstration sketched in Fig. ) Three Useful Concepts: Work, Energy, and Momentum 5 Exercise 5.5 ferent masses are shown. ( http://tiny.cc/xrqujy Collisions between balls having the same and dif- Video 5.4: “Elastic collisions” 80

Part I 5.9 Momentum Conservation in Inelastic Collisions and the Ballistic Pendulum 81

Figure 5.19 A ballistic pendulum as a force meter.

Measuring the velocity of a bul- Part I let (suspension by two threads of length ca. 4.2 m, natural period of oscillation T D 4:1s). (Video 5.5) Video 5.5: “Measuring the velocity of a bullet” http://tiny.cc/qrqujy For safety, the pendulum swings along two light metal rails. Its period is 2 s, and its mass is 2.2 kg. The mass of the bullet is 2.6 g. The pen- make a prediction (since there is only one common velocity after the dulum exhibits an amplitude collision). Momentum conservation requires of around 12 cm (6 marks on the scale; compare the sil- Left-hand cart Both carts together houette). From these data, we mu D u2.m C M/ (5.33) ﬁnd the bullet’s velocity to be before the collision after the collision 320 m/s. or m u D u : (5.34) 2 M C m

The velocity u2 of the target body after the collision is thus only half as great as in an elastic collision for equal masses; cf. Eq. (5.32). This means that twice as much momentum is transferred in an elastic collision as in a completely inelastic collision. An example of an application is the measurement of the muzzle velocity of the bullet from a pistol. In Fig. 5.19, we see the bullet impacting with its velocity u onto a block of mass M, where it lodges. Bullet and block then move together to the right with their common velocity u2. This velocity is measured, and using it, the muzzle velocity u can be computed from Eq. (5.34).

The measurement of u2 can be carried out with a simple stopwatch. For that purpose, we construct a ballistic pendulum.Thismeansthat we mount the block M in such a way that it can oscillate, e.g. between two springs or as the mass of a gravity pendulum (Fig. 5.19). In either case, we guarantee a linear restoring force law: the springs or the cord of the pendulum are made sufficiently long. For a linear force law, the important equation u0 D !x0 : (4.12) holds. In words: the velocity u0,hereu2, with which a previously resting body leaves its rest position in an oscillator system is obtained in a simple manner from its amplitude x0: One need only multiply the latter by the circular frequency ! D 2=T (T is the period of oscillation of the pendulum). the line elastic not not a “loss” of me- of the pendulum. ; after one second, 2 / 0 x 55 m/s. – Explanation: .! ), we had to use a chart M 1 2 u in principle 2.2 Dissipative Forces of accelerated ndulum. They set the kinetic energy . This is demonstrated and discussed ı constant value and 180 ı “Non-central collisions”. On the glass plate of an over- : ). On the contrary, the kinetic energy of the bullet is converted on shows the velocity-time curve of a person falling from 5.4 of the bullet equal to the kinetic energy 2 5.20 mu 1 2 (Sect. impact almost completely intoenergy. heat; only about 0.16 % remains as kinetic This, however, is not permissible; the impact of the bullet is Beginners sometimes try tovelocity of apply a energy bullet with conservation the ballistic for pe measuring the Video 5.6 recorder with time marks,ter, an and electric a motor, backstop.for a With the rheostat, momentum same a conservation tachome- measurementsome at twine, we our a need disposal, balance only and a a stopwatch. cigar box full of sand, head projector, which has been dustedtwo with round fine steel lycopodium powder, disks canin glide the nearly powder without friction. arevaried, Their readily as tracks visible. welldirection). as The masses the Thus, with of velocity this thenomena, of model which disks we the play can such can projectile an demonstrate be importantnuclear (in collision role physics. in phe- magnitude atomic, molecular and and in a great height. Initially, the motion is chanical energy. “Energyversion loss” of or energy “dissipation” into describesIn heat, the all or con- the more other correctly, motions,effects; into they such were internal losses reduced energy. were to practically aperiments, negligible minimum side and by careful were design completely oftheoretical the neglected treatment ex- in of the the measurements. computations and However, many motionslosses with play continual anFig. and important unavoidable role energy in mechanics. As a first example, the velocity hashowever, reached it begins a to value increaseand much of finally more it nearly slowly reaches than a 9.8 m/s. in a vacuum, Rather soon, Inelastic collisions are an exceptionerwise among the considered, motions in we that have oth- they exhibit 5.11 Motions Against 5.10 Non-Central Collisions If two bodies approach each other along a path which is What an enormous simplification we have achievedcept by using of the con- momentum! Before (in Sect. connecting their centers oflision, gravity, there then will insteaddeviations be of between 0 a a “non-central” “central” one. col- It will produce angular ). Exercise 5.8 Three Useful Concepts: Work, Energy, and Momentum 5 Video 5.6: “Non-central collisions” http://tiny.cc/jqqujy (See also 82

Part I 5.11 Motions Against Dissipative Forces 83

Figure 5.20 The inﬂuence of air resistance on a falling person. In order

to maintain the constant terminal ve- Part I locity of around 60 m/s for a person with a mass of 70 kg, his/her falling mass has to generate a power of about 40 kW(!), provided by its potential energy.

Figure 5.21 The constant sinking velocity of balls in viscous ﬂuids

During the acceleration, a force F2 acts in the direction opposing the motion. It increases with increasing velocity, until its maximum value F2 DFG is reached, i.e. it is equal to the force of gravity FG, but opposite to it. Then the sum of the forces acting on the person, F2 C FG, is zero. As a result, no further acceleration occurs, and the velocity of the falling person has reached its constant terminal or saturation value: the person is no longer freely falling,butrather sinking with the constant terminal velocity u.

The essential point here, the increase of the velocity of falling up to a constant terminal velocity u, can be readily shown in a demonstration experiment, e.g. by letting small steel balls fall in a viscous ﬂuid (Fig. 5.21). Details are given in Sect. 10.3.

As a further example, we can consider our various means of transport: automobiles, railroads, ships and aircraft. Even on a horizontal path, a vehicle requires a driving force not only to accelerate,butalsoto maintain a constant velocity! – In Fig. 5.22, we see a cart carrying about 50 kg of mass on the horizontal ﬂoor of the lecture hall. It is being pulled via a cord with a force F1 D 9:81 N. After it has moved about 1 m, its velocity reaches a constant value of ca. 0.5 m/s, and its acceleration becomes zero. Therefore, there must be a second force acting during its acceleration, which is directed oppositely to its motion and which increases with velocity dx=dt, that is a force with a maximum value of F2 DF1. – This force may be produced in a variety of ways, for example by friction in the bearings of the cart’s wheels or by displacement and turbulence of the surrounding medium, usually air or water. ) 2 x t F . 2 (5.35) F /time W . The driv- 1 . Therefore, F must always u ound convert 1 1 F F D t along the distance = x 2 1 F F is possible due to sticking 1 D F : P W u 1 . The quotient (work x F of their velocities. requires a driving force 1 ,i.e. , the driving force u F D 2 P W kW, the engines of large ships up to more F P 3 W , and it must produce a power of u squares power 1.4 m/s on a horizontal path, one requires around 60 W; D udge the power values quoted for various technical devices. As the result of a “dissipative” or “energy consuming” force performs work against the dissipative force (resistance) , the resistance force increases proportionally to the momentary kW. The power required for the natural method of locomotion of 1 5 F 5.21 humans, walking, isspeed in of comparison 5 km/h verya modest. rapid walk at For 7 km/h a increaseswalking this to typical is ca. composed 200 walking W. of – two Theter work parts: performed of in 1. gravity A (walk periodic alongheight, a raising and wall of holding look the a at body’s piece cen- theaccelerate of our resulting chalk legs. wavy on The line!); inelastic the and impacts wall ofmost 2. at our of hip feet the this on work the energy required gr intousage. to heat, – which When is ridinggravity thus a is ‘lost’ minimized, bicycle, for and the the further up-and-down work mechanical a of motion bicycle moving of the speed the legs is of center18 also km/h, of 9 smaller. km/h, only For about one 120 consumes W.position – to a With j numbers power of of this kind, about we 30 are W, in and a at better craft engines usually several 10 than 10 Fig. velocity. In the casemation of proportionally ships to and the aircraft, it increases to a rough approxi- Examples Automobile engines deliver between 10 and 200 kW, locomotives and air- This is thetween reason force and that velocity. we In the cannot simplest case, deﬁne for a example that generally shown in valid relation be- , maintaining a constant velocity 2 perform work against the resistance force travelled; this work is equal to ing force some sort of motora or constant energy ‘cruising source speed’ must be available to maintain gives the corresponding F In the case ofbiles, humans the production and of the work driving force animals, locomotives and automo- Whatever produces the force Figure 5.22 increases with increasing velocity. Three Useful Concepts: Work, Energy, and Momentum 5 84

Part I 5.11 Motions Against Dissipative Forces 85

Figure 5.23 Production of the driving force for water-

craft (and aircraft) Part I

friction3 (Sect. 8.9). But how is the driving force for motorized air and water vehicles produced? Answer: The motor takes in some portion of the surrounding medium (water or air) with propellers, jets, paddle wheels or similar mechanisms and accelerates it backwards. This gives rise to a driving force F1 in the forward direction (momentum conservation!). As an example, Fig. 5.23 shows a boat moving at a constant speed u towards the right relative to the shoreline. The ‘motor’ is a man who uses a paddle to accelerate water backwards, giving it a velocity u0 to the left. In a time t, he accelerates an amount of water with a mass M, thereby producing a momentum Mu0 towards the left. At the same time, the man himself (and the boat in which he is sitting) receives an impulse to the right, i.e. in the direction of travel:

0 F1t D Mu : (5.36)

The force Mu0 F D (5.37) 1 t is the driving force. It is required to maintain a constant speed of the boat against the resistance of the water. The force F1 performs the work W1 D F1tuin a time t along the path tu, or, from Eq. (5.37),

0 W1 D Muu : (5.38)

At the same time, the water which was accelerated to the left receives 1 02 a kinetic energy W2 D 2 Mu . The motor has to deliver the sum of these two energies, W1 and W2, but only the fraction W1 is available as useful work (for moving the boat)C5.10. Then we find for the C5.10. This computation is propulsion efficiency relevant not only to boatsmen and other water-sports enthu- 0 W1 Mu u 1 siasts, but also to the propul- D D D : (5.39) W C W 0 1 02 1 u0 sion of all types of ships and 1 2 Mu u C 2 Mu 1 C 2 u aircraft. In most textbooks and lecture courses, this point We can see that it is advantageous to reduce the velocity u0 of the wa- is seldom discussed. ter which is accelerated backwards, in order to maximize the propulsion efficiency. Then, however, from Eq. (5.37), the mass M of the water accelerated must be large in order to obtain the required driving force F1. With

3 If we wish to apply momentum conservation to a walking person, we should con- sult Fig. 5.14 and imagine that the cart is replaced by the earth, with its enormous mass. r u M ( ,then (5.40) is thus M M f )). Then ounding u 5.27 of fuel). This ). This dynamic M /: r force. In the case u 0 of the rocket under 10.10 M u u . D ; down to a mass r M 0 u M , the ﬁnal velocity M is independent of the ejected : M r , and on the right at the time f 0 / , the load has momentum and : u t 0 2 u / u M / u M 0 u . 2 M D u M d ln M r u C r C = u u 0 C u u . / . u M D u upwardly-directed ). For heavier-than-air craft, however, M /. u D D d M u 9.4 . At constant

C ). The latter of course eject not only the air they C M ). It follows that . u (after burning a mass M . 10.11 u ; this is the surprisingly simple result! t D d u = M M 0 rocket equation , we have just considered the propulsion of vehicles ! t without Consuming Power 5.11 , when the rocket has ejected combustion products of mass t C is negative, the mass of the load is decreasing) with the relative velocity have taken in, but alsoRocket combustion products propulsion from is theirmaterial burning not fuel. which fundamentally is different; acceleratedmedium, however, backwards but none is rather of it takenvehicle. the is from part Even the of at surr the a initial constant load flight (fuel speed and oxidant) of the boats moved by propellers or paddle wheels,backwards the can water which be is clearly accelerated seen inIn the the wake case of of the boat. aircraft,powered the flight, same the considerations backwards-directed hold. streampropellers. In of air the Today, jet was early produced engines years only or of in by turbojets many of cases various (Sect. designs are employed energy as seen by ana observer quantitative treatment. at For rest. the efficiency, one This then has finds to be taken into account in determined uniquely by the ratio Integration, beginning with the initial mass for the final velocity is called the and for (oppositely directed to At the left is the momentum at the time mass current d Space rockets, however, do noterated. fly at The constant forces velocities;required they of for are resistance acceleration. accel- are To small compute in the comparison speed to the forces t yields these conditions, we start with momentum conservation (Eq. ( we have of road and railof vehicles, the this road force surface arises orstatic the from buoyant rails; elastic forces for deformations (Sect. ships and airships, it results from this upwardly-directed or lifting force must becally produced using aerodynami- lifting surfaces or airfoils (see Sect. 5.12 The Production of Forces with and In Sect. along horizontal paths.ported The in some weight manner of by the an vehicle had to be sup- Three Useful Concepts: Work, Energy, and Momentum 5 86

Part I 5.13 Closing Remarks 87 lift simply replaces the cables holding up the cabins in the case of an aerial tramway; its effect in the end is no different from that of a hook in the ceiling. However, a hook or a permanent magnet can pro- Part I duce a lifting force year in and year out without any external power source. This is in contrast to an airfoil, which requires a steady input of power. The production of a lifting force by an airfoil is thus fundamentally similar to the production of a force by an electromagnet or a muscle: An electromagnet draws power from its source of electric current, a muscle requires an input of chemical energy from the metabolism and tires even from “holding” (isometric force), i.e. even without performing work in the physical or technical sense. For physical work requires not only a force, but also a displacement along which the force acts. – All types of force which require input power have a common characteristic: They involve “losses” of mechanical, chemical or electrical energy; that is, some part of this energy is converted into heat (more precisely: into internal energy). Heat- ing by electric currents and by muscular activity is well known. In the case of airfoils, heat is produced by various mechanisms; one of them is the formation of turbulence at the tips of the aircraft wings. Physicists are inclined to consider only those forces which perform “Physicists are inclined to work even in economic applications. That is of course not correct. consider only those forces Often enough, even the provision of forces which do not perform any which perform work even work requires a considerable economic expenditure. in economic applications. That is of course not cor- Example rect.” A vehicle is required to use a very short towrope for towing another; otherwise, when the road is curvy, large force components perpendicular to the towing direction can occur. These do not perform work, but they use additional fuel and strain the suspensions of both vehicles.

5.13 Closing Remarks

The logical path of our considerations has taken us from the equation of motion (3.3) to the equation for momentum, (5.24). Of course, the reverse path would be equally valid (and it was indeed that fol- lowedbyNEWTON)C5.11. We could start with the definition of the C5.11. This reverse path, momentum, mu, and require that the time variation of the momentum starting with momentum is proportional to the net force acting, or, mathematically, instead of forces, or, more generally, starting from the d dp .mu/ D D F : (5.41) conservation laws instead dt dt of the equation of motion, is in fact seldom treated in For the limiting case of a constant mass m, one can then write for the physics courses. This is cer- acceleration tainly due not only to didactic du F m D F or a D I (3.3) considerations, but may also dt m be a result of the historically- and in the limit of a purely radial acceleration determined preferences of the teachers. F m! u D F or a D D ! u : (2.6) r m , 2 u h ,it 1 h (5.42) physics. approxima- is the velocity ) c does the engine 5.2 W ) 5.5 experimental : 2 ) remains valid, but not the c when it finally arrives at the , respectively. What is the = H v 2 is however only an 1 t drives for one kilometer up u 0 5.41 m m D 0)? (Sect. opposite to its direction of motion, and 1 t m D h d p m/s. When this correction is taken into mass h F 8 slides down a frictionless inclined plane. R D 10 m with a velocity of zero. At the height m D 3 ). In the region of extremely high velocities h p ) are at the same height, are filled with water D constant A 3.3 ) ) c 5.5 , one must write 5.2 mass, the two routes are equivalent. The path we took m ) up to a height of is the mass at zero velocity (rest mass), and % 0 An automobile of mass An object of mass A stream of water with a mass current of 1.2 kg/min strikes Two identical cylindrical containers whose bottoms (each m constant , albeit one that is justified within wide limits. Its applicability tion defines the boundaries of “classical mechanics”.tivistic velocity In dependence general, of the rela- theand mass instead must of be taken into account, It starts from a height receives an impulse which is so strong that it slides back up the incline. What height The assumption of a (density work performed by the forceconnected of so gravity that when the the water two levels containers equalize? are (Sect. 5.2 an incline with ait grade must of overcome frictional 1:25. forces;1 % they In of amount addition the to automobile’s to the weight. the equivalent How of force much work of gravity, perform? What is its power60 output, km/h? if (Sect. the car is moving at a speed of 5.3 Here, of light in vacuum, equation of motion ( account, the momentum equation ( the oh-so-simple equation of motion reaches the limits of its validity. a vertical platestream horizontally. if the force Calculatea) that The it the water exerts comes velocity on toflected a the of backwards stillstand plate at from the is the the 0.2 plate; water it N, plate and arrived. for with b) (Sect. two the the water cases: same is velocity de- with which 5.4 bottom of the inclined plane ( is better adapted to the needs of a course in with a surface area does it reach, and what is its velocity Exercises 5.1 For a Three Useful Concepts: Work, Energy, and Momentum 5 88

Part I Exercises 89

5.5 In Video 5.4, “Elastic collisions”, among other things collisions between a large ball (55 mm diameter) and a small ball (11 mm diameter) are shown. a) If the small ball (mass m) strikes the large Part I ball (mass M, initially at rest) from the right with a velocity v1,the large ball moves to the left with a velocity v2, while the small ball moves to the right with a velocity v2 D v1 v. Calculate v2 and v. b) After the second collision, the large ball is again at rest. Try to make this plausible without any further calculations. c) If the large ball comes from the left with a velocity v2 and strikes the small ball, which is initially at rest, the latter moves off to the right with a velocity v1. Compare the two velocities by measuring the amplitudes of the swings, and check them against the calculation. (Sect. 5.8)

5.6 A ballistic pendulum consists of a sandbag of mass M hanging from a cord. It suffers an inelastic collision with a small object of mass m and velocity u. To what height will the sandbag be raised by the resulting pendulum swing? (Sect. 5.9)

5.7 A bullet of mass m1 D 30 g is shot with a horizontal velocity of u1 D 300 m/s into a large wooden block with a mass of m2 D 3kg which is suspended from thin cords. What is the velocity of the block after the impact (without rotation)? What fraction of the original kinetic energy of the bullet is converted to kinetic energy of the block plus bullet? (Sect. 5.9)

5.8 Show that in a non-central collision of two objects of the same mass, the angle between their paths following the collision is 90°, presuming that the collision is elastic and that rotations can be neglected. (Sect. 5.10)

5.9 A rocket takes off with a total mass of Mo C Mp. Mp is the mass of its propellant (fuel), which is ejected out of the nozzle of the rocket at a relative velocity of ur when burned. Determine the mass Mp which would be necessary to accelerate the rocket to a velocity of ur. (The acceleration of gravity can be neglected.) (Sect. 5.11)

5.10 A rocket is to be accelerated vertically from the ground with an acceleration of ao. Its total mass on takeoff is Mt, and the combustion gases are ejected from its nozzle with a relative velocity ur. Find the mass current dM=dt which has to be ejected from the rocket in order to produce this acceleration on takeoff. (Sect. 5.11) For Sect. 5.2, see also Exercise 4.4; for Sect. 5.3, see also Exer- cise 3.3.

Electronic supplementary material The online version of this chapter (doi: 10.1007/978-3-319-40046-4_5) contains supplementary material, which is available to authorized users. h xso oainwl ntal easmdt efie nsaeby space in fixed be to assumed be initially bearings. will rotation of axis The ieyaogapt;ised t oini iie opure to limited is progres- motion moving its not instead, is chapter, body path; this A a In case: along – limiting sively orbit. other compared circular the as the consider cir- motion along we the rotational motion progressive around the the neglect motion progressive to could its we of Therefore, that cle. to compared small is ieojcso on masses. point or objects like iept,w e h ceeaigfreataogadrcinpassing direction a along act force straight- a accelerating the along the through motions let of we case the path, In line tricks: two using by motions eesd u h iei nryo hsrttoa oin(Sect. motion rotational this of energy kinetic the the full But of a curvature completes released. sling of a radius in the stone tation to a compared example for small dimensions Admittedly, the be orbit. all to chose body we path, the curved of a along motions For body. motions. tations vr ml ic fi is‘oueeeet’o pitmasses’), ‘point or elements’ ‘volume mass axis, this (its a around with it rotated each of is piece object the small When every bearings. on mounted fRgdBodies Rigid of Motion Rotational Figure Torque of Definition 6.2 namely distinguish motion, can of we way, forms arbitrary an different in two moving body a for general, In Remarks Introductory 6.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © xso oaini usin hsmasta h oc utcontain must force the that means This question. in rotation of axis in nta,tefrems eeaea effective an mo- rotational generate a must produce force to the sufficient Instead, is tion. force arbitrary horizontal. any is just object the Not in plane mass the point then each vertically; exactly of rotation rotation suppressed. of of of be axis axis to its has its orient weight around to its have position of We influence angular the arbitrary end, this any To in rotation. rest at remain to xs aldthe called axis, rudiscne fgaiyec iei ice eoebeing before circles it time each gravity of center its around u ramn pt o a enlmtdt progressive to limited been has now to up treatment Our . 6.1 Formally hw a,rgdojc ihapredclraxle perpendicular a with object rigid flat, a shows etro gravity of center ln frotation of plane ehv rae h oigbde spoint- as bodies moving the treated have we , m oae ihnapaepredclrt the to perpendicular plane a within rotates , olsItouto oPhysics to Introduction Pohl’s Experimentally as aldte“etro as)o the of mass”) of “center the called (also h beti sue ob able be to assumed is object The . progressive epeetdrotational prevented we , O 10.1007/978-3-319-40046-4_6 DOI , torque oin and motions rudthe around rotations 6.4 ro- ro- A ) . 6 91

Part I . ˇ d = that is W (6.1) d : .Thenit F ˇ D C6.1 M corresponds to )). In this case, M which acts on the 6.1 ,or . ˇ d 6.2 M , it points parallel to the D 6.1 simply for a force : ˇ d F ) through the angle d M ? , 6.1 Fr reaction torque M r D D . In order to measure the total torque x d M F is the perpendicular (i.e. shortest) distance . Thus, in Fig. ? r D r W electric motor ,where rotate a body (Fig. is also a vector. It is perpendicular to both the force F ? M M r The definition of a torque is the position vector which points from the axis of rotation to r andtotheleverarm from the line along whichcalled the the force “lever acts to arm”newton the meter of (N axis m). the of rotation, force. also Let The a unit torque of the torque is the the point where the force acts. The magnitude of Here, Figure 6.1 a component within the planepass of through the rotation, axis and of its rotation. direction mustQuantitatively, not we define the torque performs the work d the product which is directed parallel to therotation axis of parallel to the plane of rotation by means of the equation Its unit is thereforeter/rad. the The unit unit ‘rad’ of is equivalent work/unit tooften the of left number angle, 1, off. and For e.g. it this is newton reason, therefore me- a torqueThe has torque the same units as work. the equal but oppositely directed F axis of rotation. Looking along theresulting direction rotation of the as torque, clockwise. we see the Torques can also be producedto by the other plane forces of which rotation. areno not longer The parallel parallel direction to of theonly the axis the of resulting rotation torque component (see is of Eq.effective then the in ( producing torque a parallel rotation around to that the axis. Usually, axis a of number rotation is ofto different rotate, forces producing act avectorially on variety to of a give different body aexample torques. which for resultant is an overall These free torque. combine exerted by the This motor is even the during its case operation, for we can make use of housing of the motor. Details are given in Fig. is and c ,and a b ,where , ' is itself ) contains a . c .(Forfur- b sin 6.1 , whose def- a D to the direction ab and b of the vector a b D a c c a D b . The magnitude of , a rotation to the right Rotational Motion of Rigid Bodies b c is the angle between 6 vector product form a right-hand coordi- . The vector product is not product C6.1. Equation ( a inition we give briefly here: The result a vector which is perpendicular to both given by ' (clockwise) leads from the direction of of c nate system; that is, when one looks along the direction of a b commutative; instead: ther details, see mathematics textbooks.) 92

Part I 6.2 Deﬁnition of Torque 93 Part I

Figure 6.2 Measurement of the torque of an electric motor, while it is producing a vertical air current with its output power of WP 0:5kW.Theforce meter (cf. Fig. 3.9) is attached by a cord tangentially to the circumference of a round table (2r D 0:25 m), which is mounted on ball bearings so that it is free to rotate around a vertical axis. The mounting of the rotatable table is the same as that shown in Fig. 6.17. The product of the torque M and the angular velocity ! (Sect. 2.5)ofthemotorgivesthepowerWP , e.g. in N m/s or watt.

Figure 6.3 The center of gravity

In Fig. 6.1, the axis of rotation was chosen to be vertical. In this limiting case, the weight of the object or of its individual point masses m cannot give rise to any torques parallel to the axis, which would thus be able to produce a rotation around it. In the second limiting case, that of a horizontal axis, the situation is different. Here, each individual point mass m contributes a torque as shown in Fig. 6.3, proportional to r?m. In general, if the object is initially in some arbitrary position, it will begin to rotate; only in a special case will it stay at rest in any initial position. This special case arises when the axis of rotation passes through the center of gravity of the object. Thus, for an axis through the centePr of gravity, the resulting total torque and therefore also the sum r?m must be equal to zero. This equation comprises a definition of the center of gravity.C6.2 We C6.2. As a general definition will make use of it later. Otherwise, we will consider the center of of the center of gravity, we gravity of a body and its determination to be well known, as we have find forR the resulting torque done up to now. The center of gravity is usually treated in detail in M D r g dm D 0 school physics courses in connection with levers, balances and simple (g is the vector of the acceler- machines. ation of gravity).

If an axis of rotation is mounted on fixed bearings, the direction, magnitude and sense of rotation of a torque will be generally clear. In other cases (free axes), the beginner occasionally encounters difficulties. Among these for example is the child’s trick of “obedient” and “disobedient” spools of yarn: A spool of yarn has fallen to the floor and rolled under the sofa. Someone is trying to retrieve it by pulling on the end of the yarn. Some spools come out obediently, others hide further back in their lairs. Figure 6.4 shows the explanation. One must consider not the symmetry axis of the spool, ) at- (6.3) (6.2) 4.15 ( .When physics ). If the spring spiral spring 5.5 of the spring. torsion shaft (extension or compres- (Fig. x D D D : ˇ: x x D D D As is often true in life, a little *, and the Angular F M ˇ shows such a D elastic coefficient D D helical springs F 6.5 M Force Angle Torque of known magnitude and direction can be (the “momentary axis” or “instantaneous axis”). Displacement m ! A as 6.4 torques M Torques acting on the Constant Velocity of known magnitude and direction can be produced in an intu- but rather its line ofindicated in contact Fig. with the floorIf as the its yarn axis is of heldobstinate at rotation. spool can a be This sufficiently forced is “flat” tois angle obey. of to more the use floor, here even than the petulant most outbursts of temper. properly dimensioned (sufficient lengtha of torque proportional the to spring), therelation angle it of produces rotation. We again find a linear tached to an axle. Figure is called the spring constant or Analogously, produced in an intuitively clear manner by using a is properly dimensioned (sufficientlyproduces long), is proportional the to the elastic displacement forcesion) of that the it length oflaw): the spring. A linear force law applies (Hooke’s The quotient of the magnitudes will be called the “torsion coefficient” of the spring. The quotient of the magnitudes spools of yarn Figure 6.4 Forces itively clear manner by using 6.3 The Production of Known Torques, Rotational Motion of Rigid Bodies 6 “As is often true ina life, little physics is of more use here than petulant outbursts of temper.” 94

Part I 6.3 The Production of Known Torques, the Constant D*, and the Angular Velocity ! 95

Figure 6.5 A small torsion shaft, mounted vertically and carrying a sphere, is the basis of a torsional pen-

dulum. The torsion shaft makes use of the bending Part I elasticity of a spiral spring. Its torsion coefﬁcient is D D 0:055 N m/rad.

Figure 6.6 Calibration of the torsion shaft from Fig. 6.5 in a horizontal position. For example: r D 0:1m, ˇ D 180ı D D 3:14; F D 1:71 N; rF D 0:171 N m; D D 0:055 N m/rad.

Numerical examples See the captions of Figs. 6.5 and 6.6. Radian (rad) is another name for the number 1, the unit of an angle (cf. Sect. 1.5).

In analogy to helical springs of known spring constant D,inthefol- lowing we will frequently require a spiral spring (together with its axle) of known torsion coefﬁcient D. Therefore, we calibrate the torsion shaft sketched in Fig. 6.5 using the straightforward scheme shown in Fig. 6.6. A numerical example is given in the ﬁgure caption. The axle and spiral spring are often replaced by a torsion wire or band; but a torsion shaft with a spiral spring is a particularly clear-cut arrangement.

Beginners often underestimate the ability to twist even thick steel rods. Figure 6.7 shows a steel rod of 1 cm diameter and only 10 cm long which is clamped in a vise. This apparently very rigid object can be twisted visibly with just our fingertips. We simply need to use a light pointer of around 10 m in length. It is reflected from the mirrors a and b (compare Fig. 3.1). ) II 3. 2.5 ( ::: circles P P D s along the two 2 : ! t 3. We draw vectors r and 2 1 ::: ! ! P D 2 s and simultaneously the axis : 1 . Equation (2.5) gives its magni- t ˇ ! d illustrates this. A point d ! and D 6.8 ! t . Within a sufficiently short time interval 2 r ! has already been defined by the equation 1 is fully defined only when both its magni- he angular velocities ! u . C6.3 D 1 C6.4 can be obtained as the resultant from the vector ad- s s (Video 6.1) . These two vectors add vectorially to give the resultant II The angular Two fingers twist a short, with the angular velocity its direction are completely determined; it is a vector. The I ) and I and 6.31 , the point traverses the nearly linear orbital segment t with the magnitudes of t This segment dition of the individual orbits A second route can also lead us to the orbit with the angular velocity tude. The vector of theof angular the velocity is axis drawn of alongthe rotation. the axis direction Figure The orbital velocity tude axes velocity as a vector Figure 6.7 thick steel rod. Looking along the direction of the arrow, onea sees clockwise rotation. (See Fig. Figure 6.8 The angular velocity same is true of the angular velocity ities (or angu- , which de- u is useful for ! (Eq. (2.11)), which r = u Rotational Motion of Rigid Bodies D 6 “Twisting a rod” http://tiny.cc/csqujy Video 6.1: scription given here is very helpful for overcoming them. of axial vectors in general) often causes difficulties for beginners. The intuitive de- C6.4. From experience, it is found that the vector addition of angular veloc ! circular orbits around the axis of rotation, they all have the same angular velocity the description of rotational motion. Since during the rotation of a rigid body, allpoint its masses move along scribes a progressive motion along an orbit, the lar velocity C6.3. In analogy to the orbital velocity can therefore be attributed to the body as a whole. 96

Part I 6.4 The Moment of Inertia, Equation of Motion for Rotations, Torsional Oscillations 97 angular velocity !. It determines a new axis III, and around this new axis, the body rotates with the angular velocity !. It then traverses the orbit s D !rt in the time interval t. The vector addition Part I of two angular velocities can be understood readily in this example. One need only remember the similarity of the two triangles used in the construction.

6.4 The Moment of Inertia, Equation of Motion for Rotations, and Torsional Oscillations

Once we have understood the concepts of the torque M and the torsion coefﬁcient D, making the transition from progressive motion to rotational motion is not difﬁcult. We refer to Table 6.1. Its two upper

Table 6.1 Comparison of the corresponding quantities for progressive motions and rotational motions Progressive motions Rotational motions dx Velocity u D (2.2) 1 Angular velocity (2.5) dt dˇ ! D dt du Acceleration a D 2 Angular acceleration dt d! !P D dt Mass m 3 MomentZ of inertia (6.4) D r2dm

Equation of motion (3.3) 4 Equation of motion for rota- (6.7) force F tions a D torque M mass m !P D moment of inertia force F torque M D (4.15) 5 D (6.3) displacement x angle ˇ spring constant D torsion coefﬁcient D Oscillationr frequency (4.16) 6 Oscillationr frequency (6.13) 1 D 1 D D D 2 mZ 2 Z

Work W D Fxdx (5.1) 7 Work W D Mdˇ

Kinetic energy (5.13) 8 Kinetic energy (6.5) 1 1 E D mu2 E D !2 kin 2 kin 2 Momentum p D mu (5.24) 9 Angular momentum L D ! (6.14) Power WP D Fu (5.35) 10 Power WP D M! dp dL Force F D (5.41) 11 Torque M D (6.15) dt dt . ) by

u (6.5) 5.13 ( for all .Then (6.4) i r motions u = : u 2 D rotational mo- mr d ! . In the following Z progressive has a special name, P ! D 2 : on the right. : / ! 2 : 2 i 2 ) can also be transformed r 2 i i ! /! 6.1 u 2 i : 2 i i m : r 6.1 by the moment of inertia r of a body which is rotating 2 i 2 i m m . m m mu 1 2 ! 1 2 . 1 2 1 2 X tities for rotations (right): the an- D D ponding quantities for D D i D X / i 1 2 / kin

motions, on the left, is kin kin E E kin E D kinetic energy of a body rotating with the E kinetic energy . kin column the quantities for to the eighth row on the right; the correspond- ,andthemass . E ! becomes: left 6.1 and the angular acceleration ! progressive ), we use the same angular velocity . Any arbitrarily chosen point mass moves at a dis- ! m 2.11 from the axis of rotation with the orbital velocity moment of inertia C6.5 i r the This is indicated in the third line of Table The equation of motion (row 4 in Table in a corresponding manner. To see this, we imagine a rigid, rotating angular velocity Using this deﬁnition, the the angular velocity In words: For rotational motions, we replace the orbital velocity The summation which stands to the left of From Eq. ( the kinetic energy of this point mass or ‘volume element’ is: namely Taking the sum over all thethe point whole masses rotating yields body, i.e. the kinetic energy of Now refer in Table the point masses and obtain: tions, starting with the rows, we see in the with which we areintroduced. already familiar, in the order inWe which can they then were calculate the corres gular velocity rows contain the two(left) kinematic and quantities the velocity corresponding and quan acceleration ing equation for around its axis. This energy mustenergies consist of of each the of sum the ofwith point all mass the masses kinetic which constitute the body, each tance . m d 2 ? is the perpendicular r r R Rotational Motion of Rigid Bodies from the axis of rotation. D 6 m C6.5. distance of the point mass d Therefore, one often writes alternatively 98

Part I 6.4 The Moment of Inertia, Equation of Motion for Rotations, Torsional Oscillations 99 body to be composed of many point masses dm, each at a perpendicular distance r from the axis of rotation. Then in the case of an accelerated rotation, the orbital accelerations a of the individual Part I mass points all have different magnitudes;however,theangular accelerations !P D a=r are the same for all points. According to the equation of motion, the orbital acceleration a of each mass point requires a force which acts in the direction of the momentary orbit,

dF D a dm DP!r dm:

Multiplication by r yields

r dF DP!r2 dm; or, as a vector equation,

r dF DP!r2dm and, after integration over all the mass points, Z Z r dF D M DP! r2 dm DP! : (6.6)

The two integrals deﬁne the two new derived quantities, the torque M and the moment of inertia . To produce a torque experimentally, as on the left side of the equation, we need only a single force F,which acts on the rigid body at a known distance r from its axis of rotation (Eq. (6.1)). The resulting angular acceleration is directly proportional to the magnitude of the torque M and proportional to the inverse of the moment of inertia ; thus, the equation of motion for rotational motion (row 4 in Table 6.1, right) is

M ! : C6.6. Homogeneous means P D (6.7) that the mass is uniformly distributed within the body, For geometrically simple rigid bodies, the computation of the mo- i.e. % D const. ment of inertia presents no difficulties. The necessary integration can usually be carried out in a few steps. Some examples of the results: C6.7. We add that the ex- C6.6 1. A homogeneous circular ring. Mass m,radiiR and r, thickness pressions (6.8)and(6.9) hold d, density %, axis of rotation through its center pointC6.7: quite generally for a homogeneous, hollow cylinder. perpendicular to the end surface: D %d.R4 r4/; (6.8) Equation (6.9) can be simpli- 2 fied for the two limiting cases m d R (a flat disk with a hole along a diameter: D d2 C d%.R4 r4/: (6.9) 12 4 in its center) to %. 4 4/ D 4 d R r D 1 m.R2 C r2/, 2. A homogeneous sphere, with the axis of rotation through its center: 4 i.e. just half of expression (6.8), and for d R 8 5 2 2 D %R D mR : (6.10) (a long rod) to Eq. (6.11). 15 5 . . S of A

D . D 2 (6.11) ! (6.12) S in row 6

1 2 m moves along S of the body as -axis has a total A m : : 2 of a spiral spring. The a . If the body makes a full 2 2 S : ! D ml and cross-sectional area 2 from the ﬁrst axis? Answer: m and the spring constant l 1 2 1 -axis. A small arrow is drawn a 12 . This second energy (from the A ma 2

C / a D 2 C will deﬁne a known value of ! 3 S .! S for an axis of rotation which passes m Al

). 1 2 6.5 : How large is its moment of inertia 1 2 % of length m D 1 D D 12 6.12 A 2 2 -axis. As a result, the energy given above is at a distance S !

D . We can think of the mass ). The axis of rotation of this body must fall A mu A . We need only replace the mass 1 2

-axis, the body will have the kinetic energy 6.5 1 2 -axis, then this arrow, and thus the entire body, makes S A . At the same time, the center of gravity center of gravity 2 yields Eq. ( . Assuming that we know the moment of inertia ! 2 D S ! S shows a rotation around the

1 2 as shown in Fig. circular orbit 1 2 slaw by the moment of inertia ’ center of gravity S The straightforward derivation of 6.9 6.1 ’s law; -axis). Thus the rotation of the body around the A long, homogeneous rod from its rest position, release it, and observe the period of the TEINER for any other arbitrary axis which is parallel to the first axis and conserved, Dividing by the dashed motion of the center ofthe gravity) adds to thekinetic first energy (from of: the rotation around Derivation Rotating around the – Figure on the body, which starts atrotation the around center the of gravity a full rotation around the localized at its centerof of this gravity, circular and motion we then find for the kinetic energy ı A TEINER along the extension of the90 torsion shaft. We rotate the body by about 4. S torsion shaft At the upper endcharacterize of (cf. this Fig. shaft, we attach the body that we wish to passes through the point S For the measurement of moments of inertia,of one torsional in general oscillations makes use of Table Figure 6.9 However, much more important than theinertia computation of is moments their of measurement.integration For would be bodies unnecessarily with difficult. a complex shape, the of an arbitrary body ofthrough mass its 3. A a helical spring by the torsion coefficient The axis of rotation passespendicular to through the the long center axis of of the gravity rod: and is per- Rotational Motion of Rigid Bodies 6 100 Part I 6.4 The Moment of Inertia, Equation of Motion for Rotations, Torsional Oscillations 101 resulting torsional oscillations, T, with a stopwatch. We haveC6.8 C6.8. Analogously to a linear oscillation (Fig. 4.13),

T2 we obtain an expression for Part I D D : (6.13) 42 the period of oscillation of a torsional motion (6.13) The torsion coefﬁcient D of our small torsion shaft was already by inserting the “linear found to be 5:5 102 N m. Thus, we obtain restoring force law” (6.2) into the equation of motion T2 for rotational motion, (6.7) D 1:4 103 kg m2 : s2 (cf. Comment C4.5.).

Numerical examples 1. Verifying a calculated moment of inertia. Taking a circular wooden disk with m D 0:8 kg and 0.2 m radius, we calculate from Eq. (6.8) with r D 0 2 2 that the moment of inertia S should be 1:610 kg m for rotation around an axis passing perpendicularly through the center point of the disk. We 2 2 observe T D 3:37 s, and thus S D 1:58 10 kg m . 2. A disk and a sphere of equal moments of inertia. Figure 6.10 shows a disk and a sphere made of the same material, to the same scale. Their masses are in the ratio 1 : 2.9. Their moments of inertia should be equal, according to Eqns. (6.8)and(6.10). Indeed, they both exhibit the same oscillation period when attached to our small torsion shaft. 3. Moments of inertia of hollow and full cylinders of the same mass.Fig- ure 6.11 shows a hollow metal cylinder and a full wooden cylinder, both with the same masses m, the same diameters and the same lengths. How- ever, for rotations around their cylinder axes, the hollow cylinder is found to have a considerably larger moment of inertia than the full cylinder. This explains an often surprising experimental result: We set the two cylinders side by side on a ramp, for example a board propped up to give it a slope. The axes of the two cylinders are collinear. Then we release them both at the same time. The full wooden cylinder reaches the bottom of the ramp much sooner than the hollow metal cylinder. – Explanation: In rolling down the ramp, both cylinders are accelerated by the same torque, rFG (Fig. 6.12), since their masses and radii are the same. As a result, the hollow cylinder, with its larger moment of inertia, experiences a smaller

Figure 6.10 A disk and a sphere with the same moment of inertia

Figure 6.11 A full and a hollow cylinder of the same mass (wood and metal), with different moments of inertia

Figure 6.12 The torque M D rFG acting on a cylinder on a ramp . 2 is also ). – 6.1 4.3 of a man lying , and 2.3 kg m 2

) ? itions and axes of C6.9 ,8kgm 2 (row 4 in Table We determine the moment ! is a strong spiral spring. Its torsion F hung from a massless, frictionless cord. (Video 6.2) . itions. m 2 of a mathematical pendulum which would and angular velocity l P ! ’s law be taken into account here . Some of the results of the measurements are collected 6.13 TEINER is ca. 2.5 N m/rad. The moment of inertia . We will make good use of them later. A large torsion shaft for measuring the moment of inertia of The moment of inertia of a human body in three different posi- acceleration D 6.14 and the Beam Balance The moment of inertia of the human body. (Why must S of inertia for rotationrotation. of a For human body these withshown observations, in various we Fig. pos make usein of Fig. a large torsion shaft as angular 4. (Video 6.2) horizontally is about 17 kg m tions. The arrows indicateof the direction inertia of (from the the axis left of to rotation. the The right) moments are 1.2 kg m coefficient a human body in various pos 6.5 The Physical Pendulum The gravity pendulum (simple pendulum) treated in Sect. Figure 6.14 Figure 6.13 called a “mathematical pendulum”.a point-like body It of represents mass the ideal case of A real or “physical” pendulum oftenform. deviates strongly from For this ideal every physicalfined: pendulum, a It “reduced” is lengthhave the the can same length be period de- of oscillation as the given physical pendulum. ILSCH ) Exercise 6.2 Rotational Motion of Rigid Bodies 6 http://tiny.cc/gsqujy In the picture: R. H Video 6.2: “Moments of inertia” that the mass plays no roleall! at ( this question will also find the explanation of the similar, often surprising observation C6.9. Those who investigate (Dr. rer. nat. 1927) 102 Part I 6.5 The Physical Pendulum and the Beam Balance 103

Figure 6.15 The physical gravity pendulum (Axes through O or P, perpendicular to the plane

of the page) Part I

As an example, Fig. 6.15 shows a board of arbitrary shape, hung from a bearing as a gravity pendulum. O is the axis around which it can swing, S is its center of gravity, and s is the distance between the two. For the period of oscillation of this physical pendulum, the formula which holds for every torsional oscillation can be applied:

T D 2 0 : (6.13) D

0 is the moment of inertia around the axis O. D is again the torsion coefﬁcient, i.e. D D M=ˇ. The magnitude of the torque M can be read off in Fig. 6.15: M D mgs sin ˇ: (6.1) For small angles ˇ, we can again set sin ˇ D ˇ. We then obtain C6.10. l is longer than s: D D M=ˇ D mgs (6.3) Making use of STEINER’s law (Eq. (6.12)), it follows and, from Eq. (6.13), s that 2 0 S C ms 0 : l D D , T D 2 ms ms mgs l D s C S > s . For a “mathematical” gravity pendulum, i.e. a point-like body hanging ms

from a massless cord, we found the period previously to be ( S D moment of inertia s around the center of gravity.) l T D 2 0 : (4.17) g C6.11. To convince one-

For a physical pendulum, instead of the length of the cord l0 of the math- self of the correctness of = ematical pendulum, we insert the quantity 0 ms. This is the reduced this statement, use the tor- = C6.10 pendulum length. It is drawn in as a length l D 0 ms in Fig. 6.15 . sion coefﬁcients for s and Its lower end is called the midpoint of the oscillation P. We can think of (l s), D D mgs and the total mass m of the body as being concentrated at the point P, without 0 . / changing the period of oscillation of the pendulum. DP D mg l s , and, ap- The period of oscillation of an arbitrary pendulum remains unchanged plying STEINER’s law, the when one takes its axis of oscillation to pass through the midpoint of os- corresponding moments of C6.11 2 cillation P . This forms the basis for a common experimental method inertia 0 D S C ms and of measuring the reduced pendulum length (reversion pendulum). 2 P D S C m.l s/ ,in Eq. (6.13) and solve for the reduced pendulum length l. . u m D (6.14) p beam bal- to the axis of S , we use the angular ve- C6.13 u . It speciﬁes the direction : differs only in its external de- the corresponding momentum sion analytical balance without !

6.16 must be replaced by the moment of m D L . angular momentum 6.15 ), the 6.1 (fulcrum) is greatly exaggerated in the drawing. Up to angular Schematic of a beam balance as a physical pendulum. In the In the text, the sense of rotations applies to an observer is a physical pendulum. – We ﬁrst neglect the two balance O , , and instead of the orbital velocity . These two quantities deﬁne

! C6.12 clockwise. who is looking down from above. The angular momentum isand also the a sign vector orof sense the of (positive) the angular rotation. momentum vector, Looking the along sign the of direction rotation is a conservation law holds. In rotational motions, the mass inertia Momentum is a vector, and for the momentum of a “closed system”, pans might be e.g. 12 s,of corresponding 36 to m. a reduced pendulum length Adding the balance pansmoment of and inertia their of the weightslation balance period, increases beam. e.g. This the to also 18 effective s. increasesincreases it the An even oscil- additional further mass up of to 100 about g 24 s. on both sides pans. Then the schematic intails Fig. from that in Fig. The period of oscillation of a preci locity (row 9 in Table 6.6 Angular Momentum For progressive motions, the momentum was defined as Figure 6.16 One of ourance most important measuring instruments, the interest of clarity, the distance from the center of gravity oscillation deflections of several degrees, thein the deflection weights is on proportional the to balance the pans. difference . is ! is r (see )and 2 v a ma http:// 6.14 )has C ,where v ! s ). ).

r m D ) into Eq. ( ). In general, the or- D L 6.12 6.9 L Rotational Motion of Rigid Bodies 6 Exercise 6.7 obtain Fig. bital angular momentum is deﬁned by the vector equation which the body possesses due to its motion along the circular orbit of radius The ﬁrst term is again the proper angular momentum. The second term is called the “orbital angular momentum”, In other cases, one canEq. insert ( If the axis passes throughcenter the of gravity, we referthe to “proper angular momentum” or “spin” of the body. tum introduced here depends, like the moment of inertia, on the axis of rotation chosen. of a torsional pendulum ( C6.13. The angular momen- electronic analytical balances. However, it remains an instructive example largely disappeared from modern physics and chemistry laboratories and has been replaced by sensitive instrument (see e.g. en.wikipedia.org/wiki/ Analytical_balance C6.12. The mechanical beam balance as a measuring the position vector and the orbital velocity (see also Exercise 6.8 104 Part I 6.6 Angular Momentum 105

Figure 6.17 The conservation of angular momentum. The arrows represent the angular

momenta of the wheel and the experimenter. Part I (At small angular velocities, the motion is disturbed by the residual friction.) (Video 6.3) Video 6.3: “Conservation of angular momentum using a rotating chair” http://tiny.cc/4rqujy In the picture: R. HILSCH (Dr. rer. nat. 1927)

Angular momentum also obeys a conservation law, i.e. as long as no external torques act, the angular momentum of a system (magnitude and direction) remains constant (M D 0 in the equation of motion for rotations (6.7)). Just as in our treatment of progressive motions, we give some experimental examples. As an aid to visualization and for experiments, instead of the ﬂat carts that we used to illustrate progressive (linear) motions (Fig. 5.14), for rotations we will employ a swivel chair (Fig. 6.17). It can rotate with little friction (ball bearings) around a vertical axis. It thus reacts only to vertically-directed angular momenta. If the angular momentum vector is slanted relative to the vertical, the chair takes on only its vertical component.

Examples 1. A man is sitting in the swivel chair, at rest. In his left hand, at about eye level, he is holding a top (a weighted bicycle wheel), also at rest, with its axis of rotation nearly vertical. The angular momentum is initially zero. The man grasps the spokes of the wheel from below with his right hand and starts the wheel spinning. The spinning top thus obtains an angular momentum of 1 !1 in a counter-clockwise sense. Owing to the conservation of angular momentum, the man receives a (counter-) angular momentum 2 !2 of the same magnitude but opposite rotation sense. Indeed, he begins to rotate with the chair in a clockwise sense. His angular velocity !2 is much smaller than that of the top (wheel), because his moment of inertia is much larger than that of the wheel. 2. The man presses the rim of the rotating wheel against his chest and brings it to a stop. The rotation of the wheel and of the man and chair both stop simultaneously. Both angular momenta are again zero. 3. The man sits motionless on the chair holding the motionless top with its axis horizontal. He gives a spin to the top (wheel); its resulting angular momentum vector is horizontal. The chair and the man remain at rest, since they cannot react to an angular momentum with no vertical component (the counter-component of the angular momentum is taken up by the earth through the man’s arm and the shaft of the chair). to ı .As ı from the vertical. ı , thus all together by L plane and then back down to L path before beginning another C from his initial position. These ). He will try to circle the mallet vertical o maintain his angular position, he 6.18 different . He turns its lower end upwards. He thereby ı plane back to its original position. The angular A polo mallet can vertical in a clockwise sense. He then tips the top back around to its . The man himself begins to rotate with an angular momentum L L 2 the vertical and then isrotate, but given with a only spin. a small Theangular angular momentum man velocity. which They and is receive the a equal counter- chair onlythe begin to top’s angular the to momentum. vertical component of 5. We hand anot moving. (clockwise) spinning He top remainshad at to rest; an the we man, angular gave who him momentum.spinning the is top top initially by Then which 180 already the man rotates the axis of the only at the same time.let For to a its second swing, originalthen he position; has he to he loses return can the theswing angle mal- do of of the this mallet. rotation alongmust that If the return he he the same wants mallet had path, t along gained a but with the ﬁrst soon as the motionman of and the chair; mallet the stops, man so and does the the mallet rotation can of have the angular momenta be used to generate angularmentum mo- around various axes (Video 6.3) Figure 6.18 changes its angular momentum from a cone whose axis is inclined by for example 45 swing; he moves it upwards inin the another momenta of these motionsaxes in of vertical rotation) cannot planes be (i.e. takenof around on rotation by horizontal (they the are chair transferred withnal to its position, the vertical he axis earth). can From repeat the theand mallet’s circular origi- thereby motion in double the his horizontal angularthree plane individual gain motions can of course betion, combined so into that a single the mo- man’s arm and the mallet move along the mantle of 4. The initially motionless top is held with its axis inclined at 60 of 2 starting position and givesagain it motionless. back to – us. Thus,momentum for one The a can chair time play and and then with the give a man it6. back. “borrowed” are angular The manis holding is a sitting polo on mallet (Fig. the swivel chair, at rest. In his hand he around his head in theof horizontal plane, rotation. that is – aroundalthough While a with vertical he a axis smaller is angular swinging velocityThe than mallet the his and mallet, arm his and he arm the can begins mallet. be to circled rotate, through only around 180 Rotational Motion of Rigid Bodies 6 chair” http://tiny.cc/4rqujy Video 6.3: “Conservation of angular momentum using a rotating 106 Part I 6.6 Angular Momentum 107

Figure 6.19 The vector nature of angular momentum (˛ can be varied between 30ı and 100ı)

(Video 6.4) Video 6.4: Part I “Angular momentum as a vector” http://tiny.cc/ltqujy

7. The vector nature of angular momentum can be demonstrated effectively with a fan which can rotate around a vertical axis (Fig. 6.19). The propeller and the air stream receive an angular momentum L;the fan takes on the counter-angular momentum of equal magnitude but opposite rotational sense. The vertical component, i.e. L cos ˛, causes the fan itself to rotate around the vertical axis (see also Fig. 6.2.)

Examples The propeller blades of the fan are supposed to be turning in a clockwise sense when we look through them towards the housing of the fan’s motor. – First, the fan is blowing in a horizontal direction, that is ˛ D 90ı.The fan remains at rest on the vertical axis of its stand, since cos 90ı is zero. Now the fan is blowing at an angle upwards, e.g. with ˛ D 45ı.The fan rotates on its holder; seen from above, it rotates slowly in a counter- clockwise sense. The reason: We have cos 45ı 0:7; cos ˛ thus has a positive, nonzero value. (The frictional losses in the bearings of the holder present no problem, because the motor and propeller are continually transferring more angular momentum to the air stream.) When the power to the motor is shut off, this rotation initially stops, and then it begins again in the same sense of rotation as the propeller. The reason: The rotor of the motor and the propeller are slowed down by friction in the motor bearings. They give up their remaining angular momentum to the housing of the motor, and we can observe its vertical component.

8. We replace the swivel chair by the large torsion shaft shown in Fig. 6.13. A man is lying stretched out on it, holding on to two handles (Fig. 6.20). He is given an impulse and begins to exhibit torsional oscillations with a small amplitude. Problem:Wewantthe man to increase his oscillation amplitude up to full circular oscillations of 360ı without external help. Solution: He has to change his moment of inertia around the vertical axis periodically. When he passes through the rest position, he pulls in his legs and raises his torso. This reduces his moment of inertia and increases his angular velocity !1. At the extremes of his oscillations, the reversal points, he stretches out again and returns to a maximum moment of inertia. At the next passage through the rest position, he repeats this action. In a short time, he will be exhibiting torsional oscillations

1 The gain in energy comes from the work performed by the man’s muscles against inertial forces (Sect. 7.3). – 4.4 ). . 6.23 . Here, 6.22 C6.14 )) remains constant. http://www.ncbi.nlm.nih. 6.12 as a real shaft mounted on (see (Eq. ( ﬁxed . In order to explain this term, we m 2 = t , a grinding wheel is set in rotation by

. – This experiment demonstrates in an ı L technique of exercising on the bars giant swing free axes 50 Hz). The wheel is mounted at the end 6.24 D 180 ). 2 ˙ = , at all points along the orbit, the triangular area t ! 4.17 2 r shows a well-known circus trick: A ﬂat plate is rotating (Video 6.5) D Bar exercises 6.21 Oce ) is simply a special case of the conservation of angular mo- D 4.5 mentum. In Fig. Oac swing, by bending in the arms or legs or spreading the legs excellent manner the entire the horizontal axis of theand bars the is replaced torque by is thethe produced vertical bars, not torsion but shaft by instead the byThis weight the spiral has of spring the the attached gymnast, advantage tomore as that the easily on the torsion observe shaft. motions them. arelanguage of slower, The gymnastics, so experiment a that just described wegov/pubmed/10433423 is, can in the The gymnast on the barsinertia knows in how various to ways reduce at his the or right her moment. moment For of example, in the giant Figure 6.20 9. The lawand of areas for motions around a central point (Sects. 6.7 Free Axes In all of the rotational motions thatof we rotation have considered of so the far, the rotating axis body was with an amplitude of with swings, simulated on aswivelchair bearings. We now drop thismotions limitation. of This bodies will around lead us to rotational an electric motor ( will give several experimental examples. a) Figure at the tip of athe plate) bamboo serves stick. as Its a free axis axis of ofb) symmetry rotation. (perpendicular A to ﬂat plate,axis if parallel to skilfully its set diameter in (in motion, thec) plane can of We also now the rotate demonstrate plate) a around (Fig. we small an hang variation a cylindrical on rod these from oneof two end experiments: an onto the electric rapidly rotating motor. shaft a transverse Either axis its can serve cylinder as axis a or freed) axis – of For as rotation. technical in applications,“supple” Fig. free shafts. axes In can Fig. be used in the form of ). 11.20 Rotational Motion of Rigid Bodies 6 Video 6.5: “The physics of swinging” http://tiny.cc/ssqujy also Fig. tion”, in which a parameterthe of oscillator is varied, thus exciting vibrations (compare nity to study this technique, which children call “pumping up” and physicists call “parametric excitation of oscilla- C6.14. A playground swing offers another good opportu- 108 Part I 6.7 Free Axes 109

Figure 6.21 The ﬁgure axis of a plate as a free axis of rotationC6.15 C6.15. The author as a circus

performer (the photo was Part I taken by one of the students in his lecture course).

Figure 6.22 A diameter of the plate as a free axis of rotation

Figure 6.23 A rod rotating around the axis of its largest moment of inertia as a free axis of rotation (Video 6.6) Video 6.6: “Rotation around free axes” http://tiny.cc/itqujy

Figure 6.24 The supple shaft of a grinding wheel (this shaft is too thin for practical applications!) (Video 6.7) Video 6.7: “Supple shaft as stable axis of rotation” http://tiny.cc/5squjy of a ca. 20 cm long wire with a diameter of only a few millimeters. It rotates in a stable manner around the axis of its largest moment of inertia and makes a springy contact with a workpiece which is pressed against it C6.16. C6.16. A practical application: All of these examples have two points in common: The unavoidable differences 1. The bodies used have rotational symmetry. All of them could have in direction between the been fabricated on a lathe, in principle. In each case, the body is geometric axes and the an- characterized by a ﬁgure axis or body axis. gular momentum vectors, which lead to “hammering” 2. One of the free axes was a ﬁgure axis, while the other was perpen- of a rapidly-rotating gyro- dicular to it. scope or top, can be absorbed by supple shafts. B A 6.5 ). Its 6.25 . can serve as free axes of C , which is perpendicular to B and A B (with the ). Again, 2 (Video 6.8) A the moments of inertia of axes in different di- with the 6.26 A kg m 3 with the small- can serve as “free” axes, and the object can exhibit 10 C . measure the axes of a body with the largest or the smallest mo- C The axis Throwing a box with a ro- . The experiment demonstrates the following: The mid- 9 > = > ; C6.17 5 6 4 )to : : : and 6 5 1 6.23 and likewise passes through the center of gravity of the object, A D D D 6.13 C B C A

est moment of inertia of a box largest moment of inertia, axis with an intermediate moment of inertia, and axis With these and other, similarclusion experiments, one that can arrive atments the of con- inertia can serve as free axesIn of the rotation examples a)axes to are f), in each which case were readilyIn chosen visible other from for cases, the their one geometry simplicity, can ofand the the always objects. make use of a torsion shaft (Figs. rotation. The samecan side be of seen the from boxalways its remains lead color. facing to the Attempts wobbling observer, tochanging motions, as colors. spin so the that box the around observer axis sees rapidly Figure 6.26 Figure 6.25 f) We repeat this lastinto experiment the with air, a giving variation;on it we releasing a it throw spin (Fig. the by box a suitable positioning of our fingers In the following demonstrations,symmetry. the rotating As bodies an lack example, rotational we take a flat cigar box (Fig. stable rotation around them.to These each two other. free axes – are The perpendicular third mid-line three pairs of sides are each paintede) in a Eyelets different are color. mountedhung at by the a center wirein of from Fig. each the side. shaftlines The of box a is motor, then like the cylindrical rod tation around its free axis largest moment of inertia) and shows a different behavior. ItThe cannot object serve always as returns a to a freetwo stable axis axes rotation of around rotation. one of the other ). A Rotational Motion of Rigid Bodies 6 “Free rotation of a rectangular parallelepiped” http://tiny.cc/1squjy Video 6.8: of inertia (here, axis C6.17. The most stable rotation always occurs around the axis with the largest moment 110 Part I 6.8 Free Axes of Humans and Animals 111 rections. For completeness, we carried out measurements of this kind on our flat cigar box; the results are set out in the caption of Fig. 6.25. Part I

6.8 Free Axes of Humans and Animals

Free axes by no means require rotational symmetry of a body. This is shown by the experiments with the painted cigar box. Still more convincing are the uses of free axes by humans and animals.

Examples a) A jumper performs a somersault. Leaning slightly forwards, usually with raised hands, he gives himself an angular momentum. The corresponding axis is indicated in Fig. 6.27a as a white spot. It is very nearly the free axis with the largest moment of inertia. The angular velocity is still small. A moment later, the jumper pulls his body together into a crouching position as in Fig. 6.27b. The axis of rotation remains that of the largest moment of inertia, but that moment is about three times smaller than before. As a result, the angular velocity increases by a factor of three, from conservation of angular momentum. This high angular velocity permits one or two, sometimes even three complete rotations while the jumper remains in the air. He then stretches out his body at the appropriate moment to restore the large moment of inertia, so that he can land on the ﬂoor with a small angular velocity. The jumping techniques of good circus performers are physically very instructive. The ﬁrst requirement for jumping, however, is courage. Jumping is a matter of good nerves. “Jumping is a matter of The law of conservation of angular momentum automatically guar- good nerves. The law of antees the necessary rotations. conservation of angular momentum automatically b) A ballet dancer performs a pirouette on tiptoe. She rotates around guarantees the necessary the long axis of her body. She uses the axis of her smallest moment of rotations”. inertia as a free axis of rotation. She spins around this axis with a high angular velocity ! and the angular momentum !. To stop, she changes her body position at the appropriate moment to that shown in Fig. 6.14, center, and thereby increases the magnitude of her moment of inertia. This new moment of inertia is around seven times larger

Figure 6.27 Changing the magnitude of the moment of inertia when performing somersaultsC6.18 C6.18. In the picture: R. HILSCH (Dr. rer. nat. 1927) polo way. symmetric in everyday their own moment of inertia largest spinning tops are swung around to produce . In the examples shown there, addition it was mounted on bear- 6.28 . Rotations around free axes or around gyration , which is mounted in bearings. Instead of the tops, gyroscopes, or simply 6.18 Two “flattened” tops. The symmetry hind quarters and the tail and Its Three Axes ,the flattened . This case is defined in Fig. language. A decisive point for describing andcriminate understanding gyrations strictly is between to three dis- differentthrough the axes, center each of of gravity which of passes the body. These are: the symmetry axis is always(it the is axis also of called the the figure axiswe or will body axis; refer in to the followingare it sections, as the “figure axis”). In the physical sense, these top Figure 6.28 free axes, where the axisject, of but rotation was was no stillrotation fixed longer relative occurs held to when by the the bearings. ob- axisnal of The bearings rotation most or is general with not casepasses respect fixed through of to the either the center by of rotating exter- gravitychange object. of its the The object, direction but axis within can always continually theof object. rotation is This called last,shafts most with general bearings form aregyration. special cases of this moreIn general their motion, mostproblems general in all form, of mechanics. gyrationsone Even can with represent obtain a only the great approximateof mathematical most solutions. gyration effort, can be difficult But understood all using of the special the case essentials of the mallet a rotation. Humans can readilyThey imitate this can cat also trick in inducesmallest moment rotations of during inertia, i.e. a the jump body’s long around axis. their axis of 6.9 Definition of the Spinning Top In the rotations that wewith first considered, respect the to axis of theings rotation object, was outside and fixed the in rotating object. We then treated rotations around than before; as a result, herseven. angular velocity She is sets reduced the byand a sole lowering factor the of of point her of foot support. onto thec) floor, A braking cat the which spin its is feet. held by The itsof animal feet inertia; rotates and it around uses then the this droppedchair axis axis always in of as falls Fig. the its on substitute smallest for moment the shaft of the swivel or figure axis is theinertia. axis with the largest moment of Rotational Motion of Rigid Bodies 6 112 Part I 6.9 Definition of the Spinning Top and Its Three Axes 113

1. The figure axis (symmetry axis), and thus in our tops the axis of the largest moment of inertia of the rotating body; Part I 2. the momentary or instantaneous axis of rotation, the axis around which the body is rotating at a particular, given moment; and 3. the angular-momentum axis. It lies between the axis of symmetry and the axis of rotation, and in the plane defined by those two axes. For brevity, we will refer to it in the following as the spin axis,since the proper angular momentum of a body is often called its “spin”, just as one often refers to a “spinning top”. The figure axis is easily recognizable in each of our tops; but it requires a certain artifice to make the other two visible. The top shown in Fig. 6.29 is particularly well suited to this task. It is supported at its center of gravity by a point and socket, nearly without friction and without external forces, so that it remains in equilibrium in any position of the figure axis. Its center of gravity can be adjusted by moving the metal ring A. – The figure axis carries a disk at its upper end; paper stickers with various patterns can be placed on it. To start with, we want to demonstrate the instantaneous axis of rotation. We use a paper sticker with a printed text, start the top spinning, and give the figure axis a tap from the side. This sets the top into a wobbling motion, and we can observe the following: The text on the sticker merges into a uniform grey, owing to the rotation of the top. Only at one small spot, which moves around over the sticker, is the text briefly at rest and recognizable as printed letters. The midpoint of this spot is the instantaneous axis of rotation. This axis, as well as the figure axis, both move over the mantle of a cone, each with the same angular velocity !N, and these two cones have a common axis which is fixed in space. This latter axis, which is at this point invisible, is the angular momentum axis. In order to make the angular momentum axis visible, we start with a preliminary experiment. We attach a sticker with concentric circles

Figure 6.29 A top for demonstrating the three axes. To start it spinning, we clamp the figure axis between the palms of our hands and move them in opposite directions. (Video 6.9) Video 6.9: “The three axes of a top” http://tiny.cc/2tqujy ), with the 6.29 ). N ! . , lower part; again a system of concentric circles. (its angular velocity is 6.30 6.30 , upper part. – Then we shift the center of the circles Visualizing the spin axis. About nutation 6.30 , 542 (1949)) We give some moreHere, details we on simply nutation establish from inprove the the experiment to following a be fact section. useful: which willa – The later result nutations of decay after thein a unavoidable our certain friction example, time. in between therests. This the support is point point and of the socket the on top; which the top 1/10 of actual size (R.125 Hagedorn, Z. Phys. Figure 6.30 This whole phenomenon, the common rotationthe of instantaneous axis the of figure rotation around axis the and is angular momentum called axis, common center point onspinning the figure top, axis. we again Byfigure separate a axis, the tap and to three the axestates thus side from around the of each the center the other; angularmomentum of the momentum axis the axis. visible concentric inexperiment, circles, This just Fig. makes now the the ro- same angular way as in our preliminary ter of the circles itselfThis rotates yields around Fig. the axis ofThe rotation spacing of between the the disk. individualtheir circles contours are is fuzzy the and washed same out.paler as The before, circles common but center lies of on these So the much axis for the of preliminary rotation experiment. and showsFor us the where main it experiment, is.onto we the – put disk the at sticker the end with of concentric our circles top’s figure axis (Fig. onto a rotating disktation. with The the rotating center sticker ofcf. looks the just Fig. the circles same onto as the the when axis it side, of is away ro- at from rest; the axis of rotation of the disk; thus the cen- Rotational Motion of Rigid Bodies 6 114 Part I 6.10 The Nutation of a Force-Free Top and Its Fixed Spin Axis 115 6.10 The Nutation of a Force-Free Top

and Its Fixed Spin Axis Part I

The nutation which we have just observed is an immediate result of the conservation of angular momentum. Imagine that in Fig. 6.31, the plane of the paper contains the figure axis A of the top and its instantaneous axis of rotation ˝. The top is spinning around this instantaneous axis with its angular velocity !, represented by the vector arrow along the axis of rotation, ˝. This angular velocity ! can be decomposed into two components !1 and !2. !1 is the angular velocity around the axis A with the largest moment of inertia A, while !2 is the angular velocity around a perpendicular axis C with the moment of inertia C. The overall angular momentum is then given by LA D A!1 in the direction of the figure axis A and LC D C!2 in the direction of the perpendicular axis C. These two angular momenta are shown in the figure as vector arrows with thick arrowheads. Their resultant is the total angular momentum L. Its direction thus lies between the figure axis A and the instantaneous axis of rotation ˝, and within their common plane. Now the top is by assumption “force-free”. It is supported at its center of gravity by the point and socket. No torques of any kind act upon it. As a result, its angular momentum, magnitude and direction, must remain constant (see Sect. 6.6). Its spin axis must continually maintain the same direction in space. Both the figure axis A and the instantaneous axis of rotation ˝ have to rotate around this fixed spin axis with the nutation frequency (!N). To make this clearer, in Fig. 6.32 we have indicated the three axes with stiff wires and let them rotate around the center wire, i.e. the angular momentum axis (spin axis). Then we see how two cones are traced out around this axis. One of them is traced by the wire representing the figure axis; this is the cone

Figure 6.31 The three axes of atop , the polhode occurs rather 6.32 nutation requires a certain amount ). The discus flies along the . : Even when thrown to a consid- um axis of a top can be collinear ˝ 6.33 6.34 . It experiences a lifting force just like a wing ˛ ; the flat top degenerates to a spherical top, or the of its orbit through the air like the airfoil of an aircraft with ). It sinks to the ground more slowly than a stone would, The cone of nuta- . One spins the top carefully with its point support in the 10.10 6.29 a) We throw alike discus, a using top. the well-known Thementum direction hand axis of flip and its to figure remains startfalling branch axis fixed it coincides spinning (Fig. with thea angular fixed mo- angle of(Sect. attack and therefore flies further thancurve the in the free-fall figure). parabola – Naturally, would thethis word predict “force-free” case (dotted can be to considered apply in onlyproduces as an not approximation. only The adiscus; airflow lifting around both force, the are discus neglected but here. alsob) a The diabolo small top torque as on shown the in spinning Fig. erable height, its figure axis maintains a fixed direction in space. center of gravity, avoidingThen any its transverse axis forces remains fixed onwell in known the to space. figure many This non-physicists. axis. is – Variations: a phenomenon which is of study. But itfrequently is worth in the articles effort. onsome The physics understanding word of and what technology. it means. One shouldIn have certain cases, the angular moment with its figure axis axis of rotation oflatter case a can flat be topin realized Fig. coincides in with various its ways, figure e.g. with axis. the top – shown This Figure 6.32 Understanding the contents of this section cone. It is attached tocone the and figure rolls axis, along surrounds its thebetween surface fixed “pericycloidally”. these herpolhode two The cones line with ofof a contact the common instantaneous apex axis indicates of the rotation direction of nutation which we haveby already the seen. wire The representing the other instantaneousthe cone axis is herpolhode of cone. traced rotation; out The it is relationshipcan called between be these represented first by two a cones third cone as shown in Fig. tion Rotational Motion of Rigid Bodies 6 “Understanding the contents of this section requires a certain amount of study. But it is worth the effort.” 116 Part I 6.11 Tops Acted on by Torques. Precession of the Angular-Momentum Axis 117 Part I

Figure 6.33 The ﬂight path of a discus

Figure 6.34 A diabolo top

6.11 Tops Acted on by Torques. Precession of the Angular-Momentum Axis

After introducing the momentum p D mu, we were able to cast the equation of (progressive) motion in the following form:

d dp F D .mu/ D : (5.41) dt dt

Furthermore, for progressive motions, we had to distinguish between two limiting cases. In the ﬁrst case, the direction of the force F was parallel to the momentum p already present: Only the magnitude,but not the direction of the momentum was changed by the force (linear path). – In the second limiting case, the direction of the force was perpendicular to the previously-present momentum at every instant: Only the direction of the momentum changed (circular orbit). In a corresponding manner, we will now treat the effect of a torque M on a spinning top. We thus write the equation of motion for rotation (6.7) in the form (cf. row 11 in Table 6.1):

d dL M D . !/ D (6.15) dt dt and can again distinguish two limiting cases. In the ﬁrst, the direction of the torque vector is parallel to the direction of the angular momentum axis: Then the top experiences an angular acceleration !P . Only the magnitude of its angular momentum L is changed, but not its direction.

An arrangement which can be used for measurements is seen in Fig. 4.4 (MAXWELL’s wheel). The effective torque M is equal to FG times the radius of the shaft of the wheel. . , ) ), L L ı The 90 .The 6.15 ( ,asfar angular D It begins precession ˛ perpendicu- three is precession M M produces in every ! . It has two effects: ı already present in the M free of nutation D L 90 p D ˛ ;! ; L . t p L d d , within the plane defined by the ! ,for t Fl D D . We thus obtain . The angular momentum axis moves D M M ˇ R suffices for this purpose. One need only d of the figure axis around the spin axis on of the angular momentum axis in its mo- changes. L M around its figure axis (spinning); ; p perpendicular to the angular momentum N t ! D ˇ ! ! ˛ d 6.29 d L in the time d precession cone L sin ,d ˇ and combines with it to give a resultant angular D direction ;and Fl an additional component of angular momentum d . We find: L precessional motion of the angular momentum axis M L t 6.35 . However, it is clearer to use the arrangement shown . It consists of a top with a horizontal axis ( . The support of the gyroscope is at the center of gravity A and which acts perpendicular to the angular momentum axis ). This component is perpendicular to the original angu- M 6.35 6.35 whose central axis is vertical. nutation cone to the direction of the angular momentum torque gyroscope shift its center of gravitythe above or weight below the fulcrum point by sliding in Fig. of the whole system,provide in a the torque form of a point and socket. In order to a we hang a small weighthas on its the largest shaft magnitude, of the gyroscope . This torque A gives rise to a the angular momentum axis circles around a (here very squat) The weak nutations cancession be comes ignored; about: we explain The first constant how the torque pre- cone (Fig. time interval d First, a small nutation; and second, a very noticeable and, from Fig. The motions of a toprather complicated. with For simultaneous purposes nutation of and demonstration, onefore precession should separate are there- nutation and precessionpurpose, as we clearly usually as start possible. with For a this top which is tion around the fixed as possible. We thusangular momentum choose practically a coincide – top an whose exceptional axes case. The of top symmetry shown and in of Fig. lar top’s motion. Then the magnitude ofconstant, the only angular its momentum remains 2. the angular velocity In the second limiting case, the vector of the torque The angular momentum axis is no longer fixed in space. itself to rotate along aangular precession momentum cone axis which however is remainstation fixed the cone. in center space. The line spinning of top the is nu- now characterized by frequencies or velocities: 1. The angular velocity 3. the angular velocity through an angle d momentum in the direction vectors lar momentum Rotational Motion of Rigid Bodies 6 118 Part I 6.11 Tops Acted on by Torques. Precession of the Angular-Momentum Axis 119 Part I

Figure 6.35 Precession of a spinning top (gyroscope) under the action of a constant torque (Videos 6.10 and 6.11) (see Exercises 6.10 and 6.11) Video 6.10: “Precession of a spinning top” or, taking the directions into account by using vector notation, http://tiny.cc/wtqujy

M D !p L : (6.16) Video 6.11: “Precession of a rotating wheel” These equations are confirmed by experiment: increasing the torque http://tiny.cc/otqujy in Fig. 6.35 (heavier weight) increases the angular velocity ! of the p An analogous experiment precession. with a bicycle wheel hung When ˛<90ı, the torque Fl sin ˛ D M sin ˛ acts on the horizontal from a cord is shown. A sim- component ! sin ˛ of the angular momentum. It follows that the ple variant of this experiment angular velocity of the precession !p is independent of the angle ˛ ! is a toy top which dances around without falling over. This primitive demonstration of precession has left nutation out of consideration, as we emphasized. It is, however, sufficient for understanding many practical applications of precession. We give three examples here: 1. Riding a bicycle “no hands”. Figure 6.36 shows the front wheel of a bicycle. The rider leans a bit to the right. This produces a torque on the axle of the front wheel around the horizontal direction of motion, B. At the same time, the front wheel, acting as a top, begins to precess around the vertical axis C and leads into a right-hand curve. The line connecting the ground contact points of the front and rear wheels is again under the center of gravity of the rider; the point of support is again under the center of gravity. – The signs of all the rotations and angular momenta are shown in Fig. 6.36. A demonstration experiment which is especially graphic is provided by a small model bicycle. We give its wheels a spin with a high angular velocity by pressing them briefly against a rapidly rotating disk (Fig. 6.37) and then hold the long axis of the model suspended horizontally. A slight, cautious tilting to the right around this horizontal axis immediately brings the bicycle into position for a right-hand curve. If we set it on the ground, the bicycle moves off reliably on a straight-line path. – . ). Soon, C6.19 ). The 6.33 10.16 -axis, perpendic- C Throw a beer coaster with your ). In the case of the rotating beer coaster, (Video 6.12) 6.38 momentum is already present. The two angular momenta add L A model bicycle is “spun up” by Riding a bicycle The beer coaster as a discus. motion as indicated by the curved, feathered arrow in the figure. A non-rotating disk wouldairflow tip gives up the its disk an leadingular angular edge to momentum (cf. its along flight Fig. the pathan angular (Fig. vectorially, and the figure axis of the beer coaster carries out a precessional however, the angle offlat attack with a will slight increase;coaster climb, the will rises flight curve up up path, steeply.speed. on initially From At the its the maximum right same height, side, it time, will the fliesExplanation: fall abruptly to to The the the angular ground. smaller left momentum than and of that loses of theertia. all the beer heavier its The coaster discusa torque is with strong produced much precession its by of largeincrease the the moment and figure twist. of airflow axis, in- around causing the the angle disk of causes attack to The hoops which weresame physical once phenomena. popular toys clearly make2. use of the right hand in a horizontalit direction, will at tilting first it fly slightly like upwards. a Then good discus as an “airfoil” (Fig. pressing its wheels onto aon rapidly the rotating shaft disk of an electric motor Figure 6.36 The rider is completelybicycle is dispensable. very modest: he Hiswith or the or she automatic has her precessional simply motions role to of learn in not the riding to front interfere wheel the Figure 6.37 “no hands” , On et al. ”, , 339 (2011). 332 Rotational Motion of Rigid Bodies 6 with no hands” http://tiny.cc/vsqujy Video 6.12: “Physics of riding a bicycle Science structure of the bicycle, see e.g. J.D.G. Kooijman the Theory of the Top German original published by Teubner 1910, p. 863. For the influence of the in fact somewhat more complicated; see e.g. F. Klein and A. Sommerfeld: “ C6.19. The details of the physics of bicycle riding are 120 Part I 6.12 Precession Cone with Nutation 121

Figure 6.38 A beer coaster as a discus, thrown ﬂat with the right hand Part I

3. The boomerang. The moment of inertia of the beer coaster can be increased and the unwanted precession thereby reduced without changing its overall weight and lifting force. We need only to thicken the rim of the coaster while thinning (or cutting out) the middle part.

Take a cardboard ring of about 20 cm diameter and 4 20 mm proﬁle, and glue a sheet of writing paper onto its surface.

A ‘prepared’ beer coaster of this type with a larger moment of inertia experiences only a small precession, according to Eq. (6.16). It will also climb with an increasing angle of attack and lose speed, but at the top of its ﬂight curve, it still has a reasonable angle of attack. This allows it to glide, still rotating, back to where it was thrown. It exhibits the typical behavior of the sport article known as a boomerang. The usual curved-hook shape of the boomerang is not at all essential for its return ﬂight. However, a circular disk is not a good airfoil. An elongated oval disk with a slight curvature is a much better airfoil and a fairly good boomerang (and at the same time a non-rotationally-symmetric top). For demonstrations, one can take a piece of cardboard of about 5 12 cm in size and 0.5 mm thickness, and fold it slightly in the middle along its long axis.

Small boomerangs are best not thrown free hand; they can be laid on a tilted book with an overhang, and struck with a rod held parallel to the edge of the book. A slight tilting of this ‘flight ramp’ to the side produces a flight path that curves to the left or the right as desired, or with the return path in the same vertical plane as the outward flight. We can make the projectile swing back and forth around the vertical from its starting point, etc. Using the curved hook form and twisting the two blades like those of a propeller, we can produce still more exotic flight paths (e.g. a “corkscrew” flight) and display a number of amusing tricks.

6.12 Precession Cone with Nutation

Under suitable experimental conditions, the precession of the angular momentum axis of a top produced by the action of an external torque can lead to a well-formed precession cone. We give two examples: 1. The pendulum-top. A top is hung in a stable position, but so that it can be turned in any direction (around a “universal joint”) as shown in Fig. 6.39. It is made from a bicycle wheel (possibly with a lead- weighted rim). When it is not hanging straight down, a torque M . . The torque r ). A calculation pseudo-regular . Its required strength L Video 6.11 : increasing nutation as the angu- ) 6.40 which pulls on the lever arm center and right G F allows it to be L : The weak nutation of a suspended top, the approach to Left A top hung as a pendulum , left). The greater the angular momentum of the top, the 6.40 produced by the torque. In the position shown, the top begins L pseudo-regular precession; lar momentum of the top decreases (photo negatives) with three degrees of freedomend, (at a its small upper light bulb tracked in the photos shown in Fig. Figure 6.39 weaker is this nutation.ticeable. The nutation In can that be case, made oneThe practically refers opposite unno- to of the a precessionsion. as pseudo-regular precession In is that trueis case, suppressed. regular the This preces- weak is nutation achievedthe through caused moment certain when by initial it conditions. theis is At external opposite released, torque but one equal givesexternal to the torque; top the a the nutation which top nutationkick would which is has be given to caused a be by small in the the ‘kick’ direction when of released. the vector This d end of the top’salong figure a axis smooth circle, (the but(Fig. axle rather of along a the circle wheel) with does a wavy not contour move vector is shown in thed figure, as is the additionalto angular momentum rotate along avelocity. well-defined precession At cone the with same a time, small angular it exhibits a weak nutation: The lower Figure 6.40 acts, due to its weight can be found by trialwould and be error out (compare of place here. Rotational Motion of Rigid Bodies 6 122 Part I 6.13 A Top with Only Two Degrees of Freedom 123

Instead, we want to show how the nutation becomes stronger and stronger as the angular momentum of the top decreases, i.e. as the angular velocity around its figure axis (spinning) ‘runs down’. The Part I tip of the figure axis (axle of the bicycle wheel) follows the orbits that are photographed in Fig. 6.40. – With suitable initial conditions, the precession can even be completely suppressed. Then in spite of the external torque, only nutations survive. Again, going into detail here would take us too far afield. 2. The earth as a top. A famous example of precessional motion is provided by the earth itself. The earth is not a perfect sphere, but rather is somewhat flattened at the poles (and, precisely speaking, slightly pear-shaped). Its diameter at the equator is about 0.34 % greater than the figure axis of the earth, i.e. the line connecting the north and south poles. One can imagine that a waistline bulge has been added to a strictly spherical earth. The gravitational attraction of this bulge by the sun and the moon produces an external torque on the earth-top. Its figure axis moves along a precession cone with 1 ı an opening half-angle of 23 2 . A complete rotation around this cone takes about 26 000 years. At the same time, the torque produces extremely weak nutations. As a result, the axis of rotation deviates at each moment slightly from the earth’s figure axis. But the points at which the two axes pass through the surface of the earth are only about 10 m apart.

6.13 A Top with Only Two Degrees of Freedom2

Torques M acting perpendicular to an angular momentum axis change the direction of the angular momentum vector (precession, Eq. (6.16)). Conversely, a change in direction (rotation) of an angular momentum produces torques Mp perpendicular to the angular

2 A degree of freedom refers to a spatial dimension in which the motion of a body can take place. Examples: A point-like body (“point mass”) can in general experience a linear motion in any arbitrary direction in space. Its velocity can be decomposed into three components in a Cartesian coordinate system. The point mass thus has three degrees of freedom. – A point mass constrained to remain in a plane has only two degrees of freedom, and a point mass held to a linear track has only one. – A body of finite extension can experience rotations as well as progressive motions. Its angular velocity can in the most general case point in an arbitrary direction in space; it can then be decomposed into three perpendicular components. In addition to the three degrees of freedom of its progressive motions (translation), it has three degrees of freedom of rotation. If its axis of rotation is confined to a plane, only two degrees of freedom of rotation are present. A flywheel held by bearings has only a single rotational degree of freedom. – A body which is moving linearly and rotating can in addition exhibit oscillations of its individual parts relative to each other. A dumbbell-shaped object for example can oscillate along the axis (spring) connecting the two weights, and can simultaneously move progressively and rotate. Then in addition to the six degrees of freedom of translation and rotation, there is a seventh for the oscillation, etc. ’ p R .It M (6.17) KK indicates C is the additional L . Without the constraint t : p would adjust to the direction of ! A L D p M ). During a rotation of the millstones, they of the forced precession, and d 6.41 p ! direction as the torque produced by the weight. It shows a gymnastic bar mounted in ball bearings ). A man sits down on the seat. The center of gravity Demonstration model of a pan mill. The arrow over same 6.42 6.42 differ only in sign, so that we find M to this forced precession, a strong torque is produced. It acts on the in Fig. of the whole systemso (bar, that gyroscope, the system man)the is man’s is right. unstable well (labile). abovegyroscope; This the it tipping It bar, answers produces could with ain for a torque a precession. example on counter-clockwise tip the direction Assumethe axis to as that upper of seen end it of the from is thecome above. rotating gyroscope to moves away In the from this essential thegyroscope case, point: man. still Now The further we man awaygreater pushes from resistance the himself. than upper with end He a of feels gyroscope the practically at no rest. Nevertheless, due carries a motorized topswing (gyroscope) along and the a axis of seat. theof bar the The within page). a gyroscope The U-shaped can frame bearingscircle (in of the its in plane frame the are marked figure, with a and small white the outer frame is fixed to the bar (at ‘ The torques produced by forced precessionnology. play As a a major first example role of in amention tech- top the which is pan constrained mill, to a ato plane, form we the of Romans grain (Fig. millform which was a already top known withcase a in forced the precession.presses The the resulting millstones torque onto isgrinding the pressure. grinding in In table this a model, and thisbelow thus can the increases be made grinding the visible table using with aample spring a is pointer also impressive. attached. The following ex- Figure provided by the grinding table, the axis angular momentum that it produces within the time d the thick arrow.perpendicular Thus, to a the plane torque of which the points page must away arise. from the viewer and is the angular velocity Figure 6.41 momentum axis and to theand direction in which the axis is rotated. Rotational Motion of Rigid Bodies 6 124 Part I Exercises 125

Figure 6.42 Stabiliza- tion by negative-damped

precessional gyroscope Part I oscillations (monorail). There is a protective metal fender between the person and the gyroscope, and at the right on the gyroscope frame R is a balance weight. (Video 6.13) Video 6.13: “Stabilization using a spinning top” http://tiny.cc/5tqujy bearings of the gyroscope and thus on the bar. The bar moves back to The U-shaped frame R at- its original position. If it were initially tipped to the left, everything tached by bearings to the bar occurs similarly but with reversed signs of rotation. The upper axis of can be clearly seen at the be- the gyroscope approaches the man; he pulls it a bit closer, etc. In this ginning of the video. In the manner, one can balance with very little effort. The gyroscope swings picture: the author. with small amplitudes within the plane defined by its bearings. The man need only provide a negative damping or amplification of these precessional oscillations. That is, he must increase the amplitude already present in each case. Our brains can learn how to perform this “negative damping” as a pure reflex in a surprisingly short time. When the dimensions of the gyroscope are suitably chosen, there is no time to think before re- acting. But our nerve and muscle coordination very quickly registers the physical situation. After a few minutes, one feels just as secure on this unbalanced gymnastic bar as an experienced rider on his bicycle.

Chinese tightrope artists have long since empirically discovered this helpful device of negative-damped gyroscope oscillations. They use an umbrella which is rapidly twirled between the ﬁngers as the gyroscope; they hold the shaft of the umbrella nearly parallel to the rope and balance by tipping its axis slightly. – For the most part, however, they use the umbrellas without spinning as parachutes.

Tops or gyroscopes with only one degree of freedom can be most effectively understood by using the methods of the following chapter.

Exercises

6.1 An object of mass m1 D 15 kg is at the end of a uniformly- shaped beam with a mass of 5 kg. a) The beam is suspended from a rotatable bearing at a distance of 1/5 of its length from the object m1. What mass m2 be added at its other end to keep it in equilibrium? b) Where must the bearing be located if a mass m2 D 25 kg is hung from the other end of the beam and it is to remain in equilibrium? What is then the total force F acting on the bearing? (Sect. 6.2) 1 % has ' 18 s to ) , each at D 6.4 O is oscillat- 0 ) T m of the balance 6.5 and densities 50 cm. Calculate

around a parallel 1 and is at rest. The r ) S D . Which of the two l

would the suspension ˛ >

6.5 ? (The acceleration of 0 d and mass on the plate and ﬁnally 2 5 kg is vibrating around : T r 1 m, one end strikes the r a 0 and D D and 6.4 h ) 1 m r ) ) 6.5 6.4 6.5 20 cm falls freely in a horizontal po- and is standing on a symmetric, rotatable D must be performed in order to set a brass m ; see also Exercise 4.3) 6.4 l around the axis and W .) (Sect. 0 .) (Sect. 6.6 3 2

shows the schematic of a beam balance as a phys- ) . (Sects. l 15 cm.) b) By what distance 81 m/s 6.6 : 9 6.16 D are rolled down a ramp starting from the same initial ). Its reduced pendulum length is l 1 D g 6.15 >% A physical pendulum of mass At what point along its length would the rod in the previous Figure How much work A homogeneous thin rod of length Two full cylinders with radii A pencil of length A person of mass 2 24 s when 100 g of additional weights are added to each balance % D , then walks around a full circle of radius 6.6 a horizontal axis withearth’s a gravity; small the amplitude under axis(see the Fig. is influence located of 20 the cm from its center of gravity its moment of inertia gravity is exercise have to beperiod; suspended a) and to b) obtainwhen so the it that minimum is hung oscillation the from oscillation one end? period (Sects. would be the same as 6.5 points have to be lowered sotional that 100 the g oscillation weights period would remain with equal the to addi- a distance of T pan, as described inpans are the hung text. lie on (The a straight points line from with the which fulcrum the point balance axis which passes through its center of gravity. (Sect. 6.7 ical pendulum. a) Calculate the moment of inertia beam assuming that its oscillation period increases from conditions; the ramp has an inclination angle sphere 1 m in diametersity in of brass rotation is at 8.5 g/cm 20 revolutions/min? (The den- 6.3 cylinders will arrive first at the bottom of the ramp? (Sect. ing around its suspensionpendulum point length at one of its ends. Find its reduced 6.4 6.2 and edge of a table.velocity? How (Sect. does it continue to fall and what is its angular sition. After it has fallen a distance the plate turned? (Sect. 6.9 returns along a radial path to the center. Through what angle 6.8 plate whose axis of rotationpoint is of vertical and the passes plate. through the It center has a moment of inertia person first moves fromr the center radially outwards by a distance Rotational Motion of Rigid Bodies 6 126 Part I Exercises 127

6.10 Referring to Video 6.10, “Precession of a spinning top” (Fig. 6.35): Determine the angular velocities of precession with the small torque and the torque which is three times larger (produced by Part I attaching the small (15 g) and the large (45 g) weights, respectively), and explain the cause of the different angular velocities. (Sect. 6.11)

6.11 A spinning gyroscope whose rotation axis is horizontal is suspended on a sharp point at its center of gravity (Fig. 6.35). Under the inﬂuence of a torque M, produced by a weight hung on the axle, the gyroscope precesses at an angular velocity !P. If we consider this experiment in a frame of reference which is rotating at the same angular velocity, adding the weight in this frame of reference also produces the torque M, but the axle does not tip downwards. Why? A qualitative answer will sufﬁce. (Sects. 6.11 and 7.3) For Sect. 6.6, see also Exercise 4.3.

Electronic supplementary material The online version of this chapter (doi: 10.1007/978-3-319-40046-4_6) contains supplementary material, which is available to authorized users. .Ldr,RO ol(Eds.), Pohl word R.O. the Lüders, 2017 K. Switzerland to Publishing corresponds International which Springer © neologism a or appropriate. more name be dramatic would “weight” less A eration. pert naclrtdosre.Tercletv aei “inertial is name collective Their forces New observer. forces” accelerated phenomena. an physical to observed appear straight- the a give of to account us allow forward to an concepts within new observer requires an in frame of modifications view accelerated accel- noticeable of point to This The leads observations. physical reference direction. our of and frame magnitude the of in eration changing is velocity Their fReference of Frames Accelerated 1 are types all of vehicles general, In is reference of frame new the trivial cases, be a these can at In moving reference cases. of special frame some different a in to transition the Making unmoving. hall. indicated. and lecture clearly rigid our were as of exceptions treated floor occasional earth, the the The was or reference earth of the of frame of point Our surface the fixed from the processes of physical considered view have we now, to Forces Up Inertial Remarks. Preliminary 7.1 hywudi h etr alwihi trs.Btteeaerr and rare are these as But just rest. vehicle at is the which within situations hall exceptional place lecture the frame take in our phenomena would of in they motion All cases, the such “feel” In reference. cannot car. we of railroad vehicle, moving a the in is of or which interior ship vehicle the a a on in Experimentally, e.g. condition “smoothly”, very this direction. meet in sometimes nor can magnitude we in neither change, not its radial rmso eeec ihapure a with reference of path Frames radial pure purely cases. a a limiting with with two reference reference as of acceleration, of frames frames separately treat and to acceleration, want now we fashion, ewe w iiigcss pure a cases: limiting ha two we acceleration, between of treatment our In chapter. this of subject the is lso l id hnthey when kinds all of cles a oc,C force, gal hscoc fnm rspoe htteosre saaeo i own his of aware is observer the that presupposes name of choice This magnitude ceeain ..acag ftevlct nyi em of terms in only velocity the of change a i.e. acceleration, 1 oeo hmas aebe ie pca ae (centrifu- names special given been have also them of Some . ORIOLIS constant rol ntrso its of terms in only or oc) h ecito fteeieta forces inertial these of description The force). C7.1 eoiyrltv oteerh t eoiymust velocity Its earth. the to relative velocity olsItouto oPhysics to Introduction Pohl’s . tr moving start ahacceleration path accelerated path ecnitnl distinguished consistently ve or direction brake ceeainadapurely a and acceleration rmso reference: of frames ln tagtpath straight a along na analogous an In . O 10.1007/978-3-319-40046-4_7 DOI , fe cu.Vehi- occur. often accel- r fRltvt i o.2). Vol. (in Relativity of The- ory Special the to come we when cases such with sively exten- deal will We C7.1. 7 129

Part I ). The and con- 7.1 m = is quite a dif- F D a is lying (Fig. m rather thoroughly. We do this case rather briefly. We do this in radial acceleration No force is acting on it, since it lies on the In contrast, the cart and the man sitting on it carousel system maintains its radial acceleration constant for an arbitrarily Experiments in an acceler- Acceleration . ! 7.2 in the remaining sections of this chapter. To make the treatment clearer,in we the will following: make In use eachin case of the we rest a frame will special of first device heading!). the treat Then, earth an we or observer describe the “at the floor rest” view same of of phenomena the from an lecture the accelerated hall point (boldface observer,Both of moving observers with start the with accelerated the frame. equation of motion long time. The earth,will in particular, need is to just study such the a carousel. Thus, we ated frame of reference sider forces to be the causes of observed accelerations. Observer at rest The ball remains at rest. table without friction. are accelerated to the left. The man approaches the resting ball. Figure 7.1 7.2 Frames of Reference with Only Path First, some examples: 1. An observer istionless sitting table at top rest ontable on which top a a compensates ball moving the of cart, weightare of mass in the attached front ball. rigidly of The to a tableleft and fric- the along the its cart. chair long axis Theon (by cart a the kick!). is cart This to being causes approachof accelerated the each describing ball to this other. and process: the – the man Now we find two distinct ways are examples. Butand such the accelerations magnitude usually of lastseconds. only the We a acceleration can short is treat time thisSect. constant limiting at most for a few A frame of reference withferent matter. a Every purely carousel which is turningvelocity with a constant angular Accelerated Frames of Reference 7 130 Part I 7.2 Frames of Reference withOnlyPathAcceleration 131

Figure 7.2 Measurement of the inertial force Part I

Accelerated observer The ball is accelerated to the right. Thus, a force F Dma directed to the right must be acting on the ball. This force is called the inertial forceC7.2. C7.2. In the case of the iner- 2. The observer on the cart is holding the ball using a force meter tial force F Dma, a is the (Fig. 7.2). The cart is again accelerated to the left. During the ac- acceleration of the frame of celeration, the observer on the cart feels a force on his hand and arm reference. muscles. The force meter indicates the magnitude of the force F.

Observer at rest The ball is accelerated to the left. AforceF directed towards the left acts on it. For the acceleration, we have a D F=m.

Accelerated observer The ball remains at rest. It is not accelerated. Then the sum of the two forces acting on it is zero. The inertial force F Dma, acting to the right, and the muscular force acting to the left are equal and opposite. Their magnitude can be read off the force meter. 3. In Fig. 7.3, a cart is being accelerated to the left. The observer standing on the cart has to lean into the acceleration while the cart is starting up; otherwise, he would fall over. – Again we give two distinct descriptions:

Observer at rest The center of gravity S of the man must move with the same acceleration (magnitude and direction) as the cart. The force F (Fig. 7.4) required to accelerate the center of gravity, acting to the left, is produced by the man through his weight FG and an elastic deformation of the cart (force F3). He leans forward to achieve this.

Figure 7.3 A man on a cart which is accelerating to the left 1 F .It 3 .Im- F m ,sothat a . m pulls down- m . The balance G /= D 1 F F . The sum of the F rest G . The deformation of F m . pushes it upwards. The downwards Two unequal, oppositely- D of the balance is smaller 1 acts to the rear, that is to the pulls the object downwards, a F 1 a G F m F . 1 . F D 3 F F ). The balance thus indicates no weight. of the man remains at the sum of the forces acting upon it is zero. S 7.6 of the object. Therefore, a second upwardly- G , with the magnitude 0. F m ) acting on him is zero. His weight /= D 1 7.5 F F Forces in the accelerated system Forces as observed in the system at rest to zero (Fig C C 1 G G F F F . C . Then the elevator begins to accelerate D G G right in the ﬁgure. The two combine to give the resultant force deforms the cart below the man’s feet and thus produces the force Accelerated observer The object is atThe rest, upwards-directed spring force Observer at rest The object isdirected accelerated forces act downwards. on it. The weight 4. An observera is spring balance in holding anthe an spring elevator. object indicates a of force In mass whichF front is equal of but opposite him to is the weight a table with wards, while the inertial force resultant differencea force produces its downwards acceleration while the smaller force of the spring than the weight Figure 7.4 directed force is present, namely the inertial force now indicates a smaller reading, which is opposite but equal to Figure 7.5 Accelerated observer The center of gravity forces (Fig. 5. An observer jumps from abalance high in table his to hand. the groundmediately On with after the the he balance spring jumps, pan the is indicationvalue of a the weight balance of goes mass from the F Accelerated Frames of Reference 7 132 Part I 7.3 Frames of Reference with Radial Acceleration. Centrifugal and CORIOLIS Forces 133

Figure 7.6 A freely-falling frame of reference (Video 7.1) Video 7.1:

“Freely falling frame of Part I reference” http://tiny.cc/1uqujy In the first part, S. KÖSTER (doctorate 2007) jumps from a table holding the balance; filmed by T. BECKER (doctorate 2004). The second part shows a similar jump Unfortunately, this demonstration is over within a fraction of a second. by H. GRÜNDIG (1929– This disadvantage is not present in the four accelerations dealt with in 2003, doctorate in Göttingen, C7.3 Sect. 7.3 . 1959), filmed in slow motion. This part was filmed in 1960. Observer at rest The object is accelerated. It falls like the man with the acceleration C7.3. POHL mentions here of gravity g. The only force acting on the object is its weight FG, experiments with “zero grav- which is pulling it downwards. Neither a spring force nor a muscular ity” in space stations. He also force is pushing it upwards. points out the microgravity experiments in the drop tower Accelerated observer at the University of Bremen. The object is at rest, the sum of the forces acting on it is zero. Its In an evacuated tube, 120 m long, an experiment capsule weight FG which is pulling it downwards is cancelled by the inertial force F acting upwards. Their magnitudes mg are equal. The object (0.8 m in diameter, 1.6 to is “weightless”. 2.4 m long) can fall freely for nearly 5 s before it is braked These examples should suffice to elucidate the meaning of the words to a stop by a plastic foam ‘inertial force’. Inertial forces exist only for an accelerated observer. cushion (see e.g. http://en. The observer must – at least in thought! – participate in the accelera- wikipedia.org/wiki/Fallturm_ tion of his frame of reference. A hand which is accelerating a bowling Bremen or https://www.zarm. ball is an accelerated frame of reference. Therefore, the hand feels uni-bremen.de/drop-tower. an inertial force. html).

7.3 Frames of Reference with Radial Acceleration. Centrifugal and CORIOLIS Forces

1. An observer is sitting on a rotating swivel chair with a vertical axis of rotation and a large moment of inertia (Fig. 7.7,cf.alsoFig.7.16). In front of him, attached to the chair, is a horizontal, smooth tabletop. The observer sitting in the chair places a ball on this tabletop. It ﬂies off the table and falls to the ground.

Observer at rest The ball is not accelerated. No forces are acting on it (its weight is compensated by the elastic resistance of the tabletop). As a result, , r 2 ac- ! m acts on ); it is which acts r D ). It swings r 2 F 7.8 ! m It moves away from inertial force D . F r (radius vector 2 r ! m angular velocity of the rotating D D F increases with increasing rotational while the chair is rotating. . This force is given the special name ! r ˛ ( 2 r ! within the plane deﬁned by the radius and ! It is not accelerated. Therefore, the sum m at rest ˛ D D u F . r . This requires the force 2 . Its formula is distance from the ball to the axis of rotation at the ! D m r Experimenting in a rotating frame of . This requires the presence of a radial force C7.4 it is accelerated ; The ball of ther pendulum is moving along a circular orbit of radius the axis of rotation. Thefrequency angle of the chair. Observer at rest The ball remains atof rest. the two forcesis acting pulling on radially it outwardsinwards is and (centripetal force) the zero. are equal muscular and Thethese force opposite. forces centrifugal pulling The is magnitude radially force of which directed towards the axistripetal of force”), Eq. rotation, (4.1). which acts on the ball (“cen- Accelerated observer 3. The observera on stand his on swivel thea chair table cord. hangs in This a front pendulum gravityoutward of will through pendulum an not him, angle on hang for vertically example (Fig. a ball hung from 2. The observer onhand the and swivel chair the holds ball. a Thetowards force horizontal the meter axis between axis of his this ofa force rotation force meter of of is magnitude directed the chair. The force meter indicates Figure 7.7 reference Observer at rest The ball follows a circular path of radius it cannot move alongthe a constant circular velocity path.swivel chair, It ﬂies off tangentially with celerated ‘centrifugal force’ moment when it is placed on the tabletop). Accelerated observer The ball is accelerated fromthe its center resting position. of rotation ofthe the ball, which tabletop. is Thus, initially an Accelerated Frames of Reference 7 author. C7.4. In the picture: The 134 Part I 7.3 Frames of Reference with Radial Acceleration. Centrifugal and CORIOLIS Forces 135

Figure 7.8 The effect of centrifugal force on a gravity pendulum Part I

Figure 7.9 Forces in the system at rest

Figure 7.10 Forces in the rotating system

horizontally towards the axis of rotation (Fig. 7.9). It is produced by the weight of the ball FG and the elastic tension of the cord (force F3).

Accelerated observer The ball of the pendulum is at rest, the sum of the forces which act at its center of gravity S is zero (Fig. 7.10). The weight of the ball 2 pulls downwards with the force FG, the centrifugal force F D m! r pulls outwards. The two combine to give a resultant force F3.This force applies a tension to the cord and thereby produces the force F1, which is equal and opposite to F3. 4. An artiﬁcial satellite (a space station) circles the earth at an altitude of several 100 km (Fig. 4.25). Inside it, all the phenomena which normally result from the force that we call ‘weight’ are absent. Here (in contrast to the falling path acceleration in Fig. 7.6), we have as much time as we wish to observe the lack of such forces. If, for example, we put a metal block onto a spring balance, the balance will indicate no weight.

Observer at rest Like the satellite itself, all objects within it are accelerated continually towards the center of the earth, i.e. in the direction of the radius of the satellite’s orbit. Otherwise, they could not take part in the circular orbital motion. The weight FG D mg of the metal block produces .) A 2 D u 8.9 m/s D .First,wefire A 2 m, the deflection to : relative to the rotating 1 . Now, a second shot is D on the target disk. The tar- A ). a moving 7.12 ). The direction of its barrel can at an altitude of 300 km o forces are equal and opposite; they 7.11 . Then the chair is set in rotation with g a .( with the radius connecting it to the axis of mg The chair is supposed always to rotate in . As a result, the sum of the forces acting on ˛ 7.5 cm (cf. Fig. . D ! at rest where it hits the target is deflected to the right by ). The gun points at a target located a distance ORIOLIS b 075 m : 0 7.12 The C relative to the targeted point, which has in the meantime D . 0 in the rotating system. We simply asked the question as to s s a C7.5 at rest 60 m/s (air gun). Thethe distance right to the target is Numerical Example The chair rotates once every 2 seconds. The velocity of the bullet is from its muzzle and is aimedget at the rotates point with the chair, held by rods at the distance make an arbitrary angle rotation (Fig. a bullet from the gunit hits while the the target, chair i.e. is the at point rest, and determine where 5. In all the experimentswas up to now, we havewhether observed the a body body would which benot. accelerated Now, out we of observe this a resting bodyswivel position which chair. or is Here, wenamely restrict a our body with considerations a to highcentrifugal velocity, a a force bullet. limiting which Then case, is we relatively can small. neglect the On the tabletop attacheding to horizontally outwards the (Fig. chair, we set up a small gun point- the angular velocity a counter-clockwise sense as seen fromfired. above The point a distance Figure 7.11 force towards the center offorce the which earth, pulls is itsatellite’s circular away compensated orbit). from by The the tw thehave earth’s the centrifugal same center magnitude (the center of the this acceleration. It is thenot only be force compensated which by acts the on oppositely-directed thespring. force block of and a need compressed Accelerated observer The metal block is the block is zero. The force that we call weight, which pulls the block moved to Accelerated Frames of Reference 7 C7.5. In the picture: The author. 136 Part I 7.3 Frames of Reference with Radial Acceleration. Centrifugal and CORIOLIS Forces 137 Part I

Figure 7.12 The experiment of Fig. 7.11 seen from above. v0 D v cos ˛ D .R A/! is the component of the bullet’s velocity w which is parallel to the target disk, while v is the velocity of the muzzle of the gun. For clarity, the angle !t is exaggerated in the drawing. This causes a minor error: the sight line from the gun to a0 appears not to meet the target disk at right angles.

Observer at rest When the chair is at rest, the bullet strikes the point a that it was aimed at. If the chair is stopped immediately after firing the bullet, the point b where it strikes the target is to the left of the targeted point. In this case, the velocity v of the muzzle of the gun adds to the velocity u of the bullet. As a result, the bullet flies in the direction w through the lecture hall. In the experiment as in fact demonstrated, the chair continues to rotate following the firing of the bullet. The bullet, in contrast, flies on a force-free, linear path after leaving the muzzle of the gun, in the direction w through the lecture hall. Therefore, the sighting line rotates relative to the path of the bullet. At the end of the bullet’s flight time t, the point aimed at is at a0. Thus the point b where the bullet strikes the target disk is deflected to the right by a distance s relative to the aiming point. From Fig. 7.12, we read off the relation

s D A!t:

For both ﬂight paths (that is in the directions of u and w), the ﬂight time of the bullet to the target is the same, namely

t D A=u: C7.6. For a simple derivation of the CORIOLIS accelera- Therefore, we find tion, let the muzzle of the s D u!.t/2: gun be on the axis of rotation and the moment of firing be t0 D 0. Then the distance Accelerated observer s D R ! t (Fig. 7.12). With During the flight of the bullet, it is accelerated in a transverse direc- R D ut,wehave tion. Its path is thus curved to the right. Within its flight time t,the s D u ! t2 , 1 . /2 bullet is deflected by the distance s D 2 a t to the right. s, accord- from which we obtain by ing to the result of the observer at rest, is s D u!.t/2.Fromthis, differentiating twice with we find for the observed acceleration a D 2u!. It is named for its respect to time: 2 discoverer, CORIOLIS accelerationC7.6. There can be no acceleration d s ! a D dt2 D 2u . acts of ! u (7.1) Then the mu ). It draws 2 . n (the aiming D 7.8 F force which is accepted by both : which acts on an object in u ! en initial directio ORIOLIS != 2 ). To make it easier to observe, A u . In a second experiment, the pen- D m 7.13 2 2 t 7.14 acts on the moving body perpendicular D ! inertial force u ! F ; thus, the C Within this system, a body is moving with -vertical(!) rest position (Fig. D . mu t . ma Let a frame of reference be rotating, e.g. with 2 ! non ! (Video 7.2) D D 7.14 A . The observer in the chair is initially holding the F ! D C7.7 force is an s A simple pendulum in a rotating force F body within an accelerated frame of reference, but only for . It acts perpendicularly to the vectors of the angular velocity ORIOLIS a bullet. The equation observers provides a very simple method for measuring the velocity without a force ORIOLIS moving an path velocity u perpendicular to the axis of rotation motion and the path velocity. the angular velocity C the pendulum bob is released,amplitude it around its swings with a gradually decreasing pendulum bob and plugging thetipped ink out jet. of The its cord rest of position the within pendulum an is arbitrary vertical plane. When The C a continuous curve on the paper withorbit the ink shown jet, at tracing out the the left rosette dulum in is Fig. kicked out of itson rest the position, giving right the in rosette Fig. orbit shown Figure 7.13 a single initial directiontion of should its be independent motion. of theof The chos the magnitude gun). of But itsshown we deflec- much intentionally more did quickly not and simply demonstratein by this; the introducing experiment: it a We can small replace change be thelum. gun The by pendulum the is bob hung ofchair above a the in gravity tabletop pendu- the attached to usual thethe manner swivel (Fig. moving pendulum bobthis itself end, is a used small toink ink record jet pot its underneath. is own built A path.below into sheet the the of To pendulum pendulum white and bob, paperangular the with is velocity swivel a pinned chair fine to is the set tabletop in rotation with the 6. The previousa example demonstrates the transverse deflection of a frame of reference on the bullet transversely to its flight path, or, more generally, to its trajectory. PERBER Accelerated Frames of Reference 7 http://tiny.cc/euqujy Video 7.2: “Simple pendulum in a rotating reference frame” C7.7. In the picture: Master mechanic W. S 138 Part I 7.3 Frames of Reference with Radial Acceleration. Centrifugal and CORIOLIS Forces 139

Observer at rest The pendulum swings around its rest position. In the case of

Fig. 7.14, left, the orbit is an ellipse, since on releasing the pendulum Part I bob from its deflected position, a tangential velocity component is also present. It swings with an “elliptical polarization” on an orbit fixed in space. The plane of the paper rotates under the swinging pendulum, producing the rosette, whose center remains free of ink dots. In the second case (Fig. 7.14, right), there is no tangential velocity component at the moment the pendulum starts to swing. It swings with a “linear polarization” continually parallel within a vertical plane fixed in space. The fact that the rest position of the pendulum is not vertical was explained above under Point 3.

Accelerated observer During the motion, the pendulum bob is deﬂected by a CORIOLIS force continually in a direction transverse to the plane in which it is swinging. All the individual curves of the rosette have the same form in spite of their different orientations in the rotating chair. Therefore, the direction of motion within the accelerated system is unimportant for the magnitude of the CORIOLIS force. The deﬂection of the rest position of the pendulum from the vertical results from a centrifugal force (cf. Point 3!). Both a CORIOLIS force and a centrifugal force act on a moving body in a rotating frame of reference.

Figure 7.14 The rosette orbits of a pendulum on a carousel. In the left image, the pendulum was released from its maximum deflection above the ink dots at bottom, and it initially swung to the right (arrow). The endpoint of the rosette is coincidentally the same as its starting point. In the right image, the pendulum was given a kick out of its initial rest position (Video 7.2). A to M , like of the A at first . F (poles). The perpendicular ı 1 M component of the 1 M meridian plane shows a top mounted , left). Then he leaves circle of latitude 7.15 7.15 lies in a (equator) and 90 (‘latitude angle’) with the plane oscillations. The axis of the top A ı ' , so that it can no longer produce (Video 7.3) F can be adjusted to different latitudes. . Seen from above, it is assumed to causes a precession of the top’s figure A 1 , right). of the frame. The figure axis M globe A which is perpendicular to the symmetry axis 7.15 A of the top. The axis of the top: it lies parallel to a F . Otherwise, we could not have followed the position of the Model of a gyrocompass F ! axis. This torque axis. The observer in the rotating chair starts the top spinning around the axis A A F (In this demonstration,velocity the swivel chairplane was in given which only the pendulum a bob small is swinging.) angular to the horizon of the position of theto top the can be imagined toby be perpendicular pulling a few times on its spokes (Fig. a compass needle (Fig. the top to itself: Thethe figure meridian axis of plane the after top a takes few up a initial fixed swings position around in the axis torque lies along the figure axis axis swings out of the meridian plane;axis but quickly friction in damps the these bearings pendulum comes of the to rest in the meridian plane. Then the Both observers, for simplicity, assume thefigure same axis initial position of the Observer at rest The rotation around the chair’s (oract globe’s) axis on causes the a figure axis torque of the top. It has a component Now we let the two observers speak their piece: Figure 7.15 refer to the chairrotate in as a the counter-clockwise sense.rotate around The an frame axis of the top can itself 7. A spinningrocompass top in on a a rotating rotatingwithin frame a globe): of frame reference (gyroscope) Figure on (model the of swivel a chair. gy- For brevity, we will further precessional motions. It can thus make anof arbitrary angle rotation of the globe, between 0 (figure axis) globe. Furthermore, the axis Accelerated Frames of Reference 7 Video 7.3: “Gyrocompass” http://tiny.cc/ququjy 140 Part I 7.4 Vehicles as Accelerated Frames of Reference 141

Accelerated observer CORIOLIS forces deflect the parts of the top’s rim at ˇ to the right along their orbits. The half of the top on the right as seen by the Part I reader moves out of the plane of the page towards the reader. This moves the figure axis of the top into the meridian plane. Thereafter, CORIOLIS forces indeed continue to act on the moving rim, but they no longer exert a torque around the A-axis. So much for experiments to demonstrate the definitions of the concepts of centrifugal force and CORIOLIS force. Both forces exist only for a radially-accelerated observer. The observer must participate in the rotation of his frame of reference, at least in thought. He or she can still continue to use the equation a D F=m within the rotating frame by taking these new forces into account. The appearance or disappearance of inertial forces is thus determined by the choice of the frame of reference. For observers who move with the frame of reference, they are no less “real” than “genuine” forces are for the observer at rest. The term “fictitious forces” should therefore be avoided. What is the situation for the accelerated observer with regard to the principle of “actio D reactio”? – Answer: He or she experiences the same situation as an observer on the earth regarding the reaction force to the weight. The observer in an accelerated frame of reference cannot detect the corresponding counter-forces to the inertial forces during the free motion of bodies within that frame.

7.4 Vehicles as Accelerated Frames of Reference

The choice between an accelerated and a non-accelerated frame of reference is simply a matter of taste in some cases, for example with circular motions of bodies around fixed axes. The important thing then is just to give a clear description of the frame of reference used (cf. Sect. 4.2, beginning). – In other cases, however, the accelerated frame of reference is preferable. These include most of the physics of the vehicles used in our technological society. The acceleration of these frames of reference is often rather complicated, since path accelerations (starting up and braking) and transverse accelerations (driving around curves) occur together. Our everyday experience with inertial forces in vehicles was already treated in the examples given in Sects. 7.2 and 7.3. For example: a) Leaning in a train when it is starting up or braking, and when it is travelling around curves. This prevents falling over. b) Leaning into curves by bicycle and rider, rider and horse, aircraft and pilot. D D G ! 25 N; 2m/s; , horizon- F 1 D D 2 acceleration u 5s : ). The result 0 D 7.16 radial 25 kg m/s shows a carousel in force on the deck of D 1 2 kg; velocity 7.17 D 14 s : 3 m ORIOLIS . Figure of the carousel, the torsion pendu- 2m/s forces especially clearly on a swivel 1 ! ) 2kg 7.13 2 , D ORIOLIS 7.11 ! , mu 2 7.7 20 N)! D forces. The heavy ; mass of the metal block Exercise 7.1 ( 1 2 . (Video 7.4) force A swivel chair with A torsion pendulum 14 s C7.8 : 6.5 3 81 m/s : ORIOLIS 9 along an arbitrary straight-line path (Fig. C7.9 D ORIOLIS 7.15 C Numerical Example Rotation rate: one revolution2 every 2 seconds, so that this force is greater2kg than the weight of the moving metal block ( a side view.which On can it be isof placed a the at torsion carousel. pendulum anycontinually with desired Under point a its what distance long rod-shaped conditions axis fromdependently mass will towards of the the the all center center torsion accelerations? of the pendulum axis carousel, in- At a constant angular velocity lum remains at rest. This is because the purely chair with a large momentangular velocity. of Try inertia to and move a thereforequickly weighted a block steady, (e.g. constant with 2 kg mass) a ship which isone changing walk a course. straight line Only in by the intendedd) tacking direction. One “sideways” can can “feel” the C a large moment of inertiastrating for C demon- on a carousel (rotating table).pendulum The consists of a wooden rod (pendulum mass) on thetorsion small axle that we have already seen in Fig. be conveniently used for thements experi- shown in Figs. weights hanging at the sides can also Figure 7.17 tal torsion pendulum on a carousel Figure 7.16 The list of qualitativeexamining examples them is all very in long.a detail, More particular however, is instructive case a than which quantitative at treatment of first seems weird. It involves a c) The sideways deflection by the C is startling.current It feels within as aexperiment though viscous your liquid. arm has This gotten into is a a strong particularly important and . ABEL force. We how- Accelerated Frames of Reference 7 ORIOLIS “Torsional pendulum on a rotating table” http://tiny.cc/cuqujy Video 7.4: C as a demonstration, since only the experimenter him- or herself can “feel” the C7.8. This “particularly important experiment” can unfortunately not be shown C7.9. In the picture: Master mechanic W. N yourself on various carnival or amusement-park rides. ever want to point outnumerous the possibilities of ex- periencing this phenomenon 142 Part I 7.4 Vehicles as Accelerated Frames of Reference 143

(centrifugal acceleration) of the rotating carousel points precisely along the long axis of the pendulum mass. Such an acceleration can never produce a torque. Part I

To verify this, one could mount the pendulum on a rail so that it can be slid in a radial direction, orienting its long axis parallel to the rail. The pendulum will then not react to any accelerations along the rail.

However, any change in the angular velocity !1,thatisanyangular acceleration !P 1 of the carousel, will shift the pendulum out of its rest position. It will immediately begin to oscillate with a considerable amplitude, since now the acceleration a is perpendicular to the long axis of the pendulum mass. The task outlined above may at ﬁrst seem hopeless; but on the contrary, it is in fact quite simple. We can make the torsion pendulum completely insensitive to any angular acceleration !P 1 simply through a suitable choice of its moment of inertia 0! We require that (derivation follows immediately):

0 D msR (7.2) or, using STEINER’s law (Eq. (6.12)), we obtain an expression which is more convenient for calculations:

2 S D m.sR s /: (7.3)

0 D moment of inertia of the pendulum referred to its axis of rotation; S D moment of inertia referred to its center of gravity; m D mass of the pendulum (rod); s D distance from the pendulum’s center of gravity to its axis of rotation; R D distance from the pendulum’s axis to the axis of rotation of the carousel.

The torsion coefﬁcient D of the spiral spring of the torsion pendulum is completely unimportant. It does not enter into the calculation at all. The silhouette (Fig. 7.17) shows a pendulum mass in the form of a rod which corresponds to this calculation (its dimensions are given below). This pendulum indeed remains at rest no matter how strong the angular acceleration of the carousel. This demonstration is quite surprising. Small variations of R or s restore the original sensitivity of the pendulum to angular accelerations. To derive Eq. (7.3), we consider the situation from the viewpoint of a rest frame, see Fig. 7.18. When the carousel is accelerated, a force acts on the axis of rotation O of the torsion pendulum; it has a magnitude F (direction 1). We add to it two forces of the same magnitude which act oppositely at the center of gravity S (directions 2 and 3). The force in the direction 3 accelerates the center of gravity S.We have F D ma D m.R s/ !P 1 (7.4)

.!P 1 D angular acceleration of the carousel/:

At the same time, the forces in the directions 1 and 2 produce a torque F s. It causes a rotation around the center of gravity S,justasany ) ) 7.3 ( (7.5) 6.11 ( )andﬁnd Numerical 7.5 /: .– s / s /: )and( R . R . s 7.4 52 cm. ms 12 D D l D 2 S : P 2 2 ! l

S ml 5cm,

1 12 D or s D D ) yields s S s acceleration of the object S 7.3

50 cm, F . This condition can be fulfilled by a suitable D ,wefind l 2 between the center of gravity of the pendulum D angular R s ! / : s D DP 2 P 7.18 ! 1 F . P R ! . : and length demonstration plays a role in transportation systems s m ). m The insensitivity of a torsion pendu- for Fig. 7.5 Bob in Accelerated Vehicles O example For the pendulum bodywith mass (rod) chosen for the demonstration experiment, Inserting this value into Eq. ( a relation for and its axis of rotation. We combine Eqns. ( choice of the distance Figure 7.18 (cf. Sect. lum towards angular accelerations ofrotation its center of kinesthetic sense leave him clueless. Theyof tell the him only resultant the of direction weight and centrifugal force, but never the true This strange 7.5 The Gravity Pendulum as a Plumb Navigation of an aircraftmands a without secure knowledge visual of the contact trueof vertical direction to the at all horizontal the times, plane or ground perpendiculara to de- pilot it. lacking Without thisflight visual knowledge, path information from can’t a even straight distinguish one; a the curved tension of his muscles and his torque does when it actswe on find an otherwise free object. Quantitatively, We require that Accelerated Frames of Reference 7 “This strange demon- transportation systems”. stration plays a role in 144 Part I 7.5 The Gravity Pendulum as a Plumb Bob in Accelerated Vehicles 145 vertical direction, which is identical to the local radius vector of the earth. At rest on the ground, one can use a gravity pendulum as a plumb bob Part I to find the vertical direction. In accelerated vehicles, this would at first seem useless; we have all watched a pendulum in an accelerated vehicle. Think of a strap hanging from the baggage rack in a railroad car. It swings back and forth, at the mercy of inertial forces. Never- theless, one can in principle use a pendulum as a plumb bob even in arbitrarily accelerated vehicles! We can see this by first carrying out a thought experiment. To this end, we modify the experiment described in Fig. 7.17 so that the center of rotation does not move on a fixed circle, but instead on the surface of a sphere of radius R. Every motion of the center point O thus takes place momentarily on a great circle, and in our thought experiment, we replace the circle from Fig. 7.17 momentarily by a great circle (with the torsion axis perpendicular to it). When Eq. (7.3) is fulfilled in this arrangement, the pendulum points continually towards the center of the sphere, even when the direction of the great circle changes in the course of the motion. Now, in order to carry out this thought experiment in fact, we move the point O into the vehicle and attach a physical gravity pendulum as in Fig. 6.15 at this point; it is suspended at only a single point, so that it can swing in the horizontal plane. Every motion of the vehicle now takes place momentarily on a great circle whose radius R is the earth’s radius (D 6:4 106 m). The gravity pendulum can move freely in the plane of this great circle. To keep it at rest, we have only to ensure that Eq. (7.3) is fulfilled. Then the pendulum points continually towards the center of the earth, even when the vehicle drives or flies through a sharp curve! We have thus achieved our goal. For a gravity pendulum, in contrast to an elastic torsion pendulum (mass-and-spiral-spring system), the moment of inertia is directly related to the torsion coefficient D. The choice of D is no longer free; it is determined by the weight of the pendulum mass, mg. According to Sect. 6.5,wehave

D D mgs .g D 9:81 m/s2/:

As a result, the period of oscillation of this pendulum, from Eq. (6.13), is r s s msR R T D 2 0 D 2 D 2 D 84:6min: (7.6) D mgs g

It is called the SCHULER period, corresponding to a mathematical pendulum (Sect. 6.5) whose length is equal to the earth’s radius R ! (Exercise 7.5). For a physical pendulum of dimensions suitable for installing in an airplane, that is of order of 0.1 m, a period of this magnitude would require a distance of s D 1 nm (Exercise 7.5), which D r (7.7) (7.8) ). 7.19 . Then the 2 (Fig. : frequency are in- ' 2 m/s : ' 1 s m/sec 5 . ' CHULER 03 cos 2 : 10 circle of latitude 0 , and the corresponding cen- c C7.10 Š/: 3 : D 7 ma be used as plumb bobs. This is an 03 cos ' : 0 D D cos D F R rounded off ' 2 . 2 ! produces a centrifugal force on every body on the cos 86 164 s D c . ' ! c r a D 2 a ! D ! R D a c a takes 86 164 s. The angular velocity of the globe is thus Gravitational atof Reference. Centrifugal Acceleration of Bodies at Rest of the earth, only one component of this centrifugal acceler- 2 is the radius of the associated R ' D ı cos This angular velocity at rest on the earth’s surface, dependent of accelerations and can important fact for “inertial navigation” trifugal acceleration traction and centrifugal force as a function of the latitude earth’s surface Figure 7.19 Consider a body located atR the geographic latitude centrifugal acceleration is This centrifugal acceleration isdius directed of outwards, the circle parallel of to latitude.radius the In ra- the vertical, i.e. in the direction of the As our ﬁnal“Earth accelerated Carousel”. frame We ofthe now earth reference, take relative to we into the360 account will system of the consider ﬁxed daily stars. the rotation A complete of rotation by is practically impossible to construct.which are But excited even to physical oscillations pendulums with the S 7.6 Earth as an Accelerated Frame ation acts, namely small; it has the value oduc- Accelerated Frames of Reference 7 tion to Avionics Systems”, Kluwer Academic Press, Boston, 2nd ed. (2003), Chaps. 5 and 6. C7.10. See for example R. P.G. Collinson, “Intr 146 Part I 7.6 Earth as an Accelerated Frame of Reference. Centrifugal Acceleration of Bodies at Rest 147

It is directed outwards from the center of the earth and is thus opposite to the acceleration of gravity g which results from gravitational attraction. On the rotating earth, therefore, the acceleration of gravity Part I at the geographic latitude ' is somewhat smaller than it would be if the earth were at rest. We ﬁnd

2 2 g' D g 0:03 cos ' m/s : (7.9)

Here, g, the value of the acceleration of gravity, applies to the earth at rest. Now we encounter a difficulty: The centrifugal force acts by no means only on bodies on the surface of the earth; indeed, every particle of the earth itself is subject to a centrifugal force, directed radially outwards from its circle of latitude. The sum of all these forces produces an elastic deformation of the earth; the globe is somewhat flattened at the poles, where its axis is about 1/300 shorter than its diameter at the equator. As a result of this polar flattening of the earth, the change in the acceleration of gravity g’ at the latitude ' is somewhat greater than one would calculate from Eq. (7.9). The experimentally-measured acceleration of gravity is described by the expression 2 2 g' D .9:832 0:052 cos '/m/s : (7.10) At sea level and a latitude of 45ı, one finds g D 9:806 m/s2.For ' D 0ı, i.e. on the equator, the correction term is maximal; it is then 0.5 %. This small correction can be neglected in many experiments. But a precise pendulum clock at the equator in fact loses about 3.5 minutes per day as compared to a similar clock at the poles. The polar flattening of about 1/300 mentioned above holds for the solid body of the earth. The deformation of its liquid sheath, the oceans, due to centrifugal force is much greater. But this latter deformation never occurs alone; it is combined with the diurnally and periodically varying attraction of the oceans’ water by the sun and the moon. The hydrosphere is thus deformed much more strongly by these tidal forces than is the solid body of the earth. The superposition of centrifugal forces and gravitational tidal forces yields the complicated phenomenon of the tides. It is an example of “forced oscillations” (Sect. 11.10). We can give only a rough sketch here. Similar conclusions hold for our atmosphere, the ‘ocean of air’. The tides in the atmosphere cause only small pressure variations at its bottom, on the surface of the earth, just like the ocean tides at the bottom of the oceans. But around 100 km above the surface, the air tides cause motions of the air of the order of kilometers! Up there, the tidal waves of the air are much higher than those of the water on the surface of the oceans. (7.12) (7.11) on the ' ORIOLIS component per hour. In component ı of the earth was 24 360 0 ! ' vertical . Its principle was Force . sin 1 horizontal s D 5 ˛ ' ': 10 s pendulum ’ sin cos 3 0 : ORIOLIS 0 7 ! of the earth, the ! . ı R D using a pendulum on a rotating chair: D D shows a reliable setup. Its essential 12 0 v h acceleration . The angular velocity ! ! of moving bodies. The most well-known 7.3 OUCAULT ! bodies. We start with the influence of the 7.3 7.21 ˛ . In the northern hemisphere, it always leads ), ;itis v ı ! 5 ORIOLIS : 7.6 moving 51 indicates his/her horizontal plane. At this location, ). to the right D H , an observer is shown at a location of latitude The two components of the ' 7.14 of Reference. C on Moving Bodies 7.20 H computed in Sect. Göttingen ( to a deviation example is provided by F already explained in Sect. In Fig. These two components of the angular velocity produce C (Video 7.5) The experimentallecture demonstration room. presents no Figure difficulties in any angular velocity of the earthfor on determining its the surface, C Figure 7.20 accelerations of vertical component a counter-clockwise sense. Itour thus swivel has chair in the Sect. same sense of rotation as 7.7 Earth as an Accelerated Frame For an observer looking down on the north pole, the earth rotates in and one parallel to the horizontal plane, the The pendulum followed a rosetteright (Fig. orbit, which always curved to the A corresponding rosette path isdulum, followed consisting by of a every cord long and gravityearth. a pen- weighted The ball, end on points therest surface of of point the the of rosette the move forward, pendulum, as by seen an from angle the earth. the angular velocity of the earthnents, can one be parallel decomposed to into the two radius compo- ’s pendulum” Accelerated Frames of Reference OUCAULT 7 “F Video 7.5: http://tiny.cc/luqujy 148 Part I 7.7 Earth as an Accelerated Frame of Reference. CORIOLIS Force on Moving Bodies 149 Part I

Figure 7.21 FOUCAULT’s pendulum experiment (Video 7.5) component is a good astronomical objective lens. The lens projects a strongly enlarged image of the cord of the pendulum at the turning points of the rosette loops. The ﬁgure contains the necessary numerical data. With the dimensions chosen, one can see the loops of the rosettes with their turning points spaced at around 2 cm in the enlarged image. Thus, in a single cycle of the pendulum’s swings, the earth’s rotation around its axis can be veriﬁed! (Exercise 7.6).

Still more transparent, but unfortunately more difﬁcult to demonstrate, is an experiment to detect the earth’s rotation carried out by J. G. HAGEN, S. J.C7.11. In Fig. 7.22, we explain this experiment using a swivel chair. C7.11. J.G. Hagen: “La An axle R held at a tilted angle carries a dumbbell-shaped object with rotation de la terre, ses

a moment of inertia 1. (We can initially ignore the spiral spring on the preuves mécaniques an- torsion axle.) The object is at rest on the rotating chair, and thus it has ciennes et nouvelles”, Ti- an angular velocity of !0 sin '. When the string F is burnt through, two helical springs S pull the dumbbell weights close to the axle R and thereby pograﬁa Poliglotta Vaticana, decrease the moment of inertia to the value 2. During this motion, the Roma 1912. J.G. HAGEN, two weights experience a CORIOLIS acceleration and are deﬂected to the S.J. (1847–1930) was direc- right. This causes the dumbbell to begin moving; it rotates relative to the tor of the Vatican Observa- ! ! swivel chair with an angular velocity 2. The magnitude of 2 can be tory from 1907–1930. (The calculated for an observer at rest relative to the ground by making use of letters S. J. (Societa Jesu) the conservation of angular momentum. We require indicate that he was a Jesuit.) 1!0 sin ' D 2.!0 sin ' C !2/

or 1 2 !2 D !0 sin ': (7.13) 2 ı !2 attains its maximum value for ' D 90 , thus at the “pole”. To stabilize the rest position, one attaches a spiral spring to the axle R as shown in Fig. 7.22 (torsion axle). Then the angular velocity !2 causes an oscillation, not a continuous rotation. In the original experiment of HAGEN, the axle and the spiral spring were replaced by a long torsion band, and the “dumbbell-shaped object” itself was 9 m long.

These two experiments can be neatly and quantitatively demonstrated. We mention also some more qualitative observations. In these, also, the vertical component of the angular velocity of the earth is relevant. In the northern hemisphere, it produces a deflection of moving bodies to the right due to the CORIOLIS force: ). For ships . The axis of 1 force due to the 7.15 ! , which adds vec- 2 component of the ! (Fig. , can also be detected ). But even perfectly- ' ORIOLIS 7.5 ’s pendulum experiment. cos ound curves. This can be ac- 0 ! . horizontal gyrocompass D ngular velocity C7.13 h ! OUCAULT . This plane is coincident with a meridian 84 min; cf. Sect. 2 and ! D ,i.e. 0 T 6.5 ! and 1 , and by damping the precession oscillations whose ! ı .Thesametor- forces due to the earth’s rotation play no significant ). AGEN Model experiment to accelerations due to the ORIOLIS be insensitive to yawinglanding and or rolling braking, and and to to flyingcomplished accelerations or by sailing on ar using takeoff three and are gyroscopes, at whose angles horizontally-mounted of axes 120 and aircraft, it is a serviceable compass only when it is constructed to earth’s rotation in modern times is the constructed gyrocompasses exhibit errors inon their the indication direction which and depend velocity ofthat the all moving vehicles vehicle. are The moving reasonsurface; they for momentarily thus this on have is a a momentary greattorially a circle to on the thethe earth’s rotational gyroscope angular therefore velocity doesmoves not of to remain the in a earth, the planefrom meridian the containing plane, addition the of but rather resultantplane vector only when of the thement vehicle angular C7.10). is velocity moving along the equator (see also Com- periods are long (ideally, Among the most important applications of the C C7.12 ORIOLIS Exercise 7.3 C experimentally. ButwhichisasstraightforwardasF there is no experiment to demonstrate them earth’s angular velocity A further qualitativestones. example But is its theheights, demonstration preferably easterly in requires a dropping deflection mine from shaft of considerable falling sion axis is used as6.6 in Figs. Figure 7.22 detect the earth’s rotation, asout carried by J. G. H the C a) The air in thepressure earth’s regions atmosphere to flows from the the equatorialern subtropical low-pressure hemisphere, high- trough. this In flow moves the(deflection from north- to the the northeast right); to this the iswhich southwest the are origin important of the for northeast sailing trade ships winds andb) aircraft; Bullets always deviate to the right; c) For the wearing of railroad tracks and the erosion of river banks, role. These examples, often( cited in earlier times, can be discarded . OHL Accelerated Frames of Reference 7 C7.13. . . . ortower in at the the drop University of Bremen (cf. comment C7.2). a small person as theimenter. exper- In the picture: The author’s son, R. O. P C7.12. The shadow image of the small apparatus required 150 Part I Exercises 151

Exercises Part I

7.1 A pistol is mounted on a rotating platform, whose rotational frequency is D 10 s1. The barrel of the pistol points from the center of the platform radially outwards. The platform rotates, as seen from above, in a counter-clockwise sense (and thus as in Sect. 7.3, Point 5). The mass of the bullet is m D 4 g and its velocity is u D 100 m/s; for simplicity, we assume it to be constant. a) Find the Coriolis force FC which the bullet exerts against the barrel of the pistol. b) Determine the force FH in the rest frame of the lecture room which the barrel of the pistol exerts against the bullet. (Sect. 7.3)

7.2 A pendulum hangs motionless from a cord attached to the ceiling of the lecture room and is observed by an experimenter who is sitting on a swivel chair that is rotating with the angular frequency !. How does this observer describe the motion of the pendulum and the forces which act on the pendulum in the horizontal plane? (Sect. 7.3)

7.3 Find the horizontal component aC of the Coriolis acceleration of a railroad train which is moving at a velocity of 60 km/h in the direction of the line of longitude through London (the latitude is ' D 51:4°). (Sect. 7.3)

7.4 A satellite circles the earth near its surface, i.e. on a circular orbit with a radius R RE D 6371 km (earth’s radius). Determine its period T. The acceleration of gravity is g; neglect air friction. (Sects. 2.5 and 5.9)

7.5 A physical gravity pendulum is to be mounted in such a way that it will oscillate with the Schuler period. How large must the spacing s be, i.e. the distance between the suspension point O of the pendulum and its center of gravity S (cf. Fig. 6.15)? In order not to have to take the details of the pendulum’s shape into account, we describeR its size by its radius of inertia %, deﬁned in the equation D r2dm D m%2 (see e.g. K. Magnus, “Kreisel”, Springer (1971), p. 12; English see e.g. W. Wrigley, W.M. Hollister, and W.G. Den- hard (1969), “Gyroscopic Theory, Design, and Instrumentation” (MIT Press, Cambridge, MA).). Assume that % D 0:1 m. (Sect. 7.5)

7.6 The pendulum shown in Video 7.5, “Foucault’s pendulum”, swings with an amplitude of A D 1 m (measured at the point of the pendulum wire which is projected onto the scale on the wall of the lecture room). The pendulum wire has a diameter of d D 0:4 mm. Göttingen lies at a latitude of ' D 51:5°. With these three items of in- ) 7.7 , see also Exer- 7.5 The online version of this chapter (doi: , see also Exercise 6.11; for Sect. 7.3 Electronic supplementary material 10.1007/978-3-319-40046-4_7) contains supplementary material,able which to is authorized users. avail- formation and the results of thevideo, measurements derive which the are angular shown velocity in of the the earth’s rotation. (Sect. For Sect. cise 2.7. Accelerated Frames of Reference 7 152 Part I oePoete fSolids of Properties Some a edfre yfre cigo t ntesmls ae,the called re- cases, is then simplest “load” is the the deformation In when The disappears It it. moved. on permanent. acting not forces is deformation by body deformed solid Every be chapters: preceding can the from points few a repeat We and Flow Deformation, Elastic 8.2 on, early From Remarks Preliminary 8.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © nodrt ecieti n iia esrmns edfiethe define we measurements, similar and this by describe ratio stretched to is order diameter, In measured. in is elongation mm its 0.4 and force and constant long a meters several wire, ebgnwt h ipearneetsoni Fig. in shown arrangement simple the with begin We and detail more quantitatively. somewhat it in treat bodies solid of deformation the discuss eoemr n oepoiet ihu asn hog well- a through passing point melting without defined prominent, properties more liquid ex- and their flow more layman temperature, which become the increasing liquids With even viscous thick, but slowly. very tremely and solids, to resinous like similarity as the broken their such for notices be life, biology can everyday in substances, in as glassy encounter of cases, we groups limiting Major that of plants: materials and matter animals a between dif- distinction not more conceptual is is contrast, This much in The only bodies, ficult. complete. liquid clear rather and is becomes solid gases between ‘gaseous’ of distinction understanding word our physics, the In of later. meaning the ies; D elongation Solidification (for hlrndsigihbtensldadlqi bod- liquid and solid between distinguish children "> C8.1 )or 0) eghchange Length rgnllength Original olsItouto oPhysics to Introduction Pohl’s . compression l l (for D elastic " "< .Nowwewantto O 10.1007/978-3-319-40046-4_8 DOI , 0). 8.1 copper A . (8.1) iealiquid. a like weight, own its under slowly, floor the on outwards and spreads flows it rest, at Left ball. elastic an like bounces floor,it the onto thrown is it When children: for toy a putty silly is example Another C8.1. nymc later” much only clear becomes ‘gaseous’ word the of meaning the bodies; liquid and solid between distinguish dren Fo al n chil- on, early “From elkonas known well , 8 153

Part I .Ini- (8.2) 8.2 .Tostart 8.7 – 8.3 A to the tensile stress , in general ‘normalized than proportionally. Up to this of the wire or rod tension A proportionally faster ; it disappears when the stress (load) is (for short: perpendicular to the area . C8.2 increases F (Video 8.1) " 2C8.3 ). ). D reversible 2 , it increases 8.7 d Force ˇ N/mm Cross-sectional area 5 tensile stress 126 mm 10 : The relation- The stretching of a metal wire by a ten- D 0 , i.e. up to an elongation of about 1/1000, the deformation D ˇ =" A D 4 mm, cross-sectional ; later, around : We will treat exact and careful observations in Sects. with, we describe somethese, we quick obtain and the lesstially, relatively precise the simple elongation results experiments. shown in From Fig. 0 area stress’; see Sect. remains elastic, or is called the point Modulus of elasticity E ship between elongation and tensile stress for a copper wire (diameter Figure 8.1 Figure 8.2 In addition, the quotient sile force. The scale whichend of is the attached long to wire theprojected is lower onto magniﬁed a by screen. about 15x and .In ,where A can also be ) pressure , 2005), Chap. 3.) D " F ’s law” ) as a vector equation , John Wiley and 8.2 Some Properties of Solids OOKE 8 Exercise 8.1 Eq. ( becomes this chapter, we will consider mainly tensile stresses. C8.3. Frequently, in “strain- stress plots” of this type,stress the is plotted against the strain (elongation). Sons, Heidelberg, New York, 8th edition ( general case. (For more details, see e.g. Charles Kittel, Introduction to Solid-State Physics can be a tensor in the most oppositely. In this case, is called the negative; i.e. the force and the area vector are directed Video 8.1: “Elastic deformation: H C8.2. Like http://tiny.cc/cvqujy ( 154 Part I 8.3 HOOKE’s Law and POISSON’s Relation 155 removed. Beyond ˇ, the elongation increases quickly with further increasing stress. This deformation is no longer reversible; at ˇ,the stretching or ﬂow limit is reached. Further stretching hardens the pre- Part I viously soft wire. Through heating (tempering), the hardened wire can again be softened (made ductile).

8.3 HOOKE’s Law and POISSON’s Relation

For small values of the stress, one ﬁnds a linear proportionality between the elongation " and the tensile stress .ThisisHOOKE’s law

1 " D : (8.3) E

The proportionality factor E is called the modulus of elasticity (or “Young’s modulus”). (Examples are given in Table 8.1.) Using thick wires, or rather rods, one can at the same time observe the elongation and the transverse contraction,deﬁnedbytheratio

Change in the diameter d " D : (8.4) q Original diameter d

For demonstration experiments, a rubber rod of several cm thickness is suitable. When the workpiece is sufﬁciently thick, one can demonstrate not only its elongation and transverse contraction under tensile stress, but also compression and the accompanying transverse thickening ("q < 0) under pressure. Within certain limits, the transverse contraction "q and the elongation " are mutually proportional; we ﬁnd

"q D " (8.5)

(the relation of S. D. POISSON, 1781–1840).

The proportionality factor is called POISSON’s number (Examples are found in Table 8.1). Elongation and transverse contraction cause a change in the volume, just as compression and transverse thickening do. If the body is

Table 8.1 Elastic constants Material Al Pb Cu Brass Steel Glass Granite Oak wood 4 N Young’s modulus E 7.3 1.7 12 10 20 7 2.4 10 10 mm2 Poisson’s number 0.34 0.45 0.35 0.35 0.27 0.2 – – – G 4 N Shear modulus 2.6 0.8 4.5 4.2 8.1 2 – – 10 mm2 ’s one > (8.8) (8.7) (8.9) OOKE increase is the “bulk always less = . 1 8.1 ) and its cross- 0). When a com- " 9.3 D C 1 (8.6) K "< , in contrast, the force (pressure), the volume : producing the stress acts 2 8.3 of the workpiece (wire or D ) combined with H F A 1 E "/ . It follows for the 8.7 /" : 2 / 1 E 2 2 1 was called the tensile stress ( / "/ 2 1 A "/. 1 . = . all directions 0). – In Fig. F 3 C D 1 . 1 D . 3 V V < D D 5 means that there is no change in the volume due : 0 V V V V D

to the cross-sectional area The deﬁnition of the shear (“cubic dilation”): ) yields 0), and always decreased by compression ( 8.3 The limiting case to stress. This limit applies generally to liquids; cf. Sect. , twice Poisson’s number, is according to Table

"> The constant factor modulus” of the material. law ( is called the “compressibility” and its reciprocal 2 in volume or, neglecting small quadratic terms, Thus far, we have assumed thatperpendicular the force Figure 8.3 8.4 Shear Stress a cube, its height is changed by the factor (1 stress than 1. As( a result, the volume is always increased by elongation 0) or compressive stress ( rod). In this case, the quotient sectional area by the factor (1 pressive stress is appliedchange from is three times as greatdimension as (unilateral with stress); a then compressive Eq. stress ( in only Some Properties of Solids 8 156 Part I 8.5 Normal, Shear and Principal Stress 157

F acts parallel to the cross-sectional area A of a body. (One can think of this body as a pack of playing cards to model the situation!). Then the body will be sheared by the force F; the cards which originally Part I formed an upright pile are tilted through the angle . In this case, we deﬁne the shear strain as the quotient x D tan : (8.10) l The quotient

Force F parallel to the area A D (8.11) Cross-sectional area A of the body is called the shear stress (see Sect. 8.5). For small stresses, one ﬁnds experimentally that the shear strain is proportional to the shear stress ,i.e.

1 D : (8.12) G The proportionality factor G is called the shear modulus. This quantity is also a characteristic property of the material (examples are giveninTable8.1). Thus, for isotropic bodies, we have all together three elastic constants, namely Young’s modulus E, defined by Eq. (8.3); the shear modulus G,definedbyEq.(8.12); and Poisson’s number ,defined by Eq. (8.5). These three constants are however not mutually independent; instead, they are related by the equation

1 G D E : (8.13) 2.1 C /

Therefore, the elasticity of an isotropic body can be fully characterized by two elastic constants; the third is then determined by Eq. (8.13). Its derivation is given at the end of Sect. 8.5.

8.5 Normal, Shear and Principal Stress

Every load placed on a body, e.g. by a tensile stress, changes the state of the interior of the body. The state is characterized by the quality of strain. In the interior of a transparent body in the stress-free state, we imagine that a number of small, spherical regions are made visible by coloring. During loading, each of these small spheres is deformed into a small, three-dimensional ellipsoid. This can be made clear by a demonstration experiment (Fig. 8.4). It deals with the special case /surface not per- F of the de- A . Therefore, the stress A by the quotient d by carrying out the follow- proportional to the lengths of stress is formed not strain 8.4 . A similar situation holds in the general case, that is 8.5 along the three principal axes of the ellipsoid. Apart stress: In the plane of the page, we have a broad rubber to the associated surface element d How the ellipse in Fig. The definition of the concepts ‘stress’ and ‘strain’ only , the force per surface area. The force is in general A 8.5 planar Figure 8.4 Figure 8.5 for the transition from aellipsoid” (strain sphere ellipsoid). to a three-dimensional “deformation We can arrive at the concept of of a pendicular the transition arrows. – Then for each surface element d is decomposed into two components, one perpendicular and one paring thought experiment: Wematerial, cut at the the ellipsoid same out timetain of applying its forces its ellipsoidal to surrounding shape, its thuspreviously surface replacing surrounding which material. the main- Or, effective expressed influence differently: We ofform trans- the the “internal forces” whichinto are due “external to forces” the and surroundingable. make material them The (at directions least ofFig. in these principle) forces measur- arefrom collinear these, with the their arrows magnitudes in are arrows in Fig. band. A circleform is of drawn 12 in dots. thewhich center a The of tensile ends force the of is unstressedthis then the band applied band loading, in in are the the the plane clampedtransition circle of into the from is frames, page. deformed to a During into circle an to ellipse. an ellipse, In the making 12 the dots moved along the area d formation ellipsoid, we define the Some Properties of Solids 8 158 Part I 8.5 Normal, Shear and Principal Stress 159

Δa Figure 8.6 Deformation of a rubber ΔD=Δa√2 plate by four equal forces, each of which Δa τ produces a shear stress . – The edge Part I σ =τ length is a, the thickness of the plate is γ a 1 d,sothat D F=A D F=ad. The ﬁgure 2 2 τ –σ =τ shows the relation between the shear D=a√2 2 and normal stresses and can be used to τ derive Eq. (8.13). 90+γ τ 90–τ allel to the surface element. The perpendicular component is called the normal stress; the parallel component is the shear stress. The three principal axes of the ellipsoid are special directions: In these directions, the force is exactly perpendicular to the surface of the ellipsoid. Along these directions, only normal stresses are present, and these are called the three principal stresses. Shear stresses cannot be produced independently of normal stresses. This can be seen from a simple observation: In Fig. 8.6, we attempt to deform a square plate of thickness d by shear forces alone. We apply four equal forces F which act parallel to the sides a of the plate. Each of them produces the shear stress D F=ad. The result is however the same as in Fig. 8.4, where a tensile load is applied; a circle is deformed into an ellipse. Thus, normal stresses are also produced. Their largest and smallest values, the principal stresses 1 and 2,occur in the directions of the diagonals. In the diagonalp directions, twop of the forces F combine to give a resultant F 2. These forcesp F 2 are each perpendicular to a diagonal cross-sectional area ad 2. As a result, the normal stresses 1 and 2 are likewise F=ad, thus exactly the same as the shear stresses . Thus, the deformation (strain) of the plate can be described in two ways: either by a displacement of the square’s sides a by a,orbyanelongation of the square’s diagonals D by D.

To calculate a, one can use the shear stress . It produces a shear strain of 1 D : (8.12) G In simple terms, this means that the 90ı-angles are changed to angles of (90ı˙), and the sides of the square are tipped by angles =2 relative to the diagonals D.FromFig.8.6, we can see the geometric relation tan =2 D 2a=a =2, or, with Eq. (8.12),

2a 1 1 D : (8.14) a 2 G

To calculate D,weusethenormal stresses, i.e. the tensile stresses 1 D and the compressive stresses 2 D . The tensile stresses lengthen the " 1 1 diagonals by an amount 2 Dtens D D D 1 E D D E D. In addition, according to POISSON’s relation (Eq. (8.5), the compressive stresses also cause a lengthening of the diagonals by the amount 2Dcomp D "D D . ,we mass which A . D M (8.16) (8.15) M m ( s gonals from ; D mg / D of the bent rod. We C r and shear stress 1 M 2 . 1 E p a / : / D / : C C C (see also the numerical example in 1 1 . comp . 1 , a thin rod with a constant cross- . ) yields 1 E D 1 E 1 E , 8.7 2 2 a 8.16 D D D C a 2 1 a G p tens 2 (Video 8.2) a D 2 2 ) with Eq. ( ; it shows a cross-section along the length of the D . The resulting overall lengthening of the dia 8.8 D 8.14 ), as given previously without derivation. D is bent by a constant torque 1 E 2 A to each other. 8.13 ı ) Bending load on a thin, ﬂat rod produced by a torque D D : The directions of the principal stresses (the diagonals) and 1 E 1 8.6 Combining Eq. ( and, introducing the edge length This is Eq. ( both sides is then or make use of Fig. rod. The deformation producesof tensile the stresses along rod, the and uppernormal compressive stresses, side stresses i.e. along they the are lower perpendicular side. to the Both cross are section We want to compute the radius of curvature of a weight). We observe its shape to be that of a circular segment. Exercise 8.2 is constant along its length Figure 8.7 8.6 Bending and Twisting (Torsion) In applying the conceptshave of thus normal far stress limitedwant ourselves to consider to the the bending of simplest a examples. rodTake a by cuboid-shaped an rubber Now eraser external between we torque the thumband and bend foreﬁnger it: ItsWe sides will neglect are this not arching, only“planar” and curved, consider state but only of also the strain. limiting archedsectional In case outward. area Fig. of a To conclude, we readFig. off another important factthe for directions later of use theangles from of largest 45 shear stresses (along the edges) are at Some Properties of Solids 8 http://tiny.cc/4uqujy Video 8.2: “Bending a rod” 160 Part I 8.6 Bending and Twisting (Torsion) 161

Figure 8.8 The derivation of Eq. (8.22) Part I

With the assumptions made above, which are well fulﬁlled for thin rods, the cross-sectional areas denoted as GH, G0H0 etc. should remain planar even during bending, that is they should simply rotate around their centers of gravity S. Then the transition from tensile to compressive stress occurs in a simply curved, and not arched, layer. It is stress free, stands perpendicular to the plane of the page, and cuts it along the line N N. One calls this line the neutral ﬁber (cf. Vol. 2, Sect. 24.9, Strain birefringence). Under these conditions, in Fig. 8.8 we have for the two radii of curvature r and .r C y/: r C y l0 D : (8.17) r l Furthermore, we have

l0 l D elongation ": (8.18) l

According to HOOKE’s law, this elongation is associated with a modulus of elasticity E : D " (8.3) Combining Eqns. (8.3), (8.17), and (8.18) yields y D E : (8.19) r R The integral y dA must be equal to the effective torque M,sothat Z y2 M D E dA ; (8.20) r or, with the abbreviation Z y2 dA D J ; (8.21) we obtain J r D E : (8.22) M ) 6.4 ( (8.24) (8.26) (8.23) (8.25) has been . For this A is also con- is explained r J of a tube whose wall : d : We need only replace 3 J c,d) a) /: R : 3 of the layer. As a result, we /: 8.9 8.9 2 4 : m dh S r 3 d were constant along its length. ) agrees well with observed val- 2 dh / 3 y 4 4 (Fig. M r 1 R 12 . DH Z 8.22 hd . 4 ) 4 D 1 D and D geometrical moment of inertia R 12 J . D ), the radius of curvature A J 8.11 2 J

D b) , J E D 8.9 8.22 , (Fig. J r twisting around the long axis 8.7 is dot-dashed is normalized here R ) to arrive at values for holds for a layer within the cross section of the rod A 8.8 . can have rather is composed formally just like the moment of inertia, ( D 6.4

J d J A . J in those formulas by the cross-sectional area Cross sections with the . It shows proﬁles with the same values of the geometrical m in Fig. as calculated from Eq. ( S 8.9 r (Video 8.2) 3. Circular, ring-shaped cross section (Fig. 4. Similar, butthickness is for Examples 1. Rectangular cross section, 2. Double-T beam (Fig. reason, the unfortunate name adopted for As a result, fromstant, Eq. i.e. ( theradius bent rod takesues the shape of a circular segment. Its Figure 8.9 the mass This value of can use the formulas forearlier the moment (Sect. of inertia which we established and is referred to the center of gravity in Fig. For the rod in Fig. The importance of the geometrical moment of inertia The quantity i.e. same values of the geometrical moment of inertia here different areas to the rectangular cross sectionThe (a)). neutral ﬁber which islar perpendicu- to the plane ofthrough the page and passes . Some Properties of Solids 8 Video 8.2: “Bending a rod” http://tiny.cc/4uqujy 162 Part I 8.6 Bending and Twisting (Torsion) 163 Part I

Figure 8.10 Bending load on a rod of length l which is held at one end, produced by a force F acting on its free end moment of inertia J, that is the same circular radius of curvature for the same loads. Below each profile is its cross-sectional area, given in arbitrary units. A smaller cross-sectional area A means that less material is required. In this regard, a tube is superior to a rod of similar diameter. This is the reason why the long bones in our appendages are formed as tubular bones. The geometric moment of inertia also plays a decisive role in many other aspects of deformation. We offer two examples, the first without derivation. In Fig. 8.10,arod is clamped at one end; a force F acts perpendicular to its long axis at its free end. Then for a moderate deflection y of the end of the rod, we have

l3 y D F : (8.27) 3EJ

Furthermore, we brieﬂy treat the twisting (torsion) of a cylindrical rodC8.4. It is also determined by a geometrical moment of inertia. We C8.4. See also the torsion ﬁnd the quotient experiment in Fig. 6.7 (Video 6.1, http://tiny.cc/ Torque M GJ D D D (8.28) csqujy)(Exercise 8.3) Angle of torsion ˛0 l

(J D geometrical moment of inertia of the rod (Eq. (8.26) with r D 0), l D length of the rod, G is the shear modulus of the material; cf. Table 8.1).

D is the “torsion coefﬁcient” introduced in Sect. 6.3. It can readily be measured, either directly or by using torsional oscillations. Equa- tion (8.28) thus provides a handy method of determining the shear modulus G, an important quantity for materials science (Table 8.1).

To derive Eq. (8.28), we make use of a special case as shown in Fig. 8.11, a thin-walled tube. The torque is produced by two cords wound around the tube. We imagine the tube to be divided into flat ring-shaped sections; these experience a shearing (angle ) relative to each other. The meaning of the angle can be seen from the figure. We find

x ˛0R tan D D : (8.29) l l This shear deformation is caused by the shear stress ,forwhichwehave

1 D : (8.12) G ) is 0 ˛ 5.35 (8.33) ( (8.32) (from (8.30) (8.31) 625 m, : J 0 we ﬁnd for ), yield D R P W 8.26 acting on the tube; : : M l (number of revolutions/ d GJ , the torsion angle 2 : D R ) M /; P M GJ W 2 6.1 2 2 D D M D 3 ) Fu 0 ), together with Eq. ( 62 m, outer diameter 2 D dR ˛ D M D ! 8.28 8.30 2 l F . It is simply a cylindrical rod which P M W 2 G orbital velocity acting tangentially on the ring-shaped sec- , material steel, with a shear modulus of 4 R D D D m 2 rotational frequency )and( shaft ), we could write: u 0 Cross-section 3 = P l . 480 m, geometrical moment of inertia W ˛ D : M . The power transmitted to the ship’s screw is 0 D 2 8.12 10 8.31 D ), ( D Torsion angle F 77 r : N/mm 9 8.29 4 10 D The derivation of Eq. ( )) At a given rotational frequency 1 : ). For transferring or conducting mechanical power (“kilo- 8 angular velocity, 8.26 D D 8.28 ! Eq. ( inner diameter 2 tions, and thus the shear stress time). G ( Numerical example The hollow drive shaft of a ship: The shear stress, init turn, produces forces is found from the torque The equations ( In words: a measure of the mechanical power transmitted along the shaft. Figure 8.11 Thus, instead of Eq. ( To conclude thisEq. ( topic, wewatts”), one mention often uses ais a loaded technical torsionally. applicationcontinuous motion of For the transmitted power and thus for a rotation (row 10 in Table for the torsional deformation of a tube Some Properties of Solids 8 164 Part I 8.7 Time Dependence of Deformation.Elastic Aftereffects and Hysteresis 165

WP D 2:4 104 kW, the rotational frequency is D 3:4/s. – Inserting these values into Eq. (8.33) yields the torsion angle ˛0 D 8:8 102 D 5ı.

This means that the front and back ends of the 62 m long shaft are twisted Part I relatively to each other by 0.014 of their circumference. Drill stems for deep vertical bore holes can have lengths of several kilometers. They must then be twisted through a number of rotations in order to transmit the power required for drilling to the drill bit at depth!C8.5 C8.5. For deep drilling projects, such as for example the deep drilling program KTB in Southeastern Ger- 8.7 Time Dependence of Deformation. many, with a depth of over Elastic Aftereffects and Hysteresis 9 km, multiple twists indeed occur in the drill stem. However, hydraulic motors For quantitative observations of elastic deformation, we have thus far are also employed, directly considered metals, as in Figs. 8.1, 8.7 and 8.10. Glasses are also suit- above the drill bit, so than able. For demonstration experiments, a polymeric, plastic material no twisting of the drill stem is often more convenient, in particular rubber. We will demonstrate occurs (see e.g. https://en. two important companion phenomena of elastic deformation using wikipedia.org/wiki/German_ rubber, namely elastic aftereffects (also called the “memory effect”), Continental_Deep_Drilling_ brief and hysteresis. In our ﬁrst demonstrations, we left these effects Program or https://en. out of consideration. wikipedia.org/wiki/Kola_ We load a 0.3 m long and ca. 5 mm thick rubber tube alternately with Superdeep_Borehole) around 1 and 6 N tensile forces (a 100-g weight and an additional 500-g weight), and record its elongation as a function of time. The result is shown in Fig. 8.12: its deformation occurs mainly at the same time as its loading or unloading; but a remainder, called the elastic aftereffect, takes a noticeable time to form after loading and to disappear after unloading. The new equilibrium values are approached to a good approximation exponentially with time. On loading, after the relaxation time has elapsed, about 1=e 37 % of the full amount of elastic aftereffect is still missing. On removing the load, after a time , about 1=e 37 % of the elastic aftereffect is still present. Unfortunately, the separation of deformations into those with and without aftereffects would be an oversimpliﬁcation, even for small

Figure 8.12 Elastic aftereffect on stretching a rubber tube (outer diameter 5 mm, inner diameter 3 mm) - " 8.14 .One . The relation 8.14 hysteresis loop deformation. It can be direction. This is called opposite permanent . In its center, its two halves are divided 2 and 1 deformation is thus not reversible; the material C8.6 every A hystere- The demonstration of mechanical hysteresis. A stretched rubber . It can be demonstrated using the apparatus shown in . . Forces acting 8.13 8.13 to the right are taken to be positive. The measurement series begins at the upper right corner. sis curve obtained with the apparatus shown in Fig. Fig. hysteresis Figure 8.14 A small part of is not completely elastic. Adrives small fraction the of elongation the is tensile always energy “lost” which in the form of heat. In an Figure 8.13 deformations. When the loadceding is elongation removed, remains some fraction aseliminated of a only the by pre- loading in the tube is attached at its ends, A rubber hose whichto is about clamped twice at its both relaxedstress ends length applied and can stepwise already be as further stretched loaded increasesright with or (weights). a decreases tensile from Betweenleast the two one left measurements, minute. or there the The is results are a indicated pause in of Fig. at between elongation and tensile stress along thecreasing increasing branch and the is de- shown bythat two they curves, enclose and a we narrowobserves can area, similar see behavior the for in nearly mechanical Fig. all solidmetals, bodies, glasses and thus etc. also for by a metal disk.increased A or pair decreased of by cables addingthe can two or exert balance removing pans. forces weights on on this one or disk, the which other can of be Some Properties of Solids 8 netic properties of materials (see Vol. 2, “Matter in electric and magnetic ﬁelds”, Sects. 13.4 and 14.4). C8.6. Hysteresis occurs also in the electrical and mag- 166 Part I 8.8 Rupture Strength and Speciﬁc Surface Energy of Solids 167 plot, the area within the hysteresis loop is the work per volume which is transformed into heat during a cycle of loading and unloading:

C8.7. A quite different kind Part I Loss per load cycle W of hysteresis is exhibited D : Volume of the deformed body V by “shape-memory alloys”, such as NiTi, when they pass (The product " ,i.e..l=l/ .F=A/, has the units of work/volume). through a structural phase transition during tempera- The origin of elastic aftereffects and of hysteresisC8.7 is related to the structure changes. This can also ture of the material. During elastic deformation, individual regions of the be produced by deformation. bulk can shift or rotate relative to one another and then act as ‘latches’. They can, for example, be Releasing the latches can occur either through thermal motions alone (aftereffects), or it may require loading in the opposite direction (hysteresis). plastically deformed at room temperature, but on warming, they return to their original shape. They are used among 8.8 Rupture Strength and Specific other things in medical technology (see e.g. http://web. Surface Energy of Solids stanford.edu/~richlin1/sma/ sma.html). When the load becomes sufficiently large, every solid body will be torn or ruptured into smaller pieces (Video 8.3). In idealized limiting Video 8.3: cases, the tearing seams (“fissures”) are found to lie either perpendic- “Plastic deformation, rup- ular to the direction of greatest normal stress, or parallel to the plane ture strength” of greatest shear stress. For that reason, one distinguishes between http://tiny.cc/kvqujy tensile strength and shear strength; when they are exceeded, the re- A 40 cm-long Cu wire (of sult is separation fractures or shear fractures. The directions of the diameter D 0.4 mm) is placed greatest tensile and shear stresses are tilted by ˙45ı relative to one under tensile stress by hang- another (see Fig. 8.6). Therefore, when brittle materials are pressed, ing weights on its end, until it we find rupture surfaces tilted at around 45ı from the direction of the breaks. The rupture strength compressive stress. of the copper used is found to be Z 240 N/mm2. Between the regime of elastic deformation and rupture, many ma- max terials exhibit some other phenomena, namely a regime of flow or slippage of their individual volume elements and an accompanying hardening. Some metals can be rolled out to thin sheets or drawn through a die to make wire even at room temperature (cold working). The gradual change of shape associated with “plastic deformation” makes the process of rupture still more complex than with brittle materials, i.e. materials that rupture without previous plastic deformation (e.g. glass or cast iron). – Plasticity and brittleness are not well-defined properties of a substance; at high temperatures, every material becomes more or less plastic (malleable, ductile). Table 8.2 contains some values of the tensile strength of materials for technical purposes. This is the term applied to the tensile stress Zmax which leads to rupture. These values were measured on standardized rods. – To properly evaluate these numbers, try the following simple experiment: Cut a strip about 20 cm long and 3 cm wide from a sheet of good-quality writing paper, hold it by its ends and try to tear it by pulling uniformly. This is seldom possible. Then cut a small notch into one side of the strip, barely 1 mm deep. Now, the paper strip can readily be made to tear: at the “tip of the notch”, a kind of leverage > D 2 max (8.35) max Z Z (8.34) one takes ; Wood fiber up to 120 N/mm (To define the be performed. C8.10 A : C8.8 2 x acting along a short Quartz glass 800 D A max Z W 1 2 max , a wire of cross-sectional Z D Mica 750 (Video 8.3) D m) fibers of glass or quartz, 8.15 . F max Z . It can be estimated from the ten- or C8.9 Steel up to 2000 Ax max Z of the newly-formed surface Brass 600 A D required to increase the surface area A Cu 400 W , and requires that the work 2 A Pb 20 have been attained Area 2 have been measured for mica. specific surface energy 2 Al 300 Technical tensile strengths Work . We thus find x is broken with a separation fracture. This produces two new D A a piece a few centimetersbends long (small between radii the of curvature) fingerThe can tips. smallest be defects on Surprisingly attained the sharp without surface,suffices however, breaking cause to premature the breakage; touch fiber. it the bent fiber with another glass fiber. For demonstrations, such fibers are loaded by bending them max Material Z 10 000 N/mm 3180 N/mm distance Table 8.2 With very thin (diameters of a few freshly prepared at high temperature, tensile strengths of up to This work is performed by the force is called the In the interior (bulk) ofall a sides by material, their the neighbors; molecules atmissing the are surface, on surrounded however, the one on neighbors side. are a As molecule from a the result, bulk work to a must surface be position. performed The to quotient move produces locally a verydeeper high and tensile deeper. stress,play Even and a microscopic decisive this role notches tears in rupture. or the other notch In defects some can cases, thecan perturbing be effects avoided. of With mica, notchesplanes for and example, parallel one surface to can defects orient theedges, the direction cleavage where of notches tensile couldable stress play clamping and a frame. also significant In role, protect this by the way, using tensile a strengths suit- of up to tensile stress, the originalreduced cross-sectional area during area elongation.) of the rods was used, not the sile strength of a material measured withoutor disturbance from surface notches defects. In the schematic view shown in Fig. area surfaces of area , 235 max Z 82 Ap- .See , 2 , 213–218 ), and can et al. A82 Z. Physik 8.6 Some Properties of Solids 8 (1933)). C8.9. This value originates from a publication by E. Orowan ( plied Physics (2006). B.O. Swanson therefore become especially large. C8.10. Bending produces tensile stresses on the outside of the bent region whichinversely are proportional to the radius of curvature of the bend (Sect. of 800–1600 N/mm C8.8. Exceptionally large values of the tensile strength have been observed in various spider silk fibers: Video 8.3: “Plastic deformation rupture strength” http://tiny.cc/kvqujy 168 Part I 8.9 Sticking and Sliding Friction 169

Figure 8.15 The derivation of Eq. (8.35) Part I

The distance x must be of the same order of magnitude as the range of the atomic attractive forces or the distance between neighboring atoms in the material. This is of the order of 1010 m. Then from Eq. (8.35), it follows that the speciﬁc surface energy of glass with 2 Zmax 10 000 N/mm is N Ws 5 109 1010 m D 0:5 : m2 m2

The high values of the tensile strengths of materials implied by Eq. (8.35) are called the theoretical tensile strengths. They can be larger than the measured technical tensile strengths by more than an order of magnitudeC8.11. The technical strength is determined C8.11. Even liquids can ex- essentially by perturbing secondary effects. “Notching” is indeed hibit large tensile strengths a rough simpliﬁcation, but it is still an appropriate generic term for (see Sect. 9.5 and Video 9.4: these effects. “The tensile strength of water”, http://tiny.cc/3vqujy).

8.9 Sticking and Sliding Friction

External friction (Sect. 3.1) occurs at the contact area between two solid bodies. It plays a fundamental role in everyday life and in technical applications. For physical experimentation, it is often a source of disturbance and of errors. Quantitative results depend strongly on the surface condition of the objects which are in contact. – We can distinguish between three forms of external friction: sticking friction, sliding friction, and rolling friction. In Fig. 8.16, a smooth, box-shaped block is pressed against a smooth horizontal board by the force of its weight, FG D Fn. A cord transmits a force F to the block, parallel to the board’s surface. This tractive force must exceed a threshold value, Fst, before the block will start to slide along the board. From this we conclude that the two objects (block and board) stick to each other; a force of magnitude Fst acts in opposite directions on each of them. It is called sticking friction. This frictional force is independent of the size of the contact area between the two objects. It depends on the properties and condition of the contact surfaces and is proportional to the . ; st F F ,we ace- 8.16 ount. With- acting on the which presses sl and not before n .Furthermore, F F . . C8.12 motions adhesion . The equation n F n n F F sl st

sliding friction D D sl st only during mutual F F of sticking friction (a number, usually be- pulling on the block larger than the sticking st surface of any solid body to a ﬁle or brush. The friction that applies here. F is always smaller than the sticking friction

friction is called the sl F sl F smooth however does not correspond to the tractive force a sticking Sticking and . Then the block begins to slide in an accelerated motion. coefﬁcient of sliding friction, a number usually between 0.2 and st F D sl

With increasing velocity, it may decreasefrictional by force up which to applies ca. at 20 very % small of velocities the original ( 0.5). The sliding friction is independent of the sliding velocity only to first order. The force we call sticking frictionments arises of through two extremely small objects displcan relative compare to the one another.microscopic parts In that the stick simplest outbe picture, deformed catch one to on permit each sliding. other,the – adsorbed so layers A that of more they other refined molecules have modelout on to would the these have surfaces layers, to into acc take thebecome sticking vanishingly friction between small. twoin An polished high example: surfaces can vacuum. A glassstrong The block an word and expression; ‘sticking a it friction’ glass includes is plate the dubious; concept stickingthey begin. of is too one speaks properly of there must be a smaller, oppositely-directed force block. This force the two bodies together, and is independentarea; of the thus: size of their contact It is, like the latter, proportional to the normal force The sliding friction it is smaller. Therefore,F while the block is sliding, besides the force tween 0.2 and 0.7). Stickingnical friction applications. plays an It important determines role thedriving in force maximum tech- possible for value locomotive wheels offor and the automobile the tires, shoe as soles wellwheel of as or a the shoe pedestrian. soleTherefore, are At momentarily it the at is point rest relative of to contact, the ground. a rolling Figure 8.16 sliding friction Continuing our observations using themake setup the shown traction force in Fig. friction magnitude of the normal force thatular presses them to together their (perpendic- contact surfaces), in this case defines the coefficient Its acceleration increases ). In that case, the . 10 5.22 Some Properties of Solids 8 note the experiment which is described together with Fig. ity, a dependence which is generally assumed in a first treatment of damped oscillations. In this connection, ing in liquids or gasesChap. (see frictional force proportionally to the veloc- which is mentioned here is quite different from that observed for objects mov- C8.12. The velocity dependence of the frictional force 170 Part I 8.9 Sticking and Sliding Friction 171

Measurements of the sliding friction are carried out at constant sliding velocities. For this purpose, one replaces the setup for the tractive force

as shown in Fig. 8.16 by an electric motor with a winch, inserting a force Part I meter in the cord which transmits the force.

In machines, external friction is usually reduced as far as possible; it is replaced by the internal friction of ﬂuids. This is termed lubrication. In some cases, one has to be satisﬁed with reducing the external friction to a minimum. This can be accomplished only for a particular direction of motion (Fig. 8.17). Perpendicular to this direction, one applies an ancillary force F2 to maintain a constant velocity. During 0 this ancillary motion, only the small component Fsl opposes the force F1. – Figure 8.18 illustrates a demonstration experiment.

When asked, ‘Why, when we are cutting something with a knife, e.g. slic- “When asked, ‘Why, when ing bread, do we not only press down on the knife, but also move it back we are cutting something and forth?’, even many physicists will give a wrong answer: ‘The back- with a knife, e.g. slicing and-forth motion converts the large sticking friction into a smaller sliding friction’. In reality, the motion has two effects: First, the knife acts as bread, do we not only press a saw; and second, the sliding friction in the direction of cutting1 is re- down on the knife, but also duced by means of an ancillary force which acts perpendicular to it. One move it back and forth?’, thus makes use of the scheme illustrated in Fig. 8.17. At the beginning even many physicists will of the cutting process, when the knife blade is resting on the crust of the give a wrong answer.” bread, only the sawing effect is present. Another example is pulling a wedge out of a crevice: One moves the wedge back and forth along the line of the crevice. Unintended, but – due to vibrations – often unavoidable relative motions between the sides of the wedge and the walls of the crevice produce the sometimes fatal loosening of screws. Screws are in fact simply “rolled-up wedges”.

Figure 8.17 An object (center of gravity S)istobe moved to the right on its horizontal supporting plate. To produce a motion with a constant velocity, an 00 ancillary force F2 compensates the component Fsl of the sliding friction Fsl. As a result, the tractive force F F'' 0 sl sl F1 need overcome only the small component Fsl of the sliding friction.

S F' F sl 1

F 2

1 The direction of cutting is the direction in which the knife penetrates like a wedge into the object being cut. n F , one 2 : The tractive 4 N, or the still larger acts as counter-force sl 2 ). F F 6.4 of rolling friction. It is found n is acting tangential to the cylinder ro mm and 1 mm. F 2 2

ro in Fig. F

m to produce the rolling motion A . D M M illary force (Video 8.4) ancillary force which performs work , it must act parallel to the track (e.g. the rails 1 between ca. 10 ) is pressed onto a horizontal track with a force F r . Nevertheless, the block slides when the crank is rotated ), and we will call it the rolling resistance in the fol- st 1 length F F A demonstration experiment showing the reduction of sliding of all the wheels (including the drive wheels). Wagons are D is much smaller than the sliding friction 2 1 2 F F F along the track with a constant velocity, one must apply a torque Around momentary axes of rotation can for example applying a (tractive) force force to the axle of the wheel. As driv- to be a small to energy losses. In order to generate the torque Rolling friction, in contrasthas to nothing at sticking all friction tobrication”. and do sliding with Rolling friction, adhesion. frictionthe It is track cannot caused and be by reduced thealong by the wheel the “lu- at elastic track their deformation and point of aroundsame of velocity the contact. as circumference of the This forward theno point ideally movement wheel elastic moves of with deformation, the the aftereffects wheel. and hysteresis Since always there lead is (length of the cylindera 1 pencil, m, held diameter atused 2 instead cm). an of angle, the For block. suffices a as ‘free-hand’ rotatable experiment, cylinder, and a ring can be (i.e. sticking force or swung, i.e. when the anc of a railroad track). The resistance force lowing. The rolling resistance is importantsuch for all vehicles, vehicles with whether wheels. theyor In trailers, are all the automobiles, motor locomotives,tance has tractors to perform work against the rolling resis- vastly superior to the sledges used before the invention of the wheel 2 roll to the wheel; its magnitude isThis practically equation independent defines of the coefficient the velocity. 8.10 Rolling Friction A wheel (of radius which acts perpendicular to the track. In order to cause the wheel to Figure 8.18 friction by means offorce an Some Properties of Solids 8 Video 8.4: “Reducing sliding friction” http://tiny.cc/mvqujy 172 Part I Exercises 173 roughly 5000 years ago: To pull a wagon, one requires a considerably smaller force than to pull a sledge with the same weight FG.

A wagon requires the force FWa D M=r D roFG=r, while a sledge Part I needs a force FSl D slFG. We thus ﬁnd for the ratio of the forces F Wa D ro : FSl sl r

Example ro D 1 mm, sl D 0:5, r D 50 cm, FWa =FSl D 1=250.

This ratio becomes very small when wheels with a large radius are used. Thus, replacing sledges by wagons with large wheels was an enormously important invention.

Exercises

8.1 In Video 8.1, “Elastic deformation: HOOKE’s law”, a copper wire of diameter 0.4 mm and length 4 m is loaded by a weight of 400 g and thereby reversibly stretched to an additional length of 1 mm. Find the modulus of elasticity E of the wire. (Sect. 8.2)

8.2 In Video 8.2, “Bending a rod”, the brass rod (with a cross- sectional area of 12 mm width and 4 mm height) is lying on a table of length L D 0:65 m. When each overhanging end is loaded by a weight of 1 kg, it is reversibly deformed into a circular segment, as shown in the video. The midpoint of the rod is raised by a height H. Measure H and the length s (s is required to determine the torque, as indicated in Fig. 8.7). From the quantities L and H, ﬁnd the radius of curvature r of the rod. Use these values to calculate the modulus of elasticity of brass. For length measurements in the video, use your knowledge that the length of the table is L D 0:65 m. (Sect. 8.6)

8.3 In Video 6.1, “Twisting a rod”, the length l of the cylindrical steel rod (more precisely, the spacing of the clamps which hold the mirror) is l D 9 cm. The diameter of the rod is d D 1 cm. The length of the light pointer, which is just about equal to the diagonal of the experimental area of the lecture hall, is L D 10 m. The torsion angle ˛ can be found by comparing with the rod of diameter dS D 1:4cm which is mounted close to the projection screen on which the light pointer is seen. The torque M applied by the experimenter, which he measured by using a torque wrench (not shown in the video), is M D 0:6 N m. From these data and the torsion angle ˛ (to be measured), calculate the shear modulus G of the steel rod. (Sect. 8.6)

8.4 A wooden block of mass m D 5 kg is pulled along a horizontal, ﬂat tabletop with a horizontal force of F D 25 N. The coefﬁcient s 25 : 0 D sl

, see also Exer- 8.9 The online version of this chapter (doi: , see also Exercise 12.5; for Sect. 8.5 ) 8.9 through which the block moves(Sect. in 3 s, if its initial velocity was zero. and is independent of the sliding velocity. Calculate the distance For Sect. cises 3.2 and 5.2. Electronic supplementary material 10.1007/978-3-319-40046-4_8) contains supplementary material,able which to is authorized users. avail- of sliding friction between the block and the tabletop is Some Properties of Solids 8 174 Part I oeue rudtheir around molecules iud n ae tRest at Gases and Liquids h setascnb umrzdi nimage an in summarized be can essentials The and more smaller becomes the volume smaller, constant at shape the change to required bodies solid of deformation be- The their always changed. on is based shape is their bodies when liquid havior and solid between distinction The Liquid of Displacements Free The 9.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © vripie u tl cuaepcueo hs hra oin in motions thermal the these of form of the greatly in picture liquids a accurate find can still We but possible. oversimplified also are motions rotational and tional nsldbde,teivsbe rno”mtoswihaeusu- are which motions “random” invisible, called the ally bodies, solid displaced freely In be loosely”. around can “slide molecules they another, the fixed one such all to contrast, exist; relative in not liquids, do In positions that positions. the rest this rest in fixed are from molecules, to conclude the bound can units, We main structural – smallest the required. bodies, volume, solid be constant in would at all deformation at slow force arbitrarily no an of case limiting otiigasseso feteeyfiecro odr(ri di- (grain powder carbon water, the fine to extremely ameter added of be suspension can a ink containing India of powder; amount in insoluble liquid, small fine, some a a conveniently, contains of liquid drop This a water. between put case we simplest microscope; glass, the cover sophisticated a fairly and slide a microscope a use We manner. ogous h eosrto fB of rather the demonstration a see of The form then We the in insects way. image. individual random coarsened rest; the a at of constantly in movement remain be turning restless will not they and will ants, twisting objects indiscernible moving, the These larger, by pulled few etc. and a paper, shoved throw feath- We downy of example snippets more: for bowl, see ers, the to into objects us light allows visible, readily trick simple the A discern at sur- brown cannot looking face. dark We structureless, are a only nearsightedly). we but (or insects, squirming ants; distance individual live certain with a filled from it bowl a Imagine simplicity: eursteato ffre.Frlqis ncnrs,teforce the contrast, in liquids, For forces. of action the requires Molecules 0 hra motions thermal : 5 m). slowly etpositions rest olsItouto oPhysics to Introduction Pohl’s ROWN rwinmlclrmotion molecular Brownian h eomto scridot nteideal the In out. carried is deformation the oss setal foclain fthe of oscillations of essentially consist a oini are u na anal- an in out carried is motion ian nlqis oee,transla- however, liquids, In . fams child-like almost of O 10.1007/978-3-319-40046-4_9 DOI , C9.1 ( ie 9.1 Video ). http://tiny.cc/xhgvjy osmosis” and diffusion on experiments “Model also (see http://tiny.cc/4wqujy 91 R C9.1. ity”. simplic- child-like almost of image an in marized sum- be can essentials “The “B 9.1: Video odn16,Vl 1. Vol. 1865, London Brown Robert of Works cal “ Hardwicke, R. in: available readily is article This ies”. bod- inorganic and organic in molecules active of general existence the on and plants; of pollen the in contained cles parti- the on 1827, August and July, June, of months the in made observations microscopical of account brief “A title, the with article an in contained is scription de- His motions! similar show grains dust inorganic even that finally and move, still also grains pollen old 100-year- that found he investigations, his of course the in water; in suspended grains pollen observing by initially motion this covered dis- (1773–1858), botanist h iclaeu Botani- Miscellaneous The ROWN OBERT ie 16.1: Video a motion” ian 9 B ROWN .) , , 175 ”,

Part I . , but no ,wesee ). The 2 Diffusion 9.1 , invisible to cloud observation of scopic observa- or a is defined in general macro swarm tting a thin, flat microscopic ian motion requires magnifi- Diffusion dissolved molecules ROWN of diffusion is the essential point. The presumes the , independently of the size of the particles between the two liquids is initially sharp, two different names for the same process by a microscope. Such a high magnification are ian motion belongs among the most important velocity diffusion ROWN or of extremely small boundary layer is prepared by pu The advance of an interface layer by diffusion. The Brownian motion ian motion. It gives us a peek behind the backdrop of na- Brownian motion To show the demonstrationwith to a a high large index of audience, refraction, for one example should the mineral use rutile a (TiO powder large index of refraction guarantees a bright image of the powder grains. dust grains ROWN longer allows us to discerndust-containing the water, e.g. individual grains. highly diluted In Indiapure Fig. ink, next water. to The a boundary layerbut of it becomes more and more washed-outslowly, in in the course the of course time.fuses” of Very into weeks, the the previously clear swarmas water. of any movement carbon of particles molecules “dif- causedand by thermal motions. of observation can savegrains us in from the this liquid error; only it as shows a the whole, suspended as a tions, we speak of cation of several 100 leads us to overestimate the velocities involved. A different method The term cork disk onto the lowercautiously layer sprayed of as water. a Pure fine water jet can onto then this be disk. phenomena observed by science. Noto description replace in the words effect can of begin first-hand observations. An effective demonstration of B Figure 9.1 Only a few physicalB phenomena can fascinate the observer like the of initially sharp observed. That is, the objects seen as a swarm or cloud can consist individual, larger particles. When dealing with hustle and bustle of avidual particles. completely Tiny incomprehensible grains number shoot like of arrowschanging across indi- direction the in field of a view, wild zig-zag.stantly Larger changing grains their jerk course. forward, con- The largest grainsforth, just stagger staying back and nearly inners the reveal their same rotations around place; constantly changingis their axes. a Nowhere trace jagged of edges a systematic and orderthis to cor- world; be seen. that is Random, blind the chance overpoweringserver. rules impression – of every The objective B ob- ture’s workings. It shows us a new world, the restless, confusing any optical microscope. For our purposes, the boundary of the swarm moves surprisingly slowly. Depending on Liquids and Gases at Rest 9 176 Part I 9.1 The Free Displacements of Liquid Molecules 177 the size of the suspended particles, a measurable displacement of the boundary requires days or even weeks. The reason for the slowness of diffusion is the close packing of the Part I swarming liquid molecules. The mean distance of the molecules in a liquid is of the same order of magnitude as in the corresponding solid material. This can be seen from two facts: The density of every substance is roughly the same in the liquid and the solid state; and furthermore, liquids have a very small compressibility. This property, with which we are familiar from everyday life, will be verified quantitatively in Sect. 9.3. After these preliminary considerations, we can replace the real liq- uidbyamodel liquid and study the properties of liquids using the model. A good model would be a container full of live ants or round beetles with hard wing covers. But a container full of small, smooth steel balls will sufficeC9.2. Then we will have to simulate the ran- C9.2. Model concepts in dom motions of these model molecules – the “thermal motions” – in physics are not only useful a somewhat clumsy way by shaking the whole container. In the fol- for didactic reasons; rather, lowing, we will not mention this shaking in every case, but take it for they represent a basic as- granted. pect of the physical method, namely reducing complex The free displacement of liquid molecules makes a number of proper- situations by simplifying as- ties of resting liquids at equilibrium understandable. They are treated sumptions to known laws and extensively in school physics courses and repeated briefly here in the descriptions (to “understand” following Sects. 9.2 and 9.3. We begin by considering the positioning them). Physical models of liquid surfaces. range from point masses A liquid surface always adjusts itself to be perpendicular to the di- in mechanics up to purely rection of the forces which act on its molecules. – In a flat, wide mathematical formalisms in dish, only the weight of the individual molecules acts. The surface theoretical physics. Besides positions itself in a horizontal plane. In the wide basin of an ocean the model liquid made of or a large lake, the forces of weight are no longer parallel at differ- steel balls described here, we ent locations; they point everywhere radially towards the center of will see a further example of the earth. As a result, the surface of the liquid (ocean or lake water) a model concept in Sect. 9.8 takes on the form of a sector of a spherical surface. (fundamental equation of the kinetic theory of gases). In a container which is rotating around a vertical axis, the liquid surface takes the form of a paraboloid of rotation (Fig. 9.2). We will explain this from the point of view of the accelerated frame of reference. Two forces act on every individual particle (molecule): its weight mg, directed vertically downwards; and the centrifugal force m!2r, directed radially outwards. The two forces add vectorially to give the total force F. The surface positions itself perpendicular to

Figure 9.2 The parabolic cross-section of a rotating steel-ball model liquid in a rectangular glass container (photographic snapshot) ), .– 8.5 (9.1) .This 1 ,wehave 9.3 manometers : C9.3 compresses the liquid volume, For the pressure in liquids, p : 2 . r 1 pascal (Pa) D const is the 2 D pressure p A = z 2 F r D p const , newton/meter z r D d d z D r The parabolic surface of a rotating tends to expand it like a tensile stress. , on the left, we see a piston which can be displaced nearly 2 r ! p mg d m r 9.4 D D z ˛ d 2 For solid bodies, all the directions which point outwards from a given closed g ! In Fig. without friction within a cylindering that a is attached liquid. toa the The vessel pointer hold- piston and is scale. itself – attached The to piston a and spring spring balance can with be combined into follows from the freeput deformability of differently, liquids the in pressurestress; every in direction; there is or, a no liquidwithin shear at a stress in rest liquid liquids at isremains at rest, always spherical rest. made under a A visible any normal sphericaldistorted external e.g. volume into force. by an coloring ellipsoid, The but it sphere rather with will at a not mostThe only unit dye, be changes of its pressure radius. For pressure measurements, thesure molecules which are produce replaced the byconverts pres- a the wall which “internal” exerts forcesforces. the within same This pressure, a is i.e. solid accomplished one body in pressure into meters “external” or Figure 9.3 sign is used byanegative convention: A positive pressure 1 volume are considered to be positive.and A a tensile pressure stress is (compressive therefore termed stress) positive is negative. In liquids, usually the opposite External forces produce stresses notbut only also in in solid liquids. bodies However, (Sect. ogy; in the liquids, we stress use is a called different the terminol- this total force at every point. Quantitatively, from Fig. 9.2 Pressure in Liquids. Manometers liquid. tan and integrating, the force is always perpendicular to the corresponding surface 9.9 D Pa. In 5 .The Pa 5 10 is used: 1953); 10 D already in pascal 2 bar introduced Pa, 1.33 hPa. hPa gravity pressure 5 1hPa. 3 )ofacolumnof 1bar. 1.013 ) is usually quoted 10 OHL D 9.4 D D 9.10 Liquids and Gases at Rest 9 100 Pa). In these units, atmospheric pressure is typically around 10 in hectopascals (1 hPa mercury (Hg) with a height of 1 mm. This is the (Sect. 1mbar “atmosphere” is deﬁned by: 1atm The previously-used unit addition, the 1bar the unit name the 12th edition ( however, he did not yet use pressure of the air (Sects. and C9.3. P 1983!): 1 mmHg Organization recommended as early as 1977 discontin- uing the use of this unit by In medicine, e.g. for quoting blood pressure, the older unit mmHg (Torr) is still used (although the World Health newton/meter the unit of pressure 178 Part I 9.3 The Isotropy of Pressure and Its Applications 179

Figure 9.4 Schematics of a piston and a membrane

manometer Part I

a single component in the construction of the manometer. We thus arrive at a membrane which may be smooth or grooved (Fig. 9.4, right). The pressure acting on it produces a curvature which moves the pointer. The membrane, which is caused to bulge by pressure, may be replaced by a tube with an elliptical cross-section (Fig. 9.6, left). The tube stretches when it is filled with a liquid under pressure (think of the rolled-up paper “party horns” which unroll when blown into!). Without calibration, such instruments can be used only to compare spatially or temporally separated pressures. But we will describe in the next section how they can be calibrated to indicate absolute pressures. C9.4. Alternatively, one As proud possessors of such an uncalibrated manometer, we will now can imagine that spheres consider the pressure distribution within liquids. For simplicity, we are drawn within the liquid; first distinguish two limiting cases: they just touch each other and thus produce a connection 1. The pressure in the liquid results only from its own weight. Key- between the piston which word: gravity pressure. produces the pressure and 2. The liquid is contained in a vessel which is closed on all sides. An any of the manometers. At attached cylinder with a smoothly-fitting piston (or “ram”) produces the contact surface between an external pressure, much larger than the gravity pressure of the two such imaginary spheres, liquid itself. Keyword: ram pressure. We will start by considering the pressure from both sides this second limiting case. mustbethesame(actio D reactio), and the same as the pressure in the interiors of the spheres. Therefore, the 9.3 The Isotropy of Pressure pressure measured by the and Its Applications manometer is just the same as that produced by the piston. – For a mathematically Figure 9.5 shows an iron vessel of a complex shape, in cross-section; correct derivation of this fact, it is filled with water and has four identical manometers attached to it. On the right, we can apply ram pressure via a screw which also known as PASCAL’s law, see A. Sommerfeld, presses a piston inwards. All of the manometers indicate the same Mechanik der deformier- readings and thus show that the pressure is the same in every direction (isotropic). – To explain this phenomenon, we imagine a model baren Medien, Akademische Verlagsgesellschaft Leipzig, liquid (steel balls) which has been filled into a container and is being 4th ed. (1957), Chap. 2, Sec- pressed by a piston through a suitable opening. The container will expand outwards on all sides. The free sliding of the steel balls in every tion 6; English see e.g. www. brighthubengineering.com/ direction will not permit a particular direction to predominateC9.4. naval-architecture/106499- We next deal with three important applications of the isotropy of the hydrostatic-pressure-and- ram pressure. pascals-law-for-static-fluids/ . K K To . C9.5 2 ough the laminar ). A tube leads 9.6 with a rotating piston R (Fig. with a tight-fitting piston Z . tion is accomplished through a trick: 10.3 of the piston. Now the essential point: A to the cylinder R The pressure distribution within a liquid when the ram pressure Calibration of a technical manometer Calibration of a technical manometer This experiment also illustrates the lubrication of bearings thr produce the rotation, the topOnce of it the is piston set ismoving in designed rotation, as flywheel it a is continues flywheel. given topointer a rotate of strong for the a kick manometer longof from returns the time. above; to indicator is the The each therefore same time, in reading. fact the determined by The the position weights alone. very small; otherwise, the forcementioned. would Eliminating be the less fric thanThe the piston is weights surrounded just byby a a thin uniform film rotation of liquid. of the This piston is guaranteed around its vertical axis The friction between the piston and the wall of the cylinder must be All of the inner spacesoil. are Pressure filled is with a force liquid, dividedis for by thus example area. equal hydraulic The to ram itsby pressure weight the of cross-sectional plus the area that piston of the mass sitting on it, divided 2 flow of a fluid, as described in Sect. Figure 9.6 Figure 9.5 1. from the manometer is dominant ) as “lubricant”; 10.3 gas Liquids and Gases at Rest 9 nar flow around the piston. (cf. Sect. ify. Here, one makes usehelium of it likewise forms a lami- expansion machine of a helium liquefier, this type of lubrication is not possible, since all lubricants solid- C9.5. At very low temperatures, such as in the 180 Part I 9.3 The Isotropy of Pressure and Its Applications 181

Figure 9.7 An improvised hydraulic press Part I

2. The hydraulic press. This important tool is used to produce large forces by means of small pressures. This is a widely-used application in modern times. As an example, we mention the hydraulic jacks used to lift automobiles in repair shops. We illustrate an hydraulic press (Fig. 9.7) in the form of an improvised setup. Its essential parts are a cylindrical cooking pot A, a thin-walled rubber bladder B, a wooden piston K, and a solid, rectangular frame R. The ﬁlling tube of the rubber bladder is attached to a water faucet. A leather sleeve M around the rim of the piston prevents the bladder from ‘bulging’ up between the piston and the side walls of the pot.

Numerical example The water supply in the lecture room in Göttingen has a pressure of around 4 105 Pa. The cooking pot used has an inner diameter of 30 cm, giving the piston a surface area of about 710 cm2. The press thus yields a force F of roughly 3 104 N. It can for example break oak blocks of 4 5cm2 cross-sectional area and 40 cm length.

3. The compressibility of water. The low compressibility of liquids can be measured readily using the isotropy of pressure in liquids. The principle is the following: One presses the liquid under high pressure into a measuring vessel, avoiding expansion of the vessel itself. To achieve this, the vessel is surrounded by a mantle of liquid C9.6. The compressibility at the same pressure as that inside. The resulting setup is sketched in D .1=V/.dV=dp/,which Fig. 9.8. Initially, a volume decrease V is found to be proportional we introduce here for liquids, to the increase of pressure, p, and to the volume V,i.e.V D and likewise its reciprocal, Vp; the proportionality factor is then measured. It is called the the modulus of compression compressibilityC9.6, K D 1=, were already dis- 5 1010 Pa1 : cussed as material constants of solids (Sect. 8.3) and will 8 Thus, the volume decrease V=V of the pressurized water at 10 Pa be applied also to gases in (about 1000 times higher than atmospheric pressure) is only about a later section (Sect. 14.10). 5 %. – This low compressibility of water can be shown in a variety Note that for liquids and of surprising demonstration experiments. They all illustrate the ap- gases, owing to the isotropy pearance of large forces and pressures when only a small degree of of pressure, a modulus of compression is produced. elasticity cannot be defined. 3 ace. A bullet ) is left out of considera- 9.9 ). It is filled with a liquid of by 9.10 M column (Video 9.2) Hg : Mercury (Fig. ’ explode within a water-filled beaker. Glass A Hg ). – The . The handwheel when the pressure is A h M . This indicates a volume p glass teardrop . The compressibility of wa- A Two glass teardrops ). A glass teardrop is very insensitive to blows and impacts; one in a Gravitational Field. Buoyancy h 9.9 D is no pain orshatters!). damage (as The when the harmlessnesssurprising safety contrast of to glass this the in complete experiment an destructionter of automobile (‘water in a hammer’). window one’s beaker hand filled with stands wa- in Example We start with atight rectangular, and lidless is filled wooden with boxis water. which shot The is through liquid suitably has theequal a water- side to free of upper that this surf ofthe box, the water compressing bullet to the (the risesplintered water time upwards). into by of kindling a its (bladder A volume shot!). passageA considerable is pressure variant too results of short –tomakea‘ this to the experiment allow box requires is teardrops only are a made modest byare allowing solid, effort. molten droplet-shaped pieces glass It(Fig. of to glass suffices drop with into enormouscan water. internal hammer stresses on They it withoutdamage effect. to In its contrast, filamentary it ‘tail’.explodes will When not into the survive fine any tail sort glass isa of broken closed shards. off, hand, the one If teardrop can a clearly feel teardrop how is the exploded fragments fly in apart, this but way there in is employed to press down the piston V The isotropic atmospheric pressure of the air (Sect. rises by a height increased by decrease of the water contained in ter. The thick-walled glass cylinder is filled with water, likewise the thin-walled measuring cuvette H (ram) by a screw drive.as – sealing liquid in the capillarycross-sectional area tube (of Figure 9.8 3 tion in this section. 9.4 The Pressure Distribution We use aa cylindrical cross-sectional container area which stands vertically and has Figure 9.9 Bologna drops. Their ICHTENBERG ”, Wallstein-Verlag Liquids and Gases at Rest Batavia glass 9 Verstreute Aufzeichnungen exhibited by L University of Göttingen, where the properties of glass teardrops were demonstrated. A similar experiment was The video shows a de- partmental seminar at the 1st Physics Institute of the Video 9.2: “The Compressibility of water” http://tiny.cc/6vqujy glass. under the name of or behavior also forms the basis for the shattering of safety 1781 Göttingen 2004, p. 48). The glass teardrops are known “ aus Georg Christoph Lich- tenbergs Vorlesungen über die Experimentalphysik in Göttingen many years earlier (see G.H. de Rogier, 182 Part I 9.4 The Pressure Distribution in a Gravitational Field. Buoyancy 183

Figure 9.10 The gravity pressure of a liquid Part I

density % up to a height h. The weight of this column of liquid is

FG D mg D Ah%g : (9.2)

The weight divided by the area gives the pressure p which acts at the bottom of the container in all directions (isotropy): F p D G D h%g : (9.3) A

Numerical example for water h D 103 m, % D 103 kg/m3, g D 9:81 m/s2, p D 103 m 103 kg/m3 9:81 m/s2 D 9:81106 N/m2, i.e. about 100 times atmospheric pressureC9.7. C9.7. The water pressure at – This pressure compresses the lowest layer of water by only 0.5 % of its the bottom of a 10 m high volume (see above). Therefore, to a very good approximation, one can column of water is about the treat the density % in Eq. (9.3) as independent of the depth h. same as the normal atmospheric pressure (see also The shape and cross-sectional area of the container do not enter into Fig. 9.31). Eq. (9.3). As a result, one can imagine that even the most strangely- shaped containers give the same pressure profiles as a simple vertical cylinder with a constant cross-sectional area. The decisive factor for the gravity pressure at a given point within a liquid is only the perpendicular distance h of the point from the surface of the liquid. C9.8 Quantitatively, we find Eq. (9.3) . C9.8. BLAISE PASCAL (1623–1662). Published in Among the various applications of this principle which are often given “Traité de l’Équilibre des in school books, we recall the familiar liquid manometers used to mea- Liqueurs” (1663); excerpts sure gas and vapor pressures. Their simplest form consists of a U-shaped from the original text are glass tube containing water or mercury as sealing liquid (Fig. 9.11). These manometers can be calibrated using Eq. (9.3) (e.g. 1 mmHg 1:33 hPa, translated into English in see Sect. 9.2). W.F. Magie, A Source Book in Physics, Harvard U. Press The best-known result of the pressure distribution in a gravitational (1963), p. 75. field is the static buoyancy of bodies in a liquid. We consider the buoyant force acting on a body which is immersed in a liquid. For simplicity, we assume that the body has the form of a flat cylinder (Fig. 9.12). The pressure of the liquid has no preferential direction (isotropy). This results from the free deformability of the liquid in every direction, due to the free displacement of its molecules. There- fore, an upwards-directed force F1 D p1A D h1%gA presses on the . (9.4) C9.9 ) which acts F /: 2 h shows the shadow image of Archimedes’ principle ( 1 h 9.13 . gA % D acts on the upper cylinder surface. All the F . Figure gA % 2 , the wooden ball rather high above the surface, h : . It yields an upwardly-directed force D of the cylinder, while a smaller, downwards-directed 2 float F A A 2 Buoyancy in p The origin of static buoyancy Liquid manometer model liquid and D The buoyant force on an immersed solid body is equal to the 1 buoyancy 2 F F The product on the right is justwhich the has weight of the that same amount ofgeneral: volume the as liquid the immersed body.weight of We the thus liquid find which it in displaces Figure 9.11 a steel-ball model liquid (Video 9.3) a glass containerviously half-filled buried with two many largeris small balls made steel within of balls. the wood,motion small the within We other balls; our of pre- one aluminum. modelvigorously. of liquid Immediately, them We buoyancy as simulate brings the the usualsurface. two thermal by large They shaking balls to thethe the container aluminum ball about at its midline. Figure 9.13 There are astead variety of demonstrating them, of we will quantitative elucidateusing the experiments origin of our buoyancy on buoyancy. In- Figure 9.12 lower surface force forces acting on theother exactly, side pairwise. surfaces There of remainsforces only the the cylinder difference of compensate the each two on the body assimply a whole. This force is called the buoyant force or of an ob- O % Liquids and Gases at Rest 9 fluid” http://tiny.cc/fwqujy Video 9.3: “Buoyancy in a model again when fully immersed in a liquid of known density. ject. One need only weighonce it outside the liquid and mass density C9.9. This principle forms the basis of a very simple method of determining the 184 Part I 9.5 Cohesion of Liquids: Tensile Strength,Specific Surface Energy, and Surface Tension 185

Figure 9.14 The metacenter Part I

Of course, one cannot expect a quantitative veriﬁcation of the formula for the buoyant force from such a model experiment. The simulation of thermal motions by shaking is too primitive for that.

The weight of an object and its buoyancy in a liquid oppose each other. When the weight is greater, the object sinks to the bottom of the liquid. When the buoyancy is greater, it rises to the surface and floats there. The transition between these two possibilities illustrates a special case: The object and the liquid which it displaces have exactly the same weight (equal densities). In this case, the object remains suspended at an arbitrary depth within the liquid. This special case can be demonstrated in a variety of ways. As a particular example, we mention a ball of amber in a solution of zinc sulfate of suitably-chosen concentrationC9.10. C9.10. As a simple demonstration, we recall the “Carte- When the buoyant force is greater, a portion of the object rises above the sian diver” which is often surface of the liquid. The object comes to equilibrium when the displaced used as a toy. A small glass water has just the same weight as the object itself. Then we say that the figure (often a little devil) object floats. For practical applications (ships), the stability of the floating containing air is put into attitude is very important. It is determined by the position of the metacenter.InFig.9.14, imagine that a ship is tilted (listing) by an angle ˛ from a test tube filled with wa- its equilibrium position. Let S1 be the center of gravity of the volume of ter; it is closed at its top end water displaced by the ship in this tilted position, i.e. the point of action of by a rubber membrane (or the buoyant force in this position. We draw a vertical line through the point a movable rubber stopper). S1. Its point of intersection with the central midline of the ship is called the The position of the figure metacenter M. This metacenter must remain above the center of gravity S can be readily projected as of the ship itself during any tilting or listing; only then will the resulting torque due to the buoyant force bring the ship back to its equilibrium rest- a shadow image onto the ing position. Only when its metacenter is above its center of gravity can wall of the lecture room. The a ship float in a stable manner. tail of the figure has a small opening, so that pushing on the membrane or stopper presses some water into the 9.5 Cohesion of Liquids: Tensile figure, thus increasing its overall density. The three Strength, Surface Energy and cases rising, sinking and sus- Surface Tension pension can be convincingly demonstrated by regulating the pressure with the mem- Our model liquid (steel balls) has yet to explain two well-known brane. properties of real liquids: The molecules of a real liquid are subject to attractive forces, they exhibit cohesion. When the liquid is poured, they do not move apart in all directions, but rather form droplets of different sizes and shapes. Furthermore, real liquids stick to solid bodies (adhesion). This sticking can even lead to wetting;thatis,the inertial (thisisthe (comparison 2 has been achieved; body is greater than 2 7N/mm D solid ). From the considerations tensile strength 4N/mm itself. – These shortcomings : max 8.2 , one would expect values at 3 Z of the U-tube is evacuated. In g occurs, the attraction between D B 15.9 liquid max Z and those of the . This column carries its own weight, and liquid column ). , tiny impurities in the liquid or on the walls of to the walls of the tube (adhesion). Only then can 8.8 illustrates the same experiment with a real liquid, here shows a sectional view along the length of an iron pipe The tensile strength of a model liquid 9.16 9.15 to act on the water column, it may require several tries before stick firmly nucleation centers which will be discussedleast 10 in times Sect. larger. Most likely, the tensile strength is reduced by values for solids can be found in Table for ethyl ether, the maximum value is Figure 9.15 liquid cannot be pulled off thespread solid it around body, and further. attempts When to wettin the do molecules so of only the that between the molecules ofof the the model liquidsteel can balls. be Then they overcome; areof one attracted an both need iron to only container. each other use and magnetic to theThe walls model liquid whichanother has important been fact modified inperience: that this is Liquids way not have leads a us well to considerable known from everyday ex- forces the column breaks apart. One point isof central for water, this andmust demonstration for of the the modela tensile liquid: strength cross-sectional constriction TheThis of molecules is the of the column reason theof of why the liquid tube; no liquid they gas would beformation bubbles immediately of avoided. act should constrictions as be nucleation (and leftFor points resulting water, for on a separation the tensile the of strength walls of the column). which is closedmodel at liquid. its top, Theform and a magnetic continuous is “molecules” filled stickthus with it to the has its a steelFigure walls. tensile balls strength of in They addition the a to column its of adhesion water. tothis the The manner, we walls. right can arm hangThey columns of have water a up tensile toThe strength many long which meters glass tube long. is can be often convenientlyboard attached surprising to is to a struck board; observers. when hard the on the floor, causing strong downwards limiting value of theration; tensile cf. Sect. stress which leads toFigure fracture or sepa- Liquids and Gases at Rest 9 186 Part I 9.5 Cohesion of Liquids: Tensile Strength,Specific Surface Energy, and Surface Tension 187

Figure 9.16 The tensile strength of a water column. The water has been puriﬁed of dissolved air

by boiling in vacuum. The volume B thus contains Part I gaseous water (water vapor) which adjusts to the corresponding vapor pressure (Sect. 15.7). At 20 °C, it is 23.2 hPa (see Fig. 15.11). From this we find, assuming that the water vapor behaves as an ideal gas, that its density is % D 1:7 102 kg/m3 (see Sect. 14.6). (Video 9.4) Video 9.4: “The tensile strength of water” http://tiny.cc/3vqujy To avoid gas bubbles on the wall of the tube, it must be swung back and forth several times in a horizontal position, which removes remaining the container, as discussed in in Sect. 15.9, for example, very small gas bubbles. – A further in- nitrogen-containing gas bubbles. teresting observation is the following: When the liquid In our treatment of solids, we have already mentioned the essential hits the end of the tube while relation between the tensile strength Zmax and the specific surface it is being swung, one may energy , i.e. the quotient hear a loud bang, as if the Work necessary to increase the surface area W tube had been hit by a ham- D : (8.34) mer. This does not occur Size A of the newly-formed surface in the air, since a stream of Here, both A and W can be either positive or negative. Positive waterissloweddownbyair means an increase in the surface area. Then, a force F must perform resistance, for example be- work, and it will be stored in the surface as potential energy. Negative fore it hits the side of a wash means that the surface area decreases; then previously stored energy basin. The water vapor at is released, performing work and giving rise to a force. In dealing the end of the tube obviously with solid bodies, we treated only the first case; to demonstrate the does not show this effect; it second case in solids, one requires very high temperatures or long immediately liquefies when times. This is quite different for liquids: The free sliding of their the vapor is compressed. The molecules allows us to implement both cases. speed of this phase transition A well-known example is shown in Fig. 9.17. A liquid film from the gas phase to the liq- (e.g. a soap solution) is held from above and on both sides by a U- uid phase is also discussed in Comment C16.2. shaped wire frame, and from below by a freely sliding wire which is attached to the frame by loops and can slide along it. This “slider” or “strap” can be adjusted to any desired height by a suitably-chosen load (force F). A displacement by ˙x produces a new surface area A D˙2lx (front and back surfaces of the film!), so that the force F performs the work ˙W D˙Fx D˙2xl: The distance ˙x cancels. The remaining result is F D 2l: (9.5)

The strength of the force F is thus independent of x,i.e.ofthe magnitude of the extension which has already taken place. This is : (9.6) liquid N/m edge is tension 3 10 . For liquids, sharp D 2 cylindrical boundary Ws/m are generally smaller. 3 2.5 75.5 72.5 62.3 29.2 12 10 500 Specific Surface Energy or Surface Tension in surface work (Video 9.5) energy or ). A ring with a 9.18 lists some numerical values, which When the liquid films are in contact . 0 C 9.1 boundary 18 20 80 18 ı 190 254 C9.11 Temperature in of the movable edge of the surface l , friction between the slider and the frame is .Table r is measured with a balance. The circumference ) yields 2 is often referred to as the F parallel to the surface and required to extend it 9.5 D F Length 2 l A soap-solution film in equilibrium; this is The specific surface energy (surface tension) of some liquids Force D Liquid hydrogen Liquid Mercury Water Benzene Liquid air film instead of a planardipped film into the (Fig. surface oflowered, the liquid. a cylindrical When the film liquidtube. is level formed, is The slowly similar force to a thin-walled, short also an example of “reversible” surface work with materials other than air, the values of think for example of a dropleta of liquid. mercury or In of both a cases, small bubble for within the filled sphere as well as for the empty Therefore, the termwould specific be more appropriate. Without external influences, liquids often form spherical surfaces; Figure 9.17 Table 9.1 an essential difference comparedpopular to comparison between stretching a a liquid surface rubber andtherefore a be sheet. rubber used sheet only must The with caution. Rewriting Eq. ( of the ring is refer to liquid surfaces in air both names are equally valid. In measurements of a perturbing effect. It is thus preferable to use a For this reason, Liquids and Gases at Rest 9 “Surface work” http://tiny.cc/tvqujy Video 9.5: drogen C9.11. except for liquid hy- 188 Part I 9.5 Cohesion of Liquids: Tensile Strength,Specific Surface Energy, and Surface Tension 189

Figure 9.18 The measurement of the speciﬁc surface energy using a spiral spring balance.

A numerical example for water: ring diameter Part I 5 cm, circumference 2l D 0:31 m, F D 2:26 102 N, D 0:072 W s/m2.

sphere, the surface tension produces a pressure inside the sphere:

2 p D : (9.7) r

Derivation The radius r of the sphere is presumed to increase by the amount dr.Then the surface area of the sphere increases by dA D 8rdr and its volume by dV D 4r2dr. During this volume expansion, the pressure performs the work 2 dWV D pdV D p4r dr : (9.8) Producing the new surface area dA requires the work

dWA D dA D 8rdr: (9.9)

Setting equal these two values for the work yields Eq. (9.7).

This important equation (9.7) is often used in a strict context, but also for approximations. Examples:

1. A mercury droplet at the limit of microscopic visibility has a radius r D 0:1 m D 107 m. for Hg D 0.5 W s/m2; thus we ﬁnd

2 0:5Ws=m2 p D D 107 Pa .i.e. 100 times atmospheric pressure!/ 107 m 2. Everyone knows the experiment sketched in Fig. 9.19. A wetting liquid is placed between two planar glass slides. It forms a concave surface, whose smallest radius of curvature r is d=2. Surface tension produces a pressure p of order of magnitude given by Eq. (9.7). The direction of p is marked in the ﬁgure by arrows4. Glass plates which are “glued” to- “This is a convenient but gether with water in this way cannot be separated by applied forces without lax way of putting it. The damaging them. They can only be slid apart very slowly under water. pressure itself has no direc- 3. A perfectly wetting liquid is pulled upwards into a capillary tube of tion, but instead only the radius r, up to a height h (Fig. 9.20). – Explanation: The liquid has a hollow (concave) upper surface (meniscus). Its smallest radius of curvature corresponding force.” is r. Then the pressure calculated from Eq. (9.7), p D 2=r, yields an upwardly-directed force of F D A2=r. The downwards-directed weight

4 This is a convenient but lax way of putting it. The pressure itself has no direction, but instead only the corresponding force. . ) g : % 9.10 Ah (9.10) causes ) corre- p D G aces within 9.10 F . – Equation ( h capillary elevation illary eleva- ; this effect also plays a role in of the surface area of a bubble N. 16 2 : 3 g m, , occurs for example behind rapidly % 10 2 ): “cap 7 r nu- W s. The resulting pressure 6 Pa ( : 6 D 10 9.10 1 5 h )). A 10 10 D D in its interior, of height cavitation 9.7 F r 16 tting liquid, e.g. Hg in glass, the meniscus is up- m, D r . Therefore, each cm 2 . p 2 2 : , 0 2 D C9.12 d . As a result, the pressure calculated from Eq. ( D : The wetted surface area is d A water layer – intentionally Ws/m Applications of Eq. ( , 08 W s/m : 2 2 capillary depression 0 convex 10 , diameter 10 cm h 8 the rise of sap in plants. 4. Strong inertialliquids. forces This can process, called produce bubbles, i.e. hollow sp is often employed for the measurement of rotating ship’s propellers or in waterof turbines. – Water has a surface tension The equilibrium between these two forces yields the sponds to a downwards-directedduces force. a A tube dipped into mercury pro- has a potential energy of 8 surfaces into a regionenergy of act only like a very high few localpellers molecules. temperature and spikes. turbine These blades As are concentrations a “eaten of result,with up” ship’s by deep pro- the holes. water; theysound become waves. pitted – The Cavitation local concentrations canisms which of also live energy in can be water, and destroy can produceddissolved small cause gases light organ- by flashes from high-frequency water containing In the case ofwardly a non-we the bubbles to collapse very quickly and compresses the energy of their of the liquid column is equal and opposite to this force, with D tion” Figure 9.19 times atmospheric pressure), drawn much too thick hereglass – plates (referring between to two Eq. ( merical example A example, one can place a slightly oily sewing needle carefully onto Among the numerous other examples of surfaceonly tension, we a give here brief selection.resemble In a the slightly stretched first, wrapping the or surface skin. of1. a liquid Water appears cannot to weton greased a or water surface oily as objects.cushion. on The surface a Such is loosely-filled objects visibly slightly cushion, can ‘dented’ for below rest the example object. an For air Figure 9.20 ) sonolumines- Physikalische Blät- has been studied in , 1087 (1995). English: https://en.wikipedia.org/ Liquids and Gases at Rest 51 9 ter e.g. wiki/Sonoluminescence detail in recent years, but is still not completely understood (see for example D. Lohse, C9.12. This phenomenon of so-called cence 190 Part I 9.5 Cohesion of Liquids: Tensile Strength,Specific Surface Energy, and Surface Tension 191 a water surface and it will rest there, similarly to the legs of a water strider. Part I Liquid fuels wet every solid object. Therefore, one never sees dust on their surfaces. Furthermore, they drip very slowly out of a container hung in the open; the completely wetted walls of the container act like a syphon. This process occurs very quickly when the liquid is the ‘ideal fluid’ 4He in its superfluid phase (found at temperatures below 2.17 Kelvin), which completely lacks internal frictionC9.13. C9.13. For more information on the fascinating phe- In our remaining examples, the surface tension produces the maxi- nomenon of the superfluidity mum possible reduction of the surface area of a free liquid surface of liquid helium, see for (minimal surfaces). example K. Lüders, “Super- 2. Mercury is injected as a fine stream into a flat watch glass, filled flüssigkeiten”, in Bergmann/ with a liquid. It initially forms numerous droplets at the bottom of Schaefer, Lehrbuch der Ex- the glass, of roughly 1 mm diameter (Fig. 9.21). The resulting overall perimentalphysik,Vol.5, surface area of the mercury is thus quite large. But then the droplets Chap. 5, Verlag de Gruyter begin to combine in a jerky manner; here and there, a small droplet 2nd ed. (2006). English: see is taken up by a larger one, which limits the lifetime of the small e.g. https://en.wikipedia.org/ droplets5. After about 1 minute, only a single large puddle of mer- wiki/Superfluid_helium-4 cury is present. The surface area of the mercury, driven by its surface tension, has been reduced to the minimum possible under the circumstances. This is an especially informative demonstration: The small “This is an especially infor- droplets are “physical individuals”. The fate of a particular individual mative demonstration.” cannot be predicted using physical methods; one can never say which of the droplets will be the next to disappear. Nevertheless, for the set of all the individuals, we can find a clear-cut physical rule: Their C9.14. This is an example number decreases according to an exponential lawC9.14 with a given of the exponential function, mean lifetime (in Fig. 9.21, D 10 s). One can thus make quite which occurs very frequently precise statements about a large set of individuals, even when such in physics; here with a nega- statements are completely meaningless for particular, isolated indi- tive exponent: t= viduals. This fact plays an important role in atomic physics (e.g. in N.t/ D N0 e . radioactive decay processes). N is the momentary number 3. A water surface is covered with a non-wetting powder. Then, for of individuals at the time example with a needle, one adds a tiny amount of a fatty acid at the t. Other examples of ex- center of the water surface. Immediately, the surface breaks up, and ponential behavior are the a clear-cut, circular spot appears, which is free of the powder. “barometric pressure formula” (Sect. 9.10)andthe Explanation: The surface tension of the water is greater than that of damping of harmonic oscilla- the fatty acid. Therefore, a droplet of a fatty acid on a water surface tions (Sect. 11.10). forms a monolayer whose thickness corresponds to that of a single molecule. If N molecules of a fatty acid, each with an area a,are placed onto a water surface, they will spread out to cover a circular C9.15. AGNES POCKELS area of A D Na. This allows us to determine the molecular cross- (1862–1935): cf. Nature 43, sectional area a if we know the number N of molecules added. This 437 (1891). She was self- apparently modest demonstration is in fact quite important. – For taught and wrote fundamental measurements, one uses a rectangular water surface and replaces the articles about films on liquid powder by a floating movable, rectangular frame (AGNES POCKELS, surfaces, for which she was 1891)C9.15. granted an honorary doctoral degree (Dr.-Ing. E.h.) from 5 Reflections of light simulate the appearance of bridges between neighboring the Technical University of droplets in some places in the image. Braunschweig in 1932.

2 100% 0 s 40 s 60 s 30 s 20 s 50 s 10 s 37% ≈1/e 14% ≈1/e . 29 droplets 78 droplets 205 droplets thermodynamics Similar processes play a role ) Exercise 9.8 ( 10 s, i.e. after each (Video 9.6) D s). The larger droplets are 3 The coalescence of mer- . That is, its magnitude becomes dependent on how much 10 37 % of the preceding number (pho- e = Figure 9.21 sometimes distorted because they have been set into oscillation by coalescingsmaller with droplets. 4. The addition offace impurity tension. molecules This changes can the besome value demonstrated water. with The of different a the faces grain of sur- thus, of the grain the camphor dissolve and surface at tension different rates; varies‘dances along around’ different on directions. the Thein water grain the surface. movements of microscopic life-forms (single-celled5. animals). “Oiling the seas”.foam-covered An crests oil into slick rollingchange converts in swells. the surface “breakers” tension, with a Tooil ship their produce in need the the only release form necessary small of amounts droplets of ontoWhen the ocean’s impurity surface. molecules aretension involved, the becomes rather phenomenon complex. of The surface abnormal surface tension is then termed the surface area hasbrane. already been Furthermore, increased, the as increasewarming. with in a Kinetic surface rubber energy mem- areavery is is interesting converted accompanied topics into properly by belong heat. to These potentially cury droplets in alcohol containingamount a of small glycerin. This isof a a good process example which proceedsa according statistical law: to with anumber sufﬁciently large of droplets (as inimages), the one upper can three determine thelifetime to mean be tographic images, each with antime exposure of 4 10 s, the number of1 droplets decreases to Liquids and Gases at Rest 9 Video 9.6: “Coalescence of Hg droplets” http://tiny.cc/mwqujy The coalescence of small mercury droplets under water is demonstrated. 192 Part I 9.6 Gases as Low-Density Liquids Without Surfaces.BOYLE’s Law 193 9.6 Gases as Low-Density Liquids

Without Surfaces. BOYLE’s Law Part I

The mass densities % of gases are considerably less than those of liquids. As an example, in Fig. 9.22 (left side), we show a measurement of the density of room air, % D 1:29 kg/m3. It is thus only about 1/800th of the density of liquid water. The molecules are the same in a gas and in the corresponding liquid. Therefore, the smaller density of the gaseous form can only result from greater average distances between the individual molecules. The following facts also support the hypothesis of large intermolecular distances in gases: 1. Gases, in contrast to liquids, have extremely high compressibilities (bicycle tire pump!). As a result, the density of a gas increases with increasing pressure. At p D 160 105 Pa, we ﬁnd for example for air a density of % 200 kg/m3, i.e. about 1/5th of the density of water (Fig. 9.22, right side).

2. The BROWNian molecular motion can be observed in gases at a much lower magnification than in liquids. To provide visible fine particles, tobacco smoke can be conveniently used. 3. The molecules of a gas fly apart in complete disorder in all directions. They distribute themselves randomly in any available volume. Imagine for example a leaking gas-line in a room, or the gaseous, fra- grant components of a perfume. In contrast to liquids, no attractive forces or cohesion of the molecules can be observed in gases without resorting to very sophisticated methods. In any case, gases do not form free surfaces. The cohesive forces between the molecules are

Figure 9.22 The dependence of the density % of air on the pressure p. Left- hand image: p D 105 Pa (normal atmospheric pressure), the glass balloon with V D 7 liter is evacuated and the balance is tared (with the weights A). Then we allow room air to flow into the balloon; in order to restore the equilibrium of the balance, we have to add 9 grams to the balance pan. Thus, % D m=V D 9 g/7 liter D 1.29 kg/m3. Right-hand image: p D 160 105 Pa in a steel pressure cylinder with V D 1 liter. After the compressed air is released, we have to remove 205 grams from the right pan, i.e. % D 205 g/liter D 205 kg/m3. . ’s ideal :The gases. (some- (9.11) volume is 9.23 real ARIOTTE ’s law occur, . Because of ’s law specific OYLE OYLE law, or M are proportional to each specific volume of the gas). relationship between pres- : ften excellent – approxima- D ), carefully keeping the tem- limiting-case law M ARIOTTE = const const (9.12) const (9.13) 9.22 V –M D gas. At normal pressures and tem- % , and chlorine behave as D V M 2 s s mass density V D D pV p real OYLE . Two other formulations are somewhat p ,NO 2 If noticeable deviations from B , that is of pressure, mass and volume for gases, has is represented over a large range of pressures and . mass density and ’s law is thus a typical M D = C9.16 The pressure p is directly proportional to the total mass V = OYLE pV gases, while CO s law is obeyed to a good – and o M ’ D % ( ideal OYLE peratures, for example air, hydrogen, theas noble gases, etc. all behave and then one is dealing with a law): times referred to as the B other, or, equivalently, the product of pressure and its importance, we willIn formulate an its ideal content gas, once pressure more and in words: This distinction loses its validitysufficiently at high sufficiently temperatures: low Then pressuresgases. all and/or substances B behave as In words: M of the confinedvolume gas V and of ismore its proportional compact: container to the reciprocal of the tion by all gases atThis sufficiently high is temperatures and demonstrated low by pressures the examples collected in Fig. product B sure and density been thoroughly investigated (Fig. clearly not very effective at theof large gases. intermolecular distances typical This completes our first overview. – The perature constant. The results ofrelation such over measurements a large follow range a of simple values; it is called B constant. temperatures by horizontal linesindependent parallel of the to pressure thepressures over abscissa; the and it range temperatures, shown. is substancesideal thus In in gases this the range gas of phase are called )) to the equa- ) in thermody- 9.13 ’s law (in the form 14.19 Liquids and Gases at Rest 9 OYLE namics). tion of state for ideal gases (Eq. ( of Eq. ( C9.16. Taking the temperature dependence into account leads directly from B 194 Part I 9.7 A Model Gas. Pressure Due to Random Molecular Motions (Thermal Motion) 195

Figure 9.23 The horizontal straight lines are examples of the range of

validity of BOYLE’s law for ideal Part I gases. The deviations which can be seen outside this range will be treated in Sect. 15.1. The vertical sections of the curves occur when part of the gas condenses (liqueﬁes).

9.7 A Model Gas. Pressure Due to Random Molecular Motions (Thermal Motions)

The facts presented above can be well illustrated by a model gas. This will be shown in the present section and in the following sections. – As molecules, we again use steel balls, as in our tried and tested model for liquids. But now, we give these “molecules” much more free space within a voluminous “gas container”. The latter is a flat box with glass side windows (Fig. 9.24). Furthermore, we now produce the random motions of the model molecules (the “thermal motions”) by means of a vibrating steel piston A. It serves as one end wall of the gas container. The other end wall B is also a piston; it can be displaced with little friction, and together with a connecting rod and a helical spring S, it serves as a pressure meter (manometer). Now, when the apparatus is in operation, all of the steel-ball molecules fly back and forth with a lively motion. They continually collide with each other and with the walls of the container. These collisions are elastic. Every “molecule” is constantly changing its velocity, in magnitude and direction. It presents a picture of truly random thermal motion. This random molecular motion produces the pressure of the model gas against the walls of the container. We start by measuring this pressure using our manometer. The pressure of a gas against its con- . 2 mu 1 2 (‘ram’) In this area of mean A (9.14) D D D molecules 2 A together form ( kin N u E S pA on the right can be :All B s 2 u V can move freely within the 1 3 S postulate D . p . The gas molecules continually (Video 9.7) 9.24 . and the helical spring or B ) against the wall. The sum of all specific volume of the gas, t B d D 2 F u s in Fig. V R % S 3 1 . Piston D C p density, D % A model gas consisting of steel balls. The piston of the Kinetic Theory ofVelocity Gases. of the Molecules pressure, . In the case of a liquid, the pressure is produced by “loading”, and serves to guide the piston D C p value of the squared velocities of the molecules). ( Then with a brief computationfollows (given immediately), we in can the derive fine-print thekinetic paragraph fundamental theory equation that of of the gases: the wall). The wallequal can but remain opposite at force, directed rest inwardsfor only towards example the if by gas, it the produced spring is acted upon by an vibrates to produce “thermal motions”, while the piston The origin of the pressurequantitatively. of a We gas need explained only above can fulfill be one described sense, gases exhibit a completelylack of new cohesion phenomenon, and related of topatter free surfaces their against the container walls.a wall Every reflection entails of anthese a collisions impulse molecule acts ( by like a steady force of magnitude 9.8 The Fundamental Equation Figure 9.24 tainer walls results from a quitea different liquid process than the pressure of e.g. by the weight ofing the a piston liquid into itself a (gravity closedWe pressure), system made containing or the no by liquid mention press- (ram of pressure). collisions a of liquid pressure its resulting molecules from against the the random walls of its container. displaced along the tube a manometer. The rod visible within the spring are presumed to haveit the is same independent kinetic of energy, the averaged volume over of time; the gas container: tube , ) Video 16.1 Liquids and Gases at Rest 9 (see also http://tiny.cc/xhgvjy “A Model Gas: Its barometric density distribution” http://tiny.cc/rwqujy Video 9.7: 196 Part I 9.8 The Fundamental Equation of the Kinetic Theory of Gases.Velocity of the Gas Molecules 197

Figure 9.25 Derivation of the pressure of a model gas Part I

Derivation In Fig. 9.25, the gas container of volume V is supposed to hold a total of N molecules, each of mass m; the total mass of the gas is M D N m.Then the density of the model gas in the container is

Nm M % D D : (9.15) V V

We wish to compute the pressure acting on the left-hand side wall of the container (of area A). A molecule with the velocity u1 travels a distance s D u1t in the time t. As a result, only those molecules which are within the shaded volume, As D Au1t, can reach the left side wall within the time t. In the whole volume, there are N1 molecules with the velocity u1;then in the smaller shaded volume, only a number Au1tN1=V is present. The molecules ﬂy around in a disordered manner. None of the six spatial directions (˙x; ˙y; ˙z) is preferred. Therefore, only 1/6 of the molecules is moving on average in the direction (x) towards the left side wall of area A. Thus, within the time t, only 1/6 of the molecules within the shaded vol- 1 N1 ume will collide with the wall A,thatis 6 V Au1t molecules. To simplify the computation, we assume that these molecules hit the wall in a direction Rperpendicular to it. Then each individual molecule transfers an impulse F1dt D 2mu1 to the wall (Sect. 5.5), since the collisions are all elastic. The sum of all these impulses within the time t is

1 N 1 N m 2mu 1 Au t D 1 Au2t : (9.16) 1 6 V 1 3 V 1

0 We can replace this sum by an impulse F1t, which acts during the time 0 t with the constant force F1. Then for the pressure produced by the N1 molecules with the velocity u1,weﬁnd

F0 1 N m p D 1 D 1 u2 : 1 A 3 V 1

We would obtain corresponding values p2 for the N2 molecules with the velocity u2, and so forth. Finally, we add up all the partial pressures p1, p2, p3 ::: from the N1, N2, N3 ::: molecules with velocities u1, u2, u3 ::: We 2 set p D p1 C p2 C p3 :::and N D N1 C N2 C N3 :::and deﬁne u as the arithmetic mean of the squared velocities, so that

.N u2 C N u2 C N u2 C/ u2 D 1 1 2 2 3 3 : N

We then obtain 1 Nm p D u2 : (9.17) 3 V According to our postulate, the kinetic energy of a molecule is on average constant, and thus so is u2, the average value of the squared velocity. Furthermore, Nm D M, i.e. it is equal to the total mass of the gas in the . C9.17 ,they .The rms ) )as u average ’s law! 9.12 9.14 weight OYLE , from the cor- of the gas 2 % u mean values p : 3 D eacher in Berlin). ). rms ); we obtain Eq. ( u 480 m/s. Likewise, for hy- 3 kg/m : 16.3 9.17 1 D and the density ) yields for the velocity of the air const ( ): To leave the earth, it would re- p rms . Each air molecule obeys the same % u 4.9 , the density of the gas. This yields with 9.14 % ;% D , all the air molecules would fall to the p D Pa weight V , 1856; a high-school t 5 = M 10 NIG Ö D R V p = Nm ) makes it possible to calculate the root-mean-squared thermal motion ) 9.14 Atmospheric Pressure in Demonstration Experiments 9.17 Eq. ( given above (A.K. K The constant likewise follows from Eq. ( container, and This means that our simple model leads quantitatively to B responding values of the pressure true (momentary) velocities ofaround the the molecules mean value are (details distributed in Sect. widely 2 km/s. In termstainly of reliable. the As order we of have magnitude, emphasized, this it calculation yields is cer- Inserting these values into Eq. ( molecules at room temperature drogen at room temperature, we find a molecular velocity of 9.9 The Earth’s Atmosphere. The air, just like ourLacking model a gas, free expands surface, to it fill hastain the no its available fixed volume. atmosphere? volume. Why How do does– the the Answer: air earth Like molecules re- all not bodies, fly thecenter out air of into molecules the are space? earth attracted towards by the their Equation ( For air under normal conditions, for example, we find velocity of the gas molecules, defined as velocity of the airsult, molecules the vast is majority much of the lessweight. air than molecules is this bound value. to the earth As byWithout a its their re- earth like stones, and – incidentallyearth’s – surface they of would around form 10 a m layer thickness. on Without the their quire a velocity of at least 11.2 km/s (escape velocity). The conditions as a projectile (Sect. would immediately fly off the earth,between thermal never to motion and return. weight The however keepssuspended competition the and air molecules leads toearth, the its formation atmosphere. of the The free solid air surface of mantle the of earth the prevents them ). 16.3 16.9 rms u ). 17.10 ) and Fig. 16.3 in later sections Liquids and Gases at Rest u 9 It will be abbreviated as C9.17. A precise statistical definition of will be given in Sect. (Eq. ( (e.g. Sect. 198 Part I 9.9 The Earth’s Atmosphere. Atmospheric Pressure in Demonstration Experiments 199 Part I

Figure 9.26 Two Magdeburg hemispheres are being pulled apart by 8 (not 16!) horses (Video 9.8) Video 9.8: “Magdeburg hemispheres” (Otto von Guericke’s exper- from falling closer to its center of gravity. Therefore, the earth’s sur- iment) face carries the full weight of the air contained in its atmosphere. The http://tiny.cc/vwqujy quotient of this weight divided by the surface area is the normal grav- The entire lecture, although ity pressure of the air, for short ‘air pressure’ or ‘surface atmospheric in German, can be seen pressure’. It is around 1000 hPa D 105 Pa (this corresponds to the at G. Beuermann, http:// gravity pressure at the earth’s surface of a mercury column 76 cm lichtenberg.physik.uni- high). goettingen.de. “We humans lead a deep-sea life at the bottom of the enormous ocean of air”. Every school child knows this today. The – at the time “Every school child knows sensational – experiments carried out a few centuries ago to prove this today.” the existence of “air pressure” are today among the most elementary C9.18 topics of school physics . Nevertheless, for reasons of historical C9.18. POHL often men- deference, we will describe a classic demonstration. The mayor of tions school physics. That Magdeburg, OTTO VON GUERICKE6 (1602–1686), pressed together reveals what sort of previ- two copper hemispheres of 42 cm diameter with a greased leather ous knowledge was expected seal and pumped out the air inside them. The hemispheres were then of beginning students dur- seen to be pressed ﬁrmly together by the air pressure of the surrounding his lifetime; and from ing atmosphere. We can compute the force as the product of the all students. His lectures cross-sectional area of the hemispheres (A 1400 cm2) and the air were required not only of pressure (p 105 N/m2): F D 1:4 104 N. Thus, GUERICKE needed physics majors, but also of 8 horses to pull the hemispheres apart. The woodcut shown to a very many non-majors including small scale in Fig. 9.26 illustrates a demonstration of this famous ex- pre-med students. This high- periment. The picture indeed shows not 8, but 16 horses; that was of lights a problem that today’s course a bluff intended to impress the spectators, who were for the physics lecturers must deal most part laymen in matters of physics. Eight of the horses could with, namely the often very just as well have been replaced by a solid wall; since even back then, different levels of knowledge force equaled counter-force. of elementary physics among beginning students, due to the Today, the Magdeburg hemispheres still carry on a modest but useful ex- different standards of their istence in a puny form – that is as the well-known canning jars, consisting schools, even among natural- of a glass jar with a lid and rubber gasket. They are evacuated not with science majors.

6 An excellent excerpt from his principal work, “Nova experimenta (ut vocantur) Magdeburgica”, was published in 1912 by R. Voigtländer, Leipzig, as a German “Since even back then, translation. No beginning physicist should miss reading this book. The experi- force equaled counter- mental skills of GUERICKEand his descriptions, which strive for a clear simplicity, force.” are exemplary. , right 9.28 the syphon It is explained in . Exactly the same prin- H In principle, a liquid syphon functions . H ). As a result, a water syphon will function 9.5 A liquid as an effect of air pressure. However, (left side). The overhanging end of the stream of water Left: Chain ‘syphon’ . A chain is hanging over a frictionless pulley; each of its syphon 9.28 Loading of a liquid sur- 9.27 a pump, but rather by usingtheir hot contents steam against to anaerobic displace bacteria!). theof air After the (this cooling steam, also and a sterilizes condensation “vacuum” results. side. In practice,making this use loading of can the be pressure most of simply the accomplished atmosphere. by It can hold together entirely without any action of the air pressure. The liquids wenever encounter completely in free our ofof daily water small lives, containing air especially airovercome water, bubbles. by will are loading separate the Therefore, liquid easily. surfacesample a equally in This on column principle both difﬁculty by ends, can using for frictionless ex- be pistons as in Fig. quite well in a vacuum,on so long the as walls no of visiblein gas the Fig. bubbles are tube. present is A marked vacuum with syphon its of length this type is shown Figure 9.27 syphon functions in a vacuum. Right: face holds together a column of liquid, even when itgas contains bubbles. down by the weight of its overhanging end In elementary physics classes,known one often alsoprinciple encounters has the nothing well- Fig. to do withends air is lying pressure. coiled upor in lowered, a the glass chain jar. runs When into one the of lower the of jars the is two raised jars; it is pulled Figure 9.28 ciple applies to liquids,tensile strength since (Sect. they, like solid bodies, have a ﬁnite Liquids and Gases at Rest 9 200 Part I 9.9 The Earth’s Atmosphere. Atmospheric Pressure in Demonstration Experiments 201

Figure 9.29 Gas syphon. On the right, a carbon dioxide

cylinder with a pressure- Part I reducing valve and hose ﬁll the upper beaker with the heavy gas

a column of length L 10 m even when bubbles interrupt the column across the whole cross-section of the pipe containing it. 10 m is however only a small fraction of the column length corresponding 2 to the tensile strength of gas-free water .Zmax D 3:4N/mm ). This can be rather clearly seen from the fact that air pressure plays only a modest supporting role in the operation of a syphon, although this role is important in technical applications. In the case of the gas syphon, the situation is quite different. Gases have no tensile strength. In contrast to liquids, gases can never form a column on their own. For this reason, gas syphons cannot operate in a vacuum. Figure 9.29 shows a gas syphon in operation. It permits the invisible gas carbon dioxide to flow through a syphon hose from the upper into the lower beaker. The arrival of the gas (and the resulting displacement of the air) in the lower beaker is indicated by a candle flame; it is extinguished by the CO2 gas, which does not burn or support combustion. The gas syphon clarifies the useful supporting role which the atmospheric pressure plays in many demonstration experiments: Gases have no free surfaces, but the presence of the atmospheric pressure provides a certain substitute! In place of the missing free surface, the diffusion boundary of the gas vs. the surrounding air acts as a ‘surface’. Therefore, we can handle for example ether vapor just as though it were a liquid. We tip a bottle containing diethyl ether, keeping the tipping angle small, so that no liquid can pour out of the bottle. But we can see the ether vapor flowing out like a stream of liquid; this stream is especially noticeable in a shadow image (‘schlieren’). We can also collect this flowing ether vapor in a beaker resting on a tared balance (Fig. 9.30). As the beaker is filled, the balance tips in the direction of “increasing weight”. This is because ether vapor has a greater density than the air which is displaced from the beaker. At the end of the experiment, we can empty out the beaker by tipping it; again, we see the ether vapor flow out like a broad stream of liquid and drop to the floor. h d Pa 5 g % in the ×10 ). The 1.0 D 9.31 p linear 0.8 0.6 within each individual Water % 0.4 contribution d 0.2 same Pressure

8 0 2 6 4

m 10 height m 10.2 to up of container Distance from the bottom of a water-filled water-filled a of bottom the from Distance makes the h hPa, apart from small variations due to weather condi- Distribution A stream of ether 3 Under Gravity. The Barometric Pressure Formula tions. It is the same aspond the which water is pressure 10.33 at m the deep. bottom of a freshwater In every liquid, the pressurethe decreases container on towards going higher fromcase levels. the of bottom of This liquids. decreaseabout is 100 hPa In for water, everyreason meter the for of pressure this increased height is decreasescompressed (Fig. by that for the lower weight example layers ofof by the of water layers of the above thickness them; liquid d thus are every layer not noticeably of the gravity pressure in a water column vapor in a shadow image Figure 9.30 Figure 9.31 9.10 The Pressure Distribution of Gases Thus far, we have considered the atmosphericon pressure the of surface the air of only theequal earth. to At 10 sea level, it is practically constant and to the overall pressure. The story is quiteible; different for the gases. lower Gases layersweight are in of strongly a compress- the gas upper column layers. are pressed The together density by the Liquids and Gases at Rest 9 202 Part I 9.10 The Pressure Distribution of Gases in the Gravitational Field.The Barometric Pressure Formula 203

Figure 9.32 Distribution of the gravity pressure in

the air at a uniform temper- Part I ature of 0 °C

layer is proportional to the pressure p acting on that layer. We thus ﬁnd % p p D or % D %0 : (9.18) %0 p0 p0

Here, %0 is the density of the gas at normal atmospheric pressure, p0. Then the contribution to the pressure of each layer of gas of height dh is given by p dp D%0 gdh (9.19) p0 .g D 9:81 m/s2/:

Integrating over height up to a maximum h yields:

%0gh p consth ph D p0e 0 D p0e : (9.20)

Inserting the values which hold at a temperature of 0 ıC, we obtain for the air pressure at a height h:

0:127h ph D p0e km :

This barometric pressure formula is represented graphically in Fig. 9.32. This is the counterpart of the distribution of gravity pressure in a column of water as shown in Fig. 9.31. The logic of this “barometric pressure formula” can be made intuitively clear by our model gas with steel balls. To show this, we set up the apparatus that we have seen already in Fig. 9.24 in a vertical position and look at it under stroboscopic illumination. On the projection screen, we then see a series of momentary images of the type showninFig.9.33. In the lowest layers, we can see a large number of molecules, which rapidly decreases as we go upwards. We can recognize the competition between weight and thermal motions. Al- ready at a height of 2 m (on the screen) above the vibrating piston, the ian ROWN 5); at about 11 km : 0 D ) may fail even as an approxi- 69 : . A well-defined upper limit 0 9.20 7 s). 5 For even at this high altitude, we even at several hundred km above 10 . hout any discernible upper boundary. ltitudes, Eq. ( C9.19 ). But 9.32 Our “artificial atmosphere” gradually thins out Snapshot image of a steel-ball Furthermore, the composition of the atmosphere and its temperature also depend molecular motions”. But yet theysurface”; always for remain the close weight to of one thethat of “earth’s of the a wood chips steel-ball is molecule. much greater than to the earth’s atmosphere cannotartificial be atmosphere. specified, any 5.4 km more above thanthe the for earth’s air our surface, has the density decreased of by about half (e the surface of thebe earth, found wandering some about molecules from the atmosphere can (Video 9.7) can observe the flashesthey from enter the meteorites; atmosphere.altitude; they they begin arise Auroras when to high-energy are particles glowthe from also earth’s when the atmosphere produced sun and penetrate at interact with a the similar gasTo conclude, molecules we in add it. some largere.g. bodies some to our wooden artificial atmosphere, We chips. see the They dust simulate particles dust dancing particles around in with the lively air. “B to 1/4, etc. (cf. Fig. Figure 9.33 through measurements. At highmate a description. 7 on the altitude. The true distribution of these quantities can be determined only molecules are rather rare.to Only a the height occasional of moleculeat 3 wanders m. increasing up altitude, but wit The situation in the earth’s atmosphere isthe completely height analogous; only scale is considerably greater model gas, demonstrating the barometric pressure formula (exposure time 3 /m 15 . At still 3 , 83rd edi- /m 19 Pa, with a number den- 5 Liquids and Gases at Rest 9 CRC Handbook http://tiny.cc/rwqujy Video 9.7: “A Model Gas: Its barometric density distribution” law. At 300 km, it is10 around greater altitudes, the pressure decreases more slowly than predicted by the exponential is around 10 the pressure has dropped to about 0.03 Pa and the number density of the molecules C9.19. In fact, the exponential law is valid up toaltitude an of ca. 100 km, where ( tion (2002), pp. 14–19). face of the earth, some molecules from the atmosphere can be found wandering about.” “But even at several hundred km above the sur- sity of the order of 10 204 Part I 9.11 Static Buoyancy in Gases 205 9.11 Static Buoyancy in Gases Part I From the results of the preceding section, we see that the gravity pressure in gases, as in liquids, decreases as we go upwards. Thus, there must also be a “buoyant force” in gases, as well as in liquids. As an example, we will describe the principle of the free balloon. It is drawn schematically in Fig. 9.34. Formally, we could again apply the principle derived in Sect. 9.4:The buoyant force of the balloon is equal to the weight of the air which it displaces. But it is expedient to consider the distribution of pressure inside the balloon. This makes the situation intuitively clearer. A free balloon is open at its bottom. There is no pressure difference between the air and the filling gas at their boundary. Naturally, this boundary is not completely sharp; it is simply a diffusion boundary between two gases. The effective pressure difference can be observed in the upper half of the balloon. There, the pressure of the filling gas on the inner side of the skin of the balloon is greater than the pressure of the air on its outer surface at the same height. Here is also where the bleed valve of the balloon is attached (a in Fig. 9.34).

The upwards-directed force acting on the skin of the balloon is proportional to the difference in density between the air and the filling gas. Both densities decrease with increasing altitude. For the filling gas, this decrease occurs within a loose balloon skin by gradually inflating the lower parts of the skin. When the filling pressure is exceeded, the excess filling gas escapes through the opening at the bottom of the balloon. As the densities decrease, the magnitude of their difference also decreases. At a certain limiting value of the densities, the upwards-directed force is equal to the weight of the balloon, and then the balloon remains suspended at a constant altitude. A further increase in altitude requires a reduction of its weight, e.g. by dropping off ballast.

The same pressure distribution as in a free balloon can be found in gas lines in buildings. Just like balloons, they are surrounded by air. Normally, the gas in the pipes is pressurized by a given level of ram pressure; sometimes, however, this pressure is too low. Then the gas doesn’t “want” to flow out of an open valve in the basement. On the fifth floor of the building, however, this disturbance is not noticeable; when a valve is opened there, the gas streams out forcefully.

Figure 9.34 The buoyancy of a free balloon. The density %0 (Eq. 9.20) of the filling gas must be smaller than that of the surrounding air (compare Fig. 9.12), (Exercise 9.10). Note that the difference in pressure (p2 p1) is greatly exaggerated by the arrows. ). 9.36 (Video 9.9) , but not at a , there is no b C9.20 in the decrease of the , we can be brief here. 7 ’s pipe) EHN difference . At this opening b ) into the tube through a filling lug 4 ). Exercise 9.9 On the carousel, we set a horizontal sealed box , 10 cm higher, there is a noticeable pressure dif- a ) With increasing height, the gravity pressure of natural gas Frames of Reference represents the gas-pipe system as a glass tube. This tube 9.35 Static buoyancy due to centrifugal forces. The principle of tech- is higher, a flame can be ignited only there. The setup is thus sur- Exercise 9.11 the same-sized lower opening at somewhere along its length; itsThen we flow is can restricted readily light by a a flame throttle at valve. the upper opening 9.12 Gases and Liquids in Accelerated Referring to our detailed treatment in Chap. This situation can be elucidatedure by a demonstration experiment: Fig- has a small openinghand flame for opening a is gasnow 10 cm pass flame natural lower at gas than (mainly each the CH left-hand of opening. its We ends; the right- Figure 9.35 We first give some exampleserence. for Thus, a in this radially-accelerated whole frameobserver section, on we of adopt a ref- the carousel point orseen of from a view above of swivel is an chair. rotating in Again, a the counter-clockwise1. sense. swivel chair as nical centrifuges. which is filled with water oriented in the radial direction (Fig. Under its lid, a ballter. is floating; When its the density carousel is is thus rotating, less than the that ball of moves wa- towards the center decreases more slowly than that of air (B ( pressure difference between the gasair. in the However, at tube and the surrounding gravity pressure in an air column and inFinally, a we column mention of the natural gas. furtherfactories. example They of contain chimneys warm in airdensity houses and than and combustion the gases surrounding withthe atmosphere. a greater lower The is higher thebetter the is chimney, pressure its difference “draft” ( at its lower opening, and the ference, which allows us toheld horizontally, ignite so a that bright both flame.can openings both are When be at the the ignited. tubeb same is height, When they the tippingprisingly is sensitive reversed, – so it thatincreasing does height, opening not but rather show only the the decrease in pressure with , 607 (1907). 13 . ’s tube” Liquids and Gases at Rest EHN 9 http://tiny.cc/0vqujy Video 9.9: “B C9.20. U. Behn, Phil. Mag 206 Part I 9.12 Gases and Liquids in Accelerated Frames of Reference 207

Figure 9.36 The centrifuge principle Part I

Figure 9.37 A ﬂame under the inﬂuence of inertial forces

of the carousel (axis of rotation). Conversely, a ball which is lying on the bottom of the box (density greater than that of water) will move outwards, towards the rim of the carouselC9.21. C9.21. A further example of buoyancy in an accelerated Explanation: The weight of the balls and their buoyancy due to the frame of reference is a toy weight of the water they displace are compensated by the bottom and balloon in an airplane during the lid of the box; the CORIOLIS forces are likewise compensated by takeoff or landing. its side walls. The only remaining force is the centrifugal force. It acts within the horizontal box just like the weight within a vertical box. For the centrifugal force, the axis of rotation at the center is “up”, and the outer rim of the carousel is “down”. An object in a liquid experiences a buoyant force in an “upwards” direction, i.e. here towards the axis of rotation. This buoyant force may be stronger or weaker than the centrifugal force which is acting on the object. When the latter predominates, the object will move towards the outer rim, that is, figuratively, it “sinks to the bottom”. When the buoyant force predominates, it will on the other hand “rise to the top”. This static buoyancy in radially accelerated liquids forms the basis of technical centrifuges, for example for separating butter-fat or cream from milk. The fat, owing to its lower density, moves towards the axis of rotation (cf. Sect. 16.6, last part.) 2. The deflection and curvature of a candle flame by centrifugal and CORIOLIS forces. On the carousel, we set up a candle in a large glass box, carefully protected against air currents. The flame tips towards the axis of rotation (Fig. 9.37). In addition, it becomes curved towards the right as seen from above. Explanation: The vector addition of the weight and the centrifugal force leads to a net force which slants downwards and outwards.The gases in the flame have a lower density than the air, therefore the force of buoyancy pushes them inwards and upwards. This buoyancy also gives the flame gases a velocity, and therefore, in addition to D 1 d ). 9.3 . and 9.2 ) in motion 9.3 N acts on the smaller 1 ). Then we give the table force acting on the flame. 10 9.38 ttom and the side walls of the D e liquid the same angular veloc- 1 ORIOLIS F In the center of our rotatable table, we which would have to press on the larger 2 F . It causes the tea leaves which had been lying on 75 cm. A force 9.38 D are rotating and, driven by the centrifugal force (heavy Radial circulation in 2 u d The two cylinders of a hydraulic press have diameters of Radial circulation in liquids when individual liquid layers have After a certain time,the the full molecules angular ofThe velocity the dashed of circulation upper is the water thusdish dish, slowed, layers is the and lowered, also water while level then attain it instationary they the rises parabolic center at surface also of shape the is the flow outer established circumference, outwards. (see until Figs. finally the cylinder. Find the force 4cmand cylinder in order to maintain equilibrium. (Sect. snippets placed at the bottom of the dish. arrows), they begin tolation flow shown outwards. by dashed This lines. flow starts It up can be the readily circu- verified using paper It will cause the flame to curve towards3. the right. different angular velocities. place a flat dish filleda with spin water (constant (Fig. angular velocity)ally and arrives observe how at the a waterthe stationary gradu- walls state. of the Theinitially dish, water, in gradually driven the takes by neighborhood on of frictiondish. the the with same bo As angular a velocity, of result, the at dish first only the water molecules near the bottom Figure 9.38 a dish filled with liquid This discussion of the circulation ofthe water topic has of already liquids led andtransition us gases beyond to at rest. our next This chapter: takes us Liquids through and a gases smooth Exercises 9.1 A reversal of this behaviorwith can a also spoon be initially observed. gives Inity. the a whol However, teacup, friction stirring withthe the angular velocity bottom of and the lowest sidesA layers of as radial soon the circulation as cup begins, one but stopsshown reduces stirring. in now Fig. in the oppositethe sense bottom from of the that cup to swirl towards the center. centrifugal force, there will be a C Liquids and Gases at Rest 9 208 Part I Exercises 209

9.2 The smaller piston of a hydraulic press has a cross-sectional area A1, and the larger piston has an area A2. The short end of a lever is connected to the smaller piston. A force F1 D 200 N is acting at Part I the other end of the lever, whose long arm is 10x longer than its short arm. Find the area A2 required if the press is designed to lift a block of mass 103 kg. (Sect. 9.3)

9.3 Determine the buoyant force F acting on a lead ball of diameter d in water. (Sect. 9.4)

9.4 A hollow cylindrical buoy of height l D 2 m and diameter 2r D 1 m is made of sheet iron with a thickness of d D 0:5cm,and is designed to be held under water. Determine the required force F. Iron has a density of % D 7:8g/cm3. (Sect. 9.4)

9.5 A rigid balloon of volume 1000 m3 is ﬁlled with hydrogen. The mass of the empty balloon skin and its basket is m D 200 kg. Find the force F which would be required to hold the balloon on the ground. (Densities: air, 1.3 kg/m3; hydrogen, 0.09 kg/m3.) (Sect. 9.4)

9.6 A hydrogen-ﬁlled, sealed, rigid balloon with a volume of 10 liters is hanging from a balance, which indicates a mass of 7 g (i.e. a weight of 0.0687 N). The air pressure is 1:013 105 Pa (D 1 bar). Which mass m would the balance indicate if the air pressure were to decrease by 2.6 % to 0:986 105 Pa while the temperature remained constant? (The density of air at the higher pressure is % D 1:3 kg/m3.) (Sect. 9.4)

9.7 Compare the energy which is necessary to bring a water molecule from the bulk to the surface of the liquid (surface work), with the energy which is required to expel a molecule from the liquid into the gas phase (energy of evaporation). The latent heat of vapor- 6 ization of water (see Sect. 13.4 and Fig. 14.3)islv D 2:5 10 Ws/kg at T D 0°C(Fig. 13.7), and the number density in the liquid phase 22 3 of water is NV D 3:3 10 cm . (Sect. 9.5)

9.8 In Video 9.6, “Coalescence of Hg droplets”, ﬁnd the number of mercury droplets as a function of time, N.t/. Show that N.t/ is an exponential function and ﬁnd the average lifetime (note that in the video, the liquid is not alcohol, as mentioned in the text, but rather water). (Sect. 9.5)

9.9 A chimney of length l D 30 m contains air at a temperature of tc D 200 °C. The air outside the chimney has a temperature to D 0°C 5 and a pressure of po D 1:013 10 Pa. Calculate the pressure difference po pc,wherepc is the pressure inside the chimney at ground level. (This pressure difference drives the gas inside the chimney upwards. For simplicity, we assume in this calculation that the gas inside the chimney is air and that effects due to its flow are negligible.) Note: Take the value of the constant in BOYLE’s law (Eq. (9.13)) ) 9.11 1m,and : 0 (these val- 3 D ). To simplify h 9.12 72 kg/m 1 bar). Compute the : 0 D ), and compare the force D ). The densities of air and 4 9.34 hPa ( 3 ) is tipped so far that the lower methane 10 30 m, both inside and outside the % . Neglect the weight of the empty h 9.35 The online version of this chapter (doi: in Fig. D D 1 and 0 h p ) p 3 2 9.11 p between the gas and the surrounding air at with p 3 kg/m : and h ’s pipe (Fig. 1 ) and its height , the upwardly-directed force which pushes up the g A D 9.6 % EHN 9.11 at the two temperatures and compute the pressure dif- 9.34 air D % p opening. Take the height difference to be 9.23 In Fig. When B upper the pressure difference chimney. (Sects. 9.10 ues hold at 0 °C, which we assume for simplicity here). (Sect. methane are assume that the gas is pure methane (CH flame nearly goes out, the pressureand difference between the the air gas outside insidethe pressure the is pipe thus at equal that to position is practically zero, and that it produces with the static buoyant force (Fig. the calculation, choose the shape ofits symmetry the axis balloon upright, to with be itssectional a bottom area cylinder surface with be open. Let its cross- balloon. (Sect. 9.11 10.1007/978-3-319-40046-4_9) contains supplementary material,able which to is authorized users. avail- Electronic supplementary material from Fig. ferences balloon is explained in termsin of the the barometric pressure difference of betweendrogen the inside the the surrounding decrease balloon air from and itsthis that bottom pressure to of difference its the ( top hy- surface. Calculate Liquids and Gases at Rest 9 210 Part I inrgta h einn n ra oin nlgto t crucial its of light in motions treat and beginning influence. the a at the place right from We differently tion bodies: them treat solid will of we motions Therefore, friction. by sively ttmnsaoti.–I h oin flqisadgss however, gases, and liquids of motions the the In even – it. about statements fetadol nSects. in side a not only as are and initially they friction but effect treated friction, therefore by We – qualitatively. less are changed or experiments basic more in – motions quantitatively bodies, perturbed solid of mechanics the In 3. et ftemdnmc.Te ilteeoeb icse ae,in later, discussed be therefore will Sect. They con- the their thermodynamics. without understood of and be cepts cannot compressed type are this of gases Processes velocities, change. high At 2. xeietwssatd h oe afo h lcrn a colored was glycerine the the Before of half glycerine. lower with the filled started, basin was glass experiment a from upwards slowly hscatr o rvt ewl s h od“iud sacollective a as “liquid” word the use term will we brevity for chapter, this nFig. In experiments: two with essentials liquids, the within occurs which friction ternal the contrast, In physically. n Gases and Liquids in Motions rcinbtensldbde,i.e. bodies, solid between Friction Boundary and Friction Internal 10.2 Sect. cf. m/s, (340 air in sound of velocity the example than for smaller can air non-compressible liq- m/s, a 70 in as around motions considered – of of be velocities manner. treatment to similar Up the a unifying gases. in and in treated uids further be even can go rest in can at One and gases surfaces and their liquids phenomena both of the of in formation aspects the many Nevertheless, in elasticity. differ volume their gases and Liquids 1. Remarks Preliminary Three 10.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © C10.1 18.7 rcin a erte lal umrzd edemonstrate We summarized. clearly rather be can friction, 10.1 . qualitative . ticue iud ohwt n ihu resurfaces. free without and with both liquids includes It Layers eso a ea plate metal flat a show we , set ftepeoeaaeiflecddeci- influenced are phenomena the of aspects olsItouto oPhysics to Introduction Pohl’s 8.9 – 8.10 external i emk oequantitative some make we did iud hsvlct ssilmuch still is velocity this liquid; uniaiedsuso ffric- of discussion quantitative B rcin shr ograsp to hard is friction, hc sbigpulled being is which O 10.1007/978-3-319-40046-4_10 DOI , temperatures 14.10 ). in- In nti okwhen book this used in be will and gases, and liquids for term collective as a used often is “fluid” word the literature, the In C10.1. xrsl meant. expressly 10 both are 211

Part I D of can A k .The (10.2) (10.1) x = from the A ,butalso x is zero, so

u 2 ). In simple D F k C 5.11 . Its thickness ,thatis F u ,i.e. x which is directed oppo- : F (or simply the “viscosity”; proportionality factor horizontal plane within the u x A : D

one u 2 which the boundary layer would boundary layer k D F or resistance (in aerodynamics, it is D u D (‘snapshot’ image) k F is proportional to the surface area B which have formed D k D F depends on the liquid; it is a material constant

is proportional to the velocity ), so that at least remains constant. The coefficient of viscosity 4 frictional force u F . The next layers, further out, are also pulled along, u ). We then have , the boundary layer from the upwards-moving plate The two dashed lines show the two is the 10.1 u 10.2 . x =@ u proportionality factor and is called the see Table plate is less than the thickness the moving plate and inversely proportional to complicated. We will give somesection. examples here and in the following In Fig. reaches out to the walls of the basin, so that their distance be expressed as a formula when the geometric situation is not too have in a largerlayer basin. is In practically this linear;whose case, lengths the this decrease velocity uniformly is gradient withmoving indicated increasing within plate. distance in the from We find the the that figure by arrows that the velocity also called ‘drag’). The sum of these two forces sitely to boundary layers of thickness Figure 10.1 (e.g. with KMnO liquid was made visible (at thecolored interface liquid). between the One colored and canplanes un- imagine could that be a similarly numberplanes of marked. on other both horizontal During sides of theregion. the motion, moving all This plate are of region distorted these is within a called broad the on either side of a moving plate plate. Thus, within the@ boundary layer, there is aWhen velocity friction gradient, plays a roleto in a accelerate motion, the one requires moving a object force not up only to its final velocity increases in theboundary course layer of sticks to the thesame motion. moving solid velocity object and The movesbut innermost with their the part velocities become of smaller the with increasing distance from the cases, this force The equally strong counter-force to maintain that velocity at a constant value (Sect. Motions in Liquids and Gases 10 212 Part I 10.2 Internal Friction and Boundary Layers 213

Figure 10.2 The definition of the viscosity for a planar flow. This means that the flow follows the same pattern in

all the planes parallel to the page. Part I

Table 10.1 The viscosities of some ﬂuids Substance Temperature Viscosity in °C in N s/m2 Air 20 1:7 105 5 CO2, liquid 20 7 10 Benzene 20 6:4 104 Water 0 1:8 103 20 1:0 103 98 0:3 103 Mercury 21.4 1:9 103 0 1:6 103 100 1:2 103 300 1:0 103 Glycerine 0 4:6 20 8:5 101 Pitch 20 107

The internal friction within liquids may be compared to the shear forces in solids. The quotient F=A could be termed the ‘shear stress ’. But there is a fundamental difference: The shear stress in solids increases with increasing strain or distortion. The internal friction in liquids, however, is proportional to the velocity of the distortion. Liquids at rest exhibit nothing at all comparable to a shear stress. In them, only normal stresses can occur (cf. Sect. 9.2). The thickness D of the boundary layer can be estimated. One ﬁnds s l D (10.3) %u

.l D length of the moving body;%D density of the liquid/: u at : 2 2 . 2 l p D ;and R R / (10.4) 2 % p , and 2 with a cir- 1 2), equa- p 1 p P p V . Ns/m 2 3 oduct of its vol- R along the length 10 ul D A D narrow tube

: F ) is being accelerated. It

m ; lu 10.1 shows an example. Just like W : is an example of a “laminar” 3 mm. l Volume current Al D required for this ﬂow

A , the coaxial liquid layers within D 8 t

10.3 10.2 D D that flows through the area (simplifying the velocity distribution). D D 10.2 which is equal and opposite to the force 2 . Due to friction, the velocity distri- V m l F m/s, Time ADu mu mu 2 % Cross-sectional area of the tube, 1 2 1 2 ku 2 1 10 D D to obtain E D m F % Vo lu m e of the accelerated liquid by the pr ) performs the work u u m D 10.2 P 1m, V equal yields : 0 W D l is often produced by two unequal pressures , ). – Numerical example for water: and Which Occur when Friction Plays a Decisive Role 3 and its density F E 10.3 AD kg/m 3 is the mean value of the flow velocity, defined by the equations receives a kinetic energy Derivation The liquid within the boundary layer (Fig. Setting 10 or, for the thickness of the boundary layer (with the prefactor of friction in Eq. ( The accelerating force, the force We replace the mass tion ( ume Mean flow velocity Volume current m u the ends of the tube or pipe. We then have the planar layers of liquid in Fig. the tube move past eachfrictional resistance other. is The found in force this necessary case to to overcome be the The force bution within the liquid isliquid not is uniform. not At moving; the itstube walls velocity and of increases is the towards maximal tube, the there. the center Figure of the is sufficiently small. We offer three additional examples: First, we consider a liquid flowing through a (i.e. ‘layered’) flow. It isbution generally which characterized is by constant a over velocity time, distri- and it occurs when the velocity and The fluid motion observed in Fig. 10.3 Laminar Flow: Fluid Motions cular cross-section and length Motions in Liquids and Gases 10 214 Part I 10.3 Laminar Flow: Fluid Motions Which Occur when Friction Plays a Decisive Role 215

Figure 10.3 The velocity distribution in laminar ﬂow through a tube with an open cross-section of 6 mm 6mmC10.2 C10.2. In a tube of circu-

lar cross-section and also Part I in a channel formed by planar parallel plates, in the stationary state of flow, one expects a parabolic velocity profile (see e.g. Noakes, Cath and Sleigh, "Real Flu- ids. An Introduction to Fluid Mechanics" University of Leeds (2009)). This is shown in Fig. 10.3, but only as an approximation. A possible explanation for this is the experimental difficulty of establishing precise initial conditions for the flow.

C10.3. The fluid flow shown in the “streamline apparatus” (Fig. 10.4) is a so-called HELE-SHAW flow, which occurs in flat channels as a laminar and practically two- dimensional flow field. The “streamline images” thus obtained (Fig. 10.5 and the later figures 10.10, 10.14, 10.16, 10.17 and 10.35)are thesameasthoseofpotential flows (i.e. ideal, frictionless flows; see e.g. Etienne Guyon, Jean-Pierre Hulin, Luc Petit, and Catalin Figure 10.4 A streamline apparatus for demonstrating flow fieldsC10.3, D. Mitescu, Physical Hydro- shown at the left as a plan view from above and on the right asaprofileview. dynamics, Oxford University The upper chambers are connected through holes with the flat channel (the Press, 2nd edition (2015), spacing of the glass plates is 1 mm). These holes are offset by one-half their Chap. 6.) spacing in the two chambers. – First, both chambers are filled with water, then some ink is added to the water in the right chamber. – Left: An example of parallel streamlines. For observing the shadow image on a screen, one can Video 10.1: easily show the flow in a horizontal direction; two right-angle prisms can be “Model experiments for used to rotate the optical ray path by 90°. (Video 10.1) streamlines” http://tiny.cc/8wqujy , . . P 10.5 V (10.6) laminar (10.7) (10.5) relative to and it plays u , flat channel ’law, . C10.4 blood reservoirs ). If the sphere or C10.5 law ) ). Quantitatively, we , spacing of the plates 5.21 TOKES B : body has a length of : : 10.4 m 2 u p Ru d Bl l Physiology , width ) finds many applications. , (Fig. l

1 OISEUILLE 3 6 p EIN -P 4 D 10.7 of their tiny water droplets, the clouds D

R . m

8 ku ku AGEN is usually the weight of the ball, reduced D D D F streamlines P F V F , we obtain in a fluid (compare Fig. ) is moving slowly enough at a velocity R km (this is of the order of magnitude of the circumference of the !). The expanded network of capillaries must then be filled with 4 4 volume current R 10 C10.6 ). In a channel of this type, we can readily visualize the paths of / earth!). Increased muscular activity demands an increase in blood flow 3 This is very effectively( accomplished by an expansion of the capillaries 2. Measurement of the radii of(droplets). small This spheres method which is are often suspendedmeasurements. more in convenient the than air making microscopic 3. Without the frictional resistance Examples 1. Measurement of the viscosity The system of blood capillaries in the human blood; the required amount(in the of splanchnicus blood region) is (cf. taken H. R out of the would fall on ourfrom heads. their As lower sides it and is, reforming they at just their sink upper quite sides. slowly, evaporating d the fluid, then the required force is given by S If we imagine it toflow be around three-dimensional, a it sphere would show the ball (of radius As a third example,flow. we The add streamlines then a show circular the obstacle picture to reproduced in this Fig. laminar fluid D individual liquid ‘volume elements’. We dyeimpressive the liquid image and of obtain an the have an important role in the physiology of our blood circulation. formed by two glass plates (of length As a second example, we replace the tube by a very for the by its static buoyancy. Equation ( When the fluid is at rest, This equation is called the H , AGEN ndorff’s OISEUILLE (Lectures on Mechanik Physiologie , Springer- refers here Pogge , 961 and 1041 , Guyton and , 423 (1839)), 11 Mechanics of De- OHL . Modern edition: 46 Comptes Rendus des Motions in Liquids and Gases EIN Textbook of Medical 10 Annalen der Physik und Chemie andbyJ.-L.-M.P Hall, 12th ed. by John E. Hall, Saunders-Elsevier, Philadelphia (2011). Physiology R.F. Schmidt, G. Thews, and F. Lang (eds.), Verlag, 28th ed. (2000), Chap. 24. English: see e.g. des Menschen “Without the frictional resistance of their tiny water droplets, the clouds would fall on our heads.” C10.6. The derivation of Eq. (10.6), and of theequations other in this section, can be found for exampleA. in Sommerfeld, der deformierbaren Medien Akademische Verlagsge- sellschaft, Leipzig 1957. English: formable Bodies Theoretical Physics), Arnold Sommerfeld, Academic Press (1950). C10.5. P (1799–1869), a physician in Paris ( Séances de l’Académie des Sciences C10.4. It was discovered independently by G. H (1797–1884), hydraulic engineer in Königsberg (published in to the physiology text by H. R (1840)). 216 Part I 10.4 The REYNOLDS Number 217

Figure 10.5 Laminar ﬂow around a sphere or a cylinder (photographic

positive image with bright-ﬁeld illumi- Part I nation) (Video 10.1) Video 10.1: “Model experiments for streamlines” http://tiny.cc/8wqujy

10.4 The REYNOLDS Number

The flow motions described in the previous section are observed only at sufficiently low velocities and/or dimensions. When velocities and dimensions are larger, the flow becomes turbulent.–Turbulence refers to a strong mixing of the liquid C10.7 with swirling and eddies C10.7. See for example (vortices). It can be most simply observed using a colored stream S.B. Pope, of water in a transparent tube. Figure 10.6 shows a liquid flow be- Turbulent Flows,Cam- fore and after the onset of turbulence. The turbulent motions produce bridge University Press an additional viscosity, called the apparent viscosity, which can be (2000). orders of magnitude larger than the normal viscosity (Eq. (10.2)). Equation (10.4) is no longer applicable, and the required force F increases approximately as the square of the velocity.

Figure 10.6 The formation of turbulence in a stream of water within a tube, and measurement of the REYNOLDS number as a demonstration. On the left is an apparatus suitable for shadow projection (tube: 15 mm 15 mm clear opening). In its center is a laminar flow. On the right, a turbulent flow is shown. The flow velocity is computed from the amount of water that flows out, the time and the cross-sectional area of the tube (Video 10.2). Video 10.2: “Turbulence of flowing water” http://tiny.cc/cxqujy (10.8) viscosity D

e boundary layer : viscosity, and, to . When the turbu- %

lu wind D kinematic is the velocity of the liquid , the turbulence is especially u viscosity. density of the liquid; . The altitude of th D blizzard dynamic % gusts is often called the Frictional work =% Work of acceleration D is then called the Re

The “sensitive flame”; at ) is quite impressive. . 1 10.7 is a length which is characteristic of the size of the body, e.g. the radius of the liquid. Thedistinguish ratio it, relative to a solid body;flow in velocity as a tube, defined above; for example, it is the mean value of the l of a tube, boundary-layer thickness, etc.; The turbulence of a stream(Fig. of gas which is burning as a “sensitive flame” In the evening, the wind “goes to sleep” (but only near the ground!). – The Figure 10.7 the left laminar, at the(and right noisy). turbulent The onset ofbe turbulence produced can by making asound soft or hissing shaking a keytance ring of (at several a meters). dis- work, which is performedwinds at near the the cost ground. of the kinetic energy of the air, slowing the 1 reason: Turbulent motions liftplacing cooler the and warmer therefore air more there dense air downwards, upwards, with dis- its lower density. Both require its atmosphere are familiar; they arelence known is as strong, we speak of The turbulent motions in the boundary layer between the earth and can be several kilometers.apparent In a In a turbulent flow, regions ofin the terms liquid which of are size randomlyhigher varying and order”. composition During their are lifetimes, which combinedsizes, depend they into strongly are on “individuals subject their of todecay common motions or and shed rotations. off Whenlived they individuals. pieces, these can again combineThe transition into from laminar new, to short- turbulentical” flow value is of determined by the a ratio “crit- Motions in Liquids and Gases 10 218 Part I 10.4 The REYNOLDS Number 219

This was discovered in 1883 by O. REYNOLDS; therefore, Re is called the REYNOLDS number. Part I To derive Eq. (10.8), we make use of “dimensional analysis”C10.8. This C10.8. Details of the deriva- means that we take all of the lengths which occur in the problem to be tion of the REYNOLDS num- proportional to a length l which characterizes the size of the body. Fur- ber can be found in textbooks thermore, numerical factors are left off. For the work of acceleration, from Sect. 5.2,wehave on hydrodynamics, e.g. 1 2 3% 2 : Etienne GUYON, Jean-Pierre Wa D 2 mu D l u (10.9) HULIN,LucPETIT,and The frictional work is found using Eq. (10.2): Catalin D. MITESCU, Phys- u ical Hydrodynamics, Oxford W D FlD A l D l2u : (10.10) r l University Press, 2nd edition (2015), Chaps. 1 and 2. Taking the ratio of these two expressions then leads to Eq. (10.8).

Small REYNOLDS numbers mean that the frictional work is predominant, while large numbers characterize the predominance of the work of acceleration. The ideal, frictionless fluid corresponds to a REYNOLDS number of 1. – The “critical” values of the REYNOLDS number which lead to turbulence can be determined only from experimentC10.9. In smooth-walled tubes, Re must be C10.9. Also the “sufficiently greater than 1160 to cause turbulence. low velocities” mentioned at the beginning of this section For small spheres in air, Re < 1 is the condition for avoiding turbulence can be determined only from and guaranteeing the validity of STOKES’law(Eq.(10.7)).–Inthestream- experiments. line apparatus (Fig. 10.4), we are dealing with REYNOLDS numbers of around 10. The air we breathe flows through our nasal channels without turbulence. In abnormally expanded noses, however, the critical values of the REYNOLDS number and the velocity can be exceeded, and then strong turbulence can occur, and it increases the frictional flow resistance. Noses which are in- ternally abnormally large thus seem to be continually blocked.

The REYNOLDS number plays an important role in every quantitative treatment of fluid flow. Experiments on particular geometric shapes can be first carried out with models of convenient dimensions, and the results can then be scaled up to full-size dimensions. To this end, one must simply choose the flow velocity, density and dimensions to guarantee the same REYNOLDS numberC10.10. For aircraft, C10.10. This holds as long the REYNOLDS numbers are on the order of several 107.Thishas as one can consider the fluid an annoying consequence for measurements; it makes the study of to be incompressible, i.e. the technically important questions on small models more difficult. By velocities must remain small lowering the temperature to that of liquid nitrogen (77 K) and increas- compared to the velocity of ing the working pressure to around 10 bar, the viscosity of the air can sound in the fluid. be decreased while simultaneously increasing its density. This allows realistic measurements on small models at around the same velocities as for the full-size aircraft. .The 2 . It consists of 10.8 , which are held by b . the object, and thus sees the and 10.34 a follows and ’s Equation ss of the boundary layer resulting a bottleneck. For photographic im- 10.32 , , we see an object with a circular profile, 10.26 , 10.8 ERNOULLI 10.9 . – the image of the flow can be stationary, i.e. in- , there are two objects, 10.9 Flow apparatus. Here, again, for projection . They indicate the overall pattern of velocity directions, and B flow field is sufficient. The eye of the observer 10.8 For observations projected on the wall, the fixed setup of the basin sketched in exposure of around 0.1streak. s, the Each path ofresents, of these briefly each stated, streaks flake the is iselement’. velocity practically vector seen With of a longer as a straight exposure a singlestreamlines times, line water the short and ‘volume streaks combine rep- tothat is show the Al flakes show theevery magnitude moment and throughout direction the of whole the basin flow in velocity the at projection. In an (Video 10.3) onto a wall or screen,by it 90°, is as advisable e.g. to in rotate Figs. the image Fig. liquid flowing past it. 2 Figure 10.8 of friction; that is,layer. we will For this try purpose, to weare reduce use the large a effects compared liquid of to container thefrom whose the the boundary dimensions thickne motion that we wish toA study. suitable “flow apparatus” is illustrated in Fig. 10.5 Frictionless Fluid Motion From now on, wetaken in will discussing follow the theto mechanics same of observe path solid motion that bodies: that we We is, first have as attempt already far as possible, free of the influence a basin which isare 1 cm added wide to and theshapes filled (profiles) water with can as water. be suspended Aluminumthe moved particles. glass flakes through walls. the In Objects basin, Fig. andinFig. of loosely touching various invisible rods and together form ages, the basin is moved along a track at a constant velocity Motions in Liquids and Gases 10 “Fluid Flow around obstacles” http://tiny.cc/fcgvjy Video 10.3: 220 Part I 10.5 Frictionless Fluid Motion and BERNOULLI’s Equation 221 Part I

Figure 10.9 Streamlines passing through a bottleneck; the observer (camera) and the tube are at rest, while the liquid flows through the latter dependent of time, or at a fixed position. Then the streamlines show in addition the whole path followed by the volume elements of the liquid one after another. The time-exposure image shows the flow field in its clearest form, as seen in Fig. 10.9. The image projected onto a wall or screen is more lively. Often, however, one is striving for an image without too many details, with only a few clear streaks. In this case, a strange circumstance comes to our aid: We can imitate the flow field of a stationary flow which is practically free of the effects of friction in a model experiment. The streamline apparatus which we have already seen in Fig. 10.4, with its laminar flow, can serve this purpose. In spite of the completely different conditions for producing the flow, the course of the streamlines is that of an ideal, frictionless liquid flow. Figure 10.10 shows an image made in this way. It corresponds to Fig. 10.9. But in contrast to Fig. 10.9, it shows only a model experiment; we should keep that in mind. Formally, however, the image is accurate, and its simplicity makes it clear and easy to remember. These liquid flow patterns, practically free of the effects of friction, which we can illustrate with such model experiments, can be maintained for only very short times. They are roughly analogous to the example from the mechanics of solid bodies of a ball moving free of forces at constant velocity; this is an idealized limiting case. But an important law holds for this case, which forms the basis for all that follows. It concerns the “static” pressure, i.e. the pressure of the liquid against a surface placed parallel to its streamlines. In a first, qualitative form, this law can be formulated as follows:

Figure 10.10 Streamlines in a model experiment (positive photographic image in bright-field illumination, like Figs. 10.14, 10.16, 10.17 and 10.35) (Video 10.1) Video 10.1: “Model experiments for streamlines” behind , a con- shows the and 10.12 10.11 Figure . in front of, within C10.11 eck is less than the surrounding air are useful 10.12 and The static pressure in a waist. It is lower than atmospheric Distribution of the static pressure around a flow through a bot- ); the influence of friction is therefore only partially re- 10.11 10.2 static pressure of thethe bottleneck. flowing liquid Theis figure not is yet schematic.(Sect. large The compared diameter to of the the thickness tube of the boundary layer tleneck or waist. The threemanometers. vertically-mounted glass side tubes serve as water pressure in this case. A mercury column serves here as manometer. duced. As a result, theattain static precisely pressure the behind same the bottleneck value does as not before it. – In Fig. The quantitative relation between theflow pressure is and found the by velocity applying of the the law of conservation of energy. We considerably higher flow velocitypressure is of the shown. water in At thepressure. bottlen this The flowing velocity, water can the “suck in” static ter mercury within and a manome- pulls up(“jet a pump” principle). mercury column of several centimeters in height In order to make this law intuitivelyin clear, the Figs. two experiments shown Figure 10.12 In regions where thevelocity streamlines is increased, are the pressed static pressurethe together, p surrounding or of regions. the the fluid flow is lower than in Figure 10.11 Motions in Liquids and Gases 10 following experiments in this section, we are dealing with laminar, stationary flows. C10.11. Here, and in the 222 Part I 10.5 Frictionless Fluid Motion and BERNOULLI’s Equation 223 sider a certain quantity of liquid of mass m, volume V and density %. Its static pressure and its velocity before the bottleneck are p0 and u0; within the bottleneck, they are p and u. The liquid must be acceler- Part I ated from u0 to u on entering the bottleneck, in order to pass through it. This requires the work . / 1 . 2 2/; V p0 p D 2 m u u0 (10.11) or, after dividing by the volume V, 1 % 2 1 % 2 p C 2 u D p0 C 2 u0 D const. (10.12) 1 % 2 2 u is added here to the pressure p; it must therefore itself represent a pressure. It is called the dynamic or stagnation pressure.Thesum on the right of the equation is constant. It must also represent a pressure, and it is called the ‘dynamic head’ or total pressure p1.Wehave thus obtained the important BERNOULLI equation,

p C 1 %u2 D p 2 1 : static pressure stagnation pressure total pressure (10.13)

Measurement of the static pressure p in a flowing liquid is accomplished by the setup shown in Fig. 10.12. The opening which leads to the manometer lies parallel to the streamlines of the fluid flow. For measurements within wide flow paths, one places the opening, usually in the form of a sieve or slit, in the side of a pressure probe.It is connected to a manometer via a hose or tube. This is illustrated in Fig. 10.13.

The total pressure p1 is measured in a stagnation region. One is illustrated in the model experiment in Fig. 10.14: At the center of the stagnation region (stagnation point), a streamline strikes an obstacle in a perpendicular direction. The connection to the manometer (PITOT tube) is made at this point. Here, the liquid is at rest, that is u D 0. The static pressure is, from Eq. (10.13), equal to the total pressure p1 at this point. The manometer indicates the total pressure p1. The stagnation pressure is determined as the difference of the total pressure p1 and the static pressure p. The stagnation pressure, from Eq. (10.13), is given by 1 % 2 . /: 2 u D p1 p

Figure 10.13 Sectional view of a pressure probe with a ring-shaped slit for measuring the static pressure within a flowing liquid p ). b), 10.15 10.16 indicates an shows a similar 10.17 tube”, Fig. tube, and the static 10.18 ITOT RANDTL in- 2 RANDTL tion of the static pressure, which and 1 . Numerous other demonstration ex- and the static u 1 tube and a pres- p between the two spheres. Therefore, the When the tilting is stronger (Fig. flow). u elucidate the decrease of the static pressure ITOT . is measured with a P tube for measuring the total pressure in a stagnation 1 a) proves to be labile; the disk swings back and HAW p 10.12 C10.12 b would be rotated in a clockwise sense. The first between the spheres is reduced, and they “attract” -S ITOT , the tube is bent at a right angle, usually with an outer shows a circular disk in model experiments with three p 10.16 and ELE AP Cross-setional view of a P 10.16 with a pressure probe. For technical measurements, it is c). We can observe this by letting let a stiff sheet of paper ,H 10.16 p . ); the contours of the tube were shaded after exposing the photo) 10.11 p in natura 10.4 10.16 10.4 with increasing flow velocity pressure expedient to combine the two devices (“P Stagnation-tube measurements aremining the the preferred velocity in method flowing fluids. for deter- Figures periments can also showthe this. flow field We give is two(Fig. imitated examples: by a In model each using one, the streamline apparatus region; diameter of only 2 to(Fig. 3 mm (model experiment with the streamline apparatus sure probe. The liquid-columnis manometer attached which on both sides to the tubes 1. Figure different orientations within acauses an flow asymmetry field. in theproduces distribu Even a the torque slightest tilting pressure the torque can be readilypress seen: against The the regions disk of onstreamlines expanded streamlines one pull side, on while thedisk the opposite in regions sides. Fig. of compressed In the real experiment, the Figure 10.15 tube, a combination of a P The total pressure Figure 10.14 dicates the stagnation pressure directly asdifference the of the total pressure position (Fig. forth, until it reaches(Fig. a stable orientation perpendicular to the flow fall to the floor. 2. Twotheir spheres midpoints are is moving perpendicularstreamlines. to within The the a model direction experiment fluid. shown of in the Fig. unperturbed The line connecting each other via “hydrodynamic forces”. Figure increased flow velocity static pressure experiment. In a water basin, the wooden ball at left stands upright AYLEIGH ). 12.24 Motions in Liquids and Gases 10 C10.12. An example of an application is the R disk (Sect. 224 Part I 10.5 Frictionless Fluid Motion and BERNOULLI’s Equation 225 Part I

Figure 10.16 Three model experiments on the ﬂow around a disk. The disk and the observer are at rest, the liquid is ﬂowing. In the center image, one can see the two stagnation points. This demonstrates the occurrence of a torque around the center of gravity of the disk. (Video 10.1) Video 10.1: “Model experiments for Figure 10.17 The streamlines between streamlines” spheres or cylinders; model experiment http://tiny.cc/8wqujy (see also Comment C10.6 on Fig. 10.4)

Figure 10.18 The attraction of a ball at rest and a moving ball in water

as an inverted gravity pendulum. A second ball is moved past the first one at a certain distance by a pushrod. The attraction of the two wooden balls is clearly visible. A buffer (at bottom) keeps them from colliding. – We can imagine two ships passing each other as a large-scale version of the two balls. In narrow navigation channels, for example in a canal, there is always a danger that the ships will be pulled together and collideC10.13. It can be reduced only by proceed- C10.13. The same attrac- ing very slowly; for, as seen in Eq. (10.13), the stagnation pressure tion affects a ship which 1 % 2 2 u increases as the square of the velocity, and with it the attractive passes too close to the bank force between the balls (or ships). of a canal. c, of 10.16 (or ‘flow productiv- tilted plate; there. They ) has spherical parallel flow . The flow fields . The two shaded (Video 10.3) sources sinks is called its 10.22 instead of Fig. avoidance flow . One need only modify q b. Corresponding images )orsink( D C 10.19 t 10.16 = by itself V . In the direction of motion, the edges of the . The quotient t 10.21 . – The source is assumed to release a liquid volume of the shaded regions; there are The parallel flow around a perpendicular and a and and Sinks. Irrotational or Potential Flows ); secondly, after the object is placed in that flow, there is one For a source, the productivity has a positive sign; for a sink, . 10.20 10.4 within the time C10.14 the method of observation: Up toand now, we the have considered observer the (camera) object to to flow past be them. at We(basin) rest, now adopt and and the the opposite the possibility: observer liquidprojection The are was of liquid at allowed the rest, observations, butforth one the motions.) makes object use – is of Withfor moving. limited this example back-and- (For second the method left-hand of part observation, of we Fig. find around obstacles’) can be observed and the right-hand part instead of Fig. end at the other shaded region,move where along there with are the objects; they are thus noThe longer flow stationary. field of asymmetry; single a point cross-section source is ( sketched in Fig. Figs. object are washed out andwas appear repaired as after half-tones the in factbegin the by at pictures. shading in This the figures. The streamlines the liquid without the objectFig. placed in it (asan shown additional in flow model around form it. in This additional for the avoidance flow around a sphere and a cylinder can be seen in areas denote the sameeach small other volume in in two time.without positions rotating which The follow liquidV thus flows in a radial direction and observer and liquid are at rest, and the plate is moving Figure 10.19 two different flows overlap. First of all, there is a 10.6 Flow Around Obstacles. Sources In the flow fields that we have considered thus far, it is clear that ity q . 10.3 leads directly to ). u Motions in Liquids and Gases 10.14 10 “Fluid flow around obstacles” http://tiny.cc/fcgvjy Video 10.3: Its connection to the flow velocity Eq. ( C10.14. The ‘productivity’ mentioned here again refers to a volume current, asready al- defined in Sect. 226 Part I 10.6 Flow Around Obstacles. Sources and Sinks. Irrotationalor Potential Flows 227

Figure 10.20 Avoidance ﬂow around a sphere; ob-

server and liquid (basin) Part I are at rest, and the sphere is moving

Figure 10.21 The avoidance ﬂow around a cylinder which is parallel to a parallel ﬂow; the observer and the liquid (basin) are at rest, and the cylinder is moving

Figure 10.22 The ﬂow ﬁeld of a source (or a sink, with reversed directions)

a negative sign. Then for the velocity u of a volume element at a distance r from the source or sink, we ﬁnd ˙ q u D : (10.14) 4r2

Irrotational flow fields can be combined by simple superpositions. This provides a considerable simplification in their mathematical treatment. Thus, in Fig. 10.23, the two radially-symmetric q dipole ˙ unchanged ) have been the elements . (left) is like ! all 10.19 (in volt seconds) will en- /s), the electric charge ˚ 3 ˙ (m angular velocity q ˙ resembles the field of an electrically- or . Third: The rotation of ) again later when we are dealing with elec- common are the same as a dipolar flow field. They ! . 10.20 10.14 ) and of a neighboring sink ( C 10.21 C10.16 The flow field which results is called a . , but rather that of an electric or magnetic field on a dis- u is thus similar to the magnetic field lines from a long coil The flow through C10.15 Measurement. The Irrotational Vortex Field angular velocity 10.21 . Instead of the productivity r 10.19 . It has many applications. At large distances, the flow fields of same magnetically-polarized sphere. ter the equations. Therefore,are the streamline formally patterns just of the theFigure avoidance same flow as thethrough which field-line an patterns electricthe in current electric electrodynamics. is stray flowing, fieldFig. and of 2.5). Fig. a parallel-plate Similarly, condenser Fig. (see Vol. 2, Chap. 2, (in ampere seconds) or the magnetic flux on a velocity tance We will encounter Eq. ( tricity and magnetism (Vol. 2). Then, it will not contain a dependence potential flow field superposed field Figs. can all be replaced by dipole fields. field of a source and a closely neighboring sink (a “dipole”) Figure 10.23 In a liquid, on the otherpast hand, each all other. the volume This elements leads can flow to freely quite different consequences from the 10.7 Rotations of Fluids and Their Within a solid body, all itstogether. parts (volume This elements) are has bound three rigidly trarily consequences: chosen First: volume The element shapeduring within of motions. the an body Second: arbi- remains the Every point within ais volume uniquely element defined has by their All these vectorwhich fields they are can be characterizeda derived by frictionless by liquid taking a is its also potentiala gradient. such field a The vector from flow field; field we of thus refer to it as fields of a source ( . Lec- , http://www. et al. , Addison- is formed by 10.23 Motions in Liquids and Gases 10 be read online at feynmanlectures.caltech.edu/ Wesley, Reading, Mas- sachusetts, U.S.A. (1964), Vol. II, Chap. 40. They can flow of frictionless fluids, can be found for example in R.P. Feynman tures on Physics C10.16. A good introduction to the theory of potential flows, which describes the a source and a sink. Thus, for example, the image of the field lines ofin a Fig. dipole superposition of the fields of are vector fields, and their superposition is subject to the rules for vector addition. C10.15. Since the velocity is a vector, flow (velocity) fields 228 Part I 10.7 Rotations of Fluids and Their Measurement.The Irrotational Vortex Field 229 case of solid bodies: First, marked regions (e.g. colored with a dye) within a liquid change their shapes during motions3; think for example of Fig. 10.1. Second, different points within a volume element Part I can have differing angular velocities. Therefore, we can not define the rotation of such an element by quoting a common angular velocity, as in solids. Instead, we must introduce a new measure for the rotation of a volume element within a fluid. It must summarize the motion by defining a reasonable mean value for the different angular velocities within the volume element. The measure of rotation used for fluids is called the curl of the path velocity u, or more simply, “curl u”. The experimental definition of the curl is simple: One puts a little float marked with an arrow into or onto the liquid, choosing its diameter to be small compared to the radius of curvature of its orbit. During the motion, the arrow on the float will change its direction C10.17 with an angular velocity !fl. Then we define C10.17. The factor 2 in Eq. (10.15) has no physical meaning. It simply results

2 !ﬂ D curl u : (10.15) from the mathematical deﬁni- tion of the curl.

To obtain the mathematical deﬁnition of the curl, one starts with the circulation . This is deﬁned as the path integral of the orbital velocity u along an arbitrary closed curve, i.e. I D u ds (10.16)

(the circle on the integral sign indicates a closed path or loop).

One then supposes that the path encloses a surface element dA,and computes the quotient d=dA for this limiting case. This defines anewvector which is perpendicular to the surface element; it is called the curl of the orbital velocity u:curlu. It describes the rotation of the fluid within this surface element. If the surface element lies e.g. in the xy plane, then we find for the z component of the curl Â Ã @u @u .curl u/ D y x : (10.17) z @x @y

Derivation Referring to Fig. 10.24, we calculate the circulation around the z axis along the four sides of a rectangular surface element dA D dx dy. The order of the summation is in the clockwise sense as seen by an observer looking along the positive z-axis direction. The circulation then consists of four individual terms, namely Â Ã Â Ã @u @u d D u dx C u C y dx dy u C x dy dx u dy x y @x x @y y Â Ã @u @u D dxdy y x D .curl u/ dA : @x @y z

3 With the exception of the special case treated in Eq. (10.20). r is x ∙d y ]in x u rr u D u ˛: ∙d ∂ ∂∂ plane. d curved r (10.18) (10.19) y x A d u u+ xy d ∂ d ∂ Ã r + u r paths. y r @ @ d A u d r d y C ∙d u x x u x y d u u or slope of the ve- Â ∂ ∂ the observer, and at D + y x r u u : ) yields straight-line 0d . (The vector [curl r r r u : of a planar flow field; the in the figure); its horizontal u @ @ ˛ gradient y α towards u x d u / @ 10.17 C @ r d r u C D r . z D the observer.) / Ã z u r / d u r u . Then we find for the quotient @ @ curl ˛ . boundary layer d y y C curl r d . d u r Â shows a planar, circular flow in the away from . C 0. Then equation ( D r u r @ @ D 0d A di- C x 10.25 C z u r u ˛ indicates d ,weseethe A D ur at the left of the plate points z ). The ar- ); the in a direction perpendicular to / 10.2 u u D 10.2 10.19 10.17 curl Furthermore, d . Derivation We compute the circulation alongconsists the of path four shown terms; by they heavy are lines. It again d its motion component is their upwardly-directed velocity (called row above d Then we have paths. Figure In general, the volume elements of the liquid move along In this case, the curl is thus simply the locity Fig. Figure 10.24 Figure 10.25 rection points from the plane of the page towards the eye of the reader The derivation of Eq. ( The curl of the orbitala velocity in somewhat its difficult general concept. (3-dimensional) formof is Therefore, its application: we offer several examples In Fig. The derivation of Eq. ( volume elements of the liquid move along the right of the plate, Motions in Liquids and Gases 10 230 Part I 10.7 Rotations of Fluids and Their Measurement.The Irrotational Vortex Field 231

We will now apply Eq. (10.19) to two limiting cases. In the ﬁrst case, the liquid is presumed to stick to a rotating solid disk and to have the same angular velocity ! as the disk in all its regions. Then we have Part I

@u u D !r and D !: (10.20) @r With this, from Eq. (10.19) we ﬁnd for the whole liquid a constant value of the curl, namely

.curl u/z D 2 !: (10.21)

This thus represents the limiting case of a “pseudo-solid rotation”. In liquids, a different limiting case is more important; it is characterized by the condition: C10.18. The magnetic field urD const : (10.22) outside a long straight wire carrying an electric current @u const u Then D D ,andEq.(10.19) yields is also an irrotational vortex @r r2 r field (see Vol. 2, Chap. 6, Eq. (6.14)). Within the wire, .curl u/ D 0 : (10.23) z the curl of the magnetic field equals the current density in Therefore, when the condition (10.22) is fulfilled, a liquid moves along a curved path in a rotation-free manner; the arrow on a small the wire (Ampère’s law). float would continually maintain its direction. This curious sort of flow field is called an irrotational vortex fieldC10.18, C10.19. It is a fur- C10.19. Such irrotational (or ther example of a potential field (see Sect. 10.6). rotation-free) vortex fields, which mathematically de- Such a flow field can however exist only when the liquid is circling scribe potential flows in the around a core. A well-known example is the hollow vortex around ideal case of frictionless flu- a bathtub drain. The core in this case is a liquid surface which is ro- ids (ideal liquids), can in fact tating about its axis like a tube. It surrounds a column of air which is occur in real liquids with fi- not moving with the revolving liquid and which narrows like a funnel nite viscosities, as long as the in the downwards direction. – The core of an irrotational vector field latter are not too great and can thus be the boundary layer at the surface of a rotating cylinder4. can be neglected. In any case, Imagine that the diameter of the core is continually decreasing. Then however, friction is necessary the flow velocity in its immediate neighborhood must be increasing, to produce the vortices. The and will approach 1 as a limit. This of course cannot occur. Instead, following paragraph will give the central parts of the liquid begin to rotate. They thus form a liquid some examples; and the ob- core, a vortex tube or, when the cross-section is very small, a vortex servation of vortex fields will fiber. Examples of this kind are given in Sect. 10.8. be the subject of the rest of this chapter. The strength of the vortex, known as its vorticity,isdefinedasthe circulation along an arbitrary path which encloses thecorejust

4 To produce and maintain a vortex in water, it suffices to set the cylinder in rotation around its symmetry axis. The thickness of the boundary layer increases without limit as a function of time (Eq. (10.3)). The velocity distribution approaches more and more that of an irrotational vortex field with increasing distance from the surface of the cylinder (i.e. curl u D 0). – In air, with its small dynamic viscosity, vortices can be generated by the process described in the caption of Fig. 10.36 (see Fig. 10.37). ). D are 0. ,or ). It ru D 10.6 u 2 circle the 10.27 and bottom D not s d 10.5 . All of the liquid u H D , and its angular velocity remains (Fig. are rapidly cast off in the A stream ). In the second limiting case, of a container, at its . When the path does which is sticking to it. The cross- separation surface Vortices and separation layers 10.4 walls once to . In the interior of a fluid, there are only initial vortices . 10.2 C10.20 10.28 0, the vector field is irrotational, and curl ). These . In this separation layer, several small vortices only at the boundary layer , and again observe flow through a bottleneck, as in D . One finds the same value along every closed path A of the liquid. These two phenomena are shown experimentally in the end 10.26 ! . 10.8 2 in Nearly Frictionless Fluids . Initially, the flow field is symmetric in front of and behind D C10.21 surface r 10.9 ! . Then the vortex has a vorticity of r ! separation layer following. – We againseen use in the Fig. broad flow apparatus which we have Fig. the bottleneck, both forThese the avoidance symmetrical flow flow andafter for fields beginning the the can overall motion; flow. after beBehind a the observed short bottleneck, time, only their two symmetryformed immediately large is (Fig. vortices lost. which rotate outwards are we tried to implement motionsfriction in or fluids boundary which layers are not (potential influenced flows, by Sects. 10.8 Vortices and Separation Surfaces Thus far, we haveids to restricted two our limiting considerations cases.a boundary of In layer. motions In the that first, in case,a we the flu- decisive internal considered friction motions role of the within (Sects. fluid played once. – Example:cylinder An and irrotational the vortex field surrounds a rotating We accomplished this byi.e. using large a compared to flowhowever, was the apparatus that boundary of we layer. restricted sufficient ourbeginning observations size, Of of to the short greatest times motion. importance, at the At longer observation times, wevery find low in viscosities all (tiny fluids, values eventhat of in new those the with internal phenomena friction, appear: e.g. gases!), formed direction of the overall flow, and a An irrotational vortex field, andforms a its core around rotation, which together the formsists liquid a of per- two “vortex”. components Every vortex thus con- which circles the core exactly core, we find sectional area of the cylinder is taken to be ‘volume elements’ which itschematically contains in must Fig. rotate. This is sketched can be clearly seen.limit of Such vanishing a thickness as separation a layer can be idealized in the Vortices can is itself delimited froma the surrounding liquid, which is at rest, by at the 2 is which bor- 0 can occur ¤ u separation layer , are thus regions in 10.8 Motions in Liquids and Gases 10 boundary layer which curl even without friction. C10.21. Examples were already given at the endthe of previous section. Note that vortices are formed in a C10.20. The liquid cores of vortices, as described in Sect. ders on a bounding surfacethe of fluid. They can thenagate prop- in a between regions of the fluid with different velocities. 232 Part I 10.8 Vortices and Separation Surfaces in Nearly Frictionless Fluids 233 Part I

Figure 10.26 Initial vortices at the beginning of stream formation

Figure 10.27 The stream of liquid, delimited by separation layers

Figure 10.28 The deﬁnition of the separation surface between two parallel ﬂowing liquids with different velocities. In the text, the velocity in one direction is zero.

vortices with closed cores; in the simplest case, they are circular (vortex rings). They can be demonstrated with the apparatus shown in Fig. 10.29. The bottom of a drum-shaped can consists of a stretched membrane M. The air within the drum is made visible using some sort of smoke. A tap on the membrane (drumhead) forces a stream of smoky air to emerge from the round opening at the top of the drum for a short time. Its outer edges immediately curve around, forming a vortex ring (a “smoke ring”, as occasionally demonstrated by

Figure 10.29 Apparatus for demonstrating vortex rings – closed, initial vortices (cast- off vortices) – in air (Video 10.4) Video 10.4: “Smoke rings” http://tiny.cc/ocgvjy the ). We sticking It can fly 10.10 . ry layer is no in front of the bottleneck, C10.22 of the vortex, which (described in this sec- ) (in Sect. is present (that is, curl irrotational vortex field towards core lift drag (or rotation or , the impeded bounda , it can blow over a playing card, snuff out 5 flow resistance the bottleneck, in contrast, all parts of the flow dynamic transverse force Behind Profiles 0). The second component of the vortex, the ¤ which surrounds the core, extendstual far influence outwards. of This two israpid vortex shown succession: by rings. The the second mu- Theyameter ring can in then for overtakes the example the process, be first,second while produced reducing ring the its in pass first di- through. ringeach expands This time and performance exchanging the slows, is roles letting repeated ofare the one the on two or rings. a two – collision times, Two vortex courseslow rings each moving which other towards down each and expand other as along they the approach. same line u through the air for severaland meters as past a the end resulta of of the candle its flow flame, energy out etc. of(in the the drum, – shape The of inflated vortex inner-tubes) rings contain that the are made visible by smoke is enclosed within the ring. In this core, Such a vortex ring is in general a rather stable object Bell-shaped jellyfish use the recoil from water vortices which they produce as field will be decelerated.longer pulled There along by theis neighboring no layers. alternative except It towall falls curve and behind; around and the there inject flow. itself Thus,separation between “the surface the and flow vortices detaches are from formed. the walls”, and the bottleneck, the original flow field,maintained. i.e. the potential flow field, will be all parts ofthe the flow stream velocity will attainsary be its layer accelerated. maximum will be value. pulled along Withinlayers The in the of the impeded the direction bottleneck, bound- of fluid flow whose by neighboring flow is unimpeded. Thus, a means of propulsion. 5 10.9 Flow Resistance and Streamline The processes that wetices have just and described, separation surfaces, i.e. canforces the lead which formation occur us of when to vor- nearly anbodies. frictionless understanding fluids These of flow are the around the solid So much forformed the here facts. by the(adhesion) – of same the The cause fluid separation to asand the surface everywhere, the solid resulting and object namely formation around of vortices by which atential it are boundary flow) is layer. should flowing, – An flow idealvelocity; around fluid every the real (po- fluid, walls however, will oflayer be which a forms. impeded bottleneck by This impedance the at will boundary actand a differently at the high the entrance exit of the bottleneck. On the way smokers with their mouths).along It its is diameter futile into two to semicircular try pieces to with divide free a ends. vortex ring tion) and the ). lift 10.11 and 10.10 Motions in Liquids and Gases 10 C10.22. It is notable that these vortex rings, at a sufficient distance from the boundary layer in which they were formed, can themselves be described in terms oftential a flow. po- This fact will prove to be particularly important for our discussion of the transverse force (the in aircraft wings, sails etc.; see Sects. 234 Part I 10.9 Flow Resistance and Streamline Profiles 235 Part I

Figure 10.30 Distortion of the avoidance flow behind a plate which is oriented perpendicular to its direction of motion; observer and liquid are at rest, and the plate is moving to the left (Video 10.3) Video 10.3: “Fluid flow around obstacles” will investigate both of them using our flow apparatus as shown in http://tiny.cc/fcgvjy Fig. 10.8. The objects which are moved relative to the liquid and are thus surrounded by a flow are again assumed to nearly touch the glass walls of the apparatus on both sides. In both cases, therefore, we are investigating a planar flow. The results can then be extended by analogy to three-dimensional flows. We start with the streamline patterns which we have already observed in the avoidance flow around a plate, Fig. 10.16 (potential flow). In these examples, the flow around the front and back (‘leading’ and ‘trailing’ edges) of the plate is completely symmetric. This means, according to Eq. (10.13), that the pressures and forces on the front and back sides are also symmetric. The sum of all the forces acting on the object is thus zero. Then the object could be moved within the liquid without any flow resistance at all! This, however, is in contradiction to our everyday experience; think for example of row- ing a boat, or stirring your soup. – In fact, this symmetry, which is also observed in the experiments shown in Figs. 10.19 through 10.21, is destroyed very soon after the start of the motion. To demonstrate this, we take a plate which is oriented perpendicular to the flow. At the very beginning of its motion, there is a symmetric avoidance flow, which we can observe by repeated back-and-forth motions of the plate (Fig. 10.19). When the motion is continued in one direction for a longer time, the symmetric flow becomes distorted: Two large initial vortices which rotate inwards are formed (Fig. 10.30). They rapidly move off in the flow direction, and in the stationary state, we can clearly observe a separation surface behind the plate. It separates a region (which extends beyond the right edge of the image) from the rest of the flow (Fig. 10.31). Within this region, the liquid is rotating in a lively manner. A number of vortices are present (this is shown impressively in Video 10.3). Video 10.3: We now have an overview of how resistance or drag is produced by “Fluid flow around obsta- flow around objects in real fluids. It comes about through the ro- cles” tational motions which form within the fluid at the trailing edge of http://tiny.cc/fcgvjy . then remains; it arises . A streamlined object of is somewhat greater than the 5 C10.23 A 2 ); other examples include the as wind turbines; they have an u 10.32 % and 10 1 2 3 5.20 1 10 : 1 D ,asinFig. F frictional resistance between 4 resistance rotors Re The production of flow resistance (drag) through vortices numbers That is the surprising experimental finding. streamlined profile EYNOLDS product of the stagnationone pressure finds and its the magnitude area to of be the plate. Experimentally, within a bell-shapedobject separation are surface. at rest,R In and this the experiment, liquid is observer flowing and to the right. The resistance for side of the shell is four times largerIn than other that of cases, thebe convex the side.) reduced resistance as is farobject. a as possible disturbing Only by factor. the carefulin small design the Then of boundary it layer theprovided between shape must us the with of many object prototypes the here. andis Their the common their characteristic fluid. Nature has into rotation. Creating the rotationaling motion in their these kinetic vortices, energy, provid- requireswhich that performs work this be work performed. isThe The the resistance force counter-force to experienced the by flow anavoidance resistance. flow object is which caused is by surroundeding rotational or by edge. vortex an motions on its trail- The resistance orfor drag technical felt applications. byparachute bodies As (it within an reduces a example, the55 m/s flow we sinking to is could around velocity often 5.5 mention ofoars m/s, used of the cf. a rowboats Fig. and person theample from paddle is wheels about provided of by steamboats. AS-shaped further profile ex- (Savonius rotors), oron hemispherical spokes, shells as mounted in an ‘anemometer’. (The air resistance of the concave Figure 10.31 the object. New regions of the liquid are continually being brought similar shape can bevortices placed will in be a formed. water Asame flow sphere flow at of velocity high about will velocity, thethe produce and same strong flow no diameter vortex is at formation initiated. the nature soon and after Streamlined in technology. profiles play an important role in cshows is a special 10.16 flow). , the liquid is 10.16 and ,also(owingtothe HAW 10.31 -S 10.31 Motions in Liquids and Gases 10.16 ELE 10 feature of the flow chosen for this model experiment (H one can choose the direction at will). The lack of vortices in Fig. flowing to the right; itbe could flowing to the rightFig. in symmetry of the streamlines, ence of the viscosity: In both figures, the object is atIn rest. Fig. very impressively the influ- C10.23. The comparison of Figs. 236 Part I 10.10 The Dynamic Transverse Force or Lift 237 Part I

Figure 10.32 A streamlined profile (in the flow apparatus from Fig. 10.8); observers and object are at rest, the liquid is flowing to the right (see also Video 10.3, in which the object is moving) Video 10.3: “Fluid flow around obstacles” 10.10 The Dynamic Transverse Force http://tiny.cc/fcgvjy One can see here very clearly or Lift the influence of the direction of flow on the formation of vortices in the boundary In general, the direction of the undisturbed flow is not identical with layer: When the streamline a symmetry axis of a solid body within the flow. An example is shown profile is moved upwards in Fig. 10.16b and in the right-hand part of Fig. 10.19. Then obser- (i.e. to the left in Fig. 10.32, vations teach us a new result: In addition to the resistance or drag F r liquid at rest), no vortices in the direction of the unperturbed flow, there is a second force which are formed (apart from those is perpendicular to the flow, as shown in Fig. 10.33. It is called the which are produced at the transverse force or lift, F . The resultant vector of these two forces is a thick end of the object at the the total force F which acts on the body within the flow6. end of the motion, when the The transverse force cannot be completely isolated and investigated displaced liquid flows back by itself. However, we can make the resistance Fr very small com- around the object). When pared to the simultaneously-present transverse force Fa. To this end, however the streamline pro- one has to use objects with the profile of an airfoil or a wing, e.g. as file is moved downwards, vortices immediately form at its thicker end. Figure 10.33 The transverse force (lift) and resistance force (drag) on a tilted plate which is moving to the left. (The resultant total force on thin plates is nearly perpendicular to the plane of the plate.)

6 For rough calculations, one can keep in mind two useful approximations: Trans- 1 % 2 verse force Fa D 3 u A, and resistance force Fr 1to10%ofFa (A D area of the plate or airfoil). . b acts .The a 10.35 10.16 F .Atfirst, lift irrotational ). or 10.35 10.8 rotating cylinder , upper part. This forms and below, an ). It has the same direction origin of the transverse force Õ 10.35 in front . . This vortex moves off with the flow. (Video 10.3) dynamic transverse force Ô only by the initial avoidance flow from be- 10.35 C10.24 10.34 ). The fluid and the observer (camera) are at rest, the 10.19 The formation of an initial vortex from the avoidance flow . Furthermore, the object must either be “infinitely” long is formed instead of an initial vortex; it circles the air- is established. With the directions of motion as shown, the : When the motion is initiated, a long-lived initial vortex 10.34 10.36 10.34 perpendicular to the direction of the unperturbed flow. The observation of theout avoidance flow contours of is the hampered airfoil.i.e. by the the Thus, avoidance one washed- flow and usually the draws parallel the flow, whole as flow, in Fig. and is carriedFig. off with the flow. Later, the flow pattern shown in The potential flows are superposedin and the lead bottom part to of the Fig. flow fieldThe shown airfoil or hydrofoil canformation of be replaced the by vortexair- field a or follows hydrofoil. the same At pattern first, as an around initial vortex an forms at the trailing edge a potential flow formsthe as irrotational seen vortex in field Fig. sketched in the center part of Fig. in Fig. or bounded by planes (as in our flow apparatus, Fig. Figure 10.34 Such an airfoil or hydrofoil(lift) shows the rather clearly. We begin with a comparison of Figs. around an airfoil orpare hydrofoil with which Fig. isairfoil is moving moving in to the the left shaded region (com- and hind and below. The senseupper of right rotation of with this an vortexFrom arrow is the marked at initial the avoidancevortex flow field foil (as its core) in a clockwise sense ( is formed as in Fig. more slowly below. This produces aabove region the of reduced airfoil, static so pressure that a above the airfoil as the flowingare fluid; oppositely below, in directed. contrast, the As two a flows result, the fluid flows faster above and Motions in Liquids and Gases 10 “Fluid flow around obstacles” http://tiny.cc/fcgvjy rod perpendicular to it isvisible not owing to the motion. Video 10.3: projects into the picture from the left is only athe holder airfoil. for The connecting C10.24. The rod which 238 Part I 10.10 The Dynamic Transverse Force or Lift 239

Figure 10.35 The origin of lift on an airfoil or hydrofoil. Top:

Potential flow without a vor- Part I tex field (model experiment); the streamlines observed in the model experiment are shown in the ideal case of a frictionless fluid (Video 10.1). Center: The Video 10.1: irrotational vector field which “Model experiments for is produced by the initial vor- streamlines” > tices in a real fluid ( 0) http://tiny.cc/8wqujy (schematic). Bottom: The superposition of the two flow fields. The irrotational vortex field cannot be observed by itself, but it can be clearly discerned in Fig. 10.34 (see also Video 10.3, “Fluid flow around obstacles”)

Figure 10.36 Streamline pattern around a rotating cylinder. On its upper side, the surface of the cylinder has the same direction of motion as the avoidance ﬂow; this prevents the formation of an initial vortex there. On its lower side, the two motions are oppositely directed, which favors the formation of an initial vortex. The difference in the pressures above and below the cylinder gives rise to the lift force. cylinder experiences a transverse force or lift in the direction of the feathered arrow.

To demonstrate this effect, we use a light cardboard cylinder about the size of a rolled-up dinner napkin (Fig. 10.37). Its ends are closed off with disks which are somewhat larger in diameter than the cylinder. A flat cloth ribbon is rolled up on the cylinder and is attached to a handle like a whip. When the handle is jerked to the side horizontally, the cylinder is accelerated across the table top, but at the same time it begins to rotate because the ribbon unrolls. Instead of flying off along a falling parabola, it rises up sharply and follows a looping path. and fric- (10.24) UTTA for example . 10.37 C10.25 . One can also not ) l , 1802–1870, Physics In- 10.8 length of the airfoil or the which opposes the motion. D l AGNUS by large planar surfaces, like l M flow around an airfoil, hydrofoil drag u ction within the boundary layer, altitude in frictionless air without % : There are two ways besides D USTAV a planar . We give a brief description of how it 10.10 F (G (drag) opposed to the direction of motion. ): At the two ends of a wing (airfoil), the and vortex strength (circulation) of the vortex which D velocity of the parallel flow, in Fig. 10.38 10.9 yields the magnitude of the dynamic transverse D (Video 10.5) induced drag u resistance The dynamic transverse force or lift on a rotating cylinder of Sects. effect) : density; a OUKOWSKI F D in a boundary layer to generate forces between a nearly friction- % ( the velocity of the cylinder on the table top; rotating cylinder; surrounds the airfoil or the rotating cylinder of length AGNUS N. J. J stitute, University of Berlin). force (M In nature and in technology, a or a rotating cylindercylinder is are never not observed. bounded onthe The both glass sides ends walls of of the our airfoil flow or apparatus in Fig. Figure 10.37 In both cases, forder, air- the or hydrofoils relation as discovered well independently as for by a M. W. rotating cylin- K construct infinitely long wingshowever or give rise cylinders. to awing – new or Their effect. cylinder finite The (potential lengths vortex) vectorforce produces field which not which provides only circles lift, the the butIt transverse also is a called the with the low-pressure regionsbelow from to the above, top producing vortices surface.the at vortex both Air field wingtips. flows around from form Together the with wing a as single coreas closed and more vortex the and initial line. more vortex,required air they to is Its set produce length into thisand continually rotation vortex its at increases, line counter-force the is must wingtips. theaircraft be “induced could The generated drag”. maintain work by a –any a constant Without propulsion. this force, drag, an Summary comes about (Fig. high-pressure regions from the bottom of the wing come into contact tion less fluid and a solidtrailing body. edge First of of thethis all, body: by creates producing a Like vortices fri at the ef- (1885– AGNUS effect” LETTNER Motions in Liquids and Gases AGNUS 10 by A. F fect, using a vertical rotating cylinder, was suggested as a propulsion method for ships Video 10.5: “M 1961). C10.25. The M http://tiny.cc/dcgvjy 240 Part I 10.11 Applications of the Transverse Force 241 Part I

Figure 10.38 The origin of induced drag: The small curved arrows indicate only the sense of rotation and not the limits of the flow field (in reality, they should be spirals). The air flows upwards to the sides in a wide region alongside the area passed over by the wing. This is the reason why many migratory birds, for example ducks and geese, prefer to fly side by side in a staggered configuration, forming a “wedge” or “string”. Then, except for the lead bird, every bird is flying in an upwards-flowing region of air, so that it requires less power to produce the lift needed to maintain its altitude. Only the bird at the point of the formation lacks this assistance, so that it must be relieved from time to time.

Second, by producing an irrotational vortex field with the body as its core: This gives rise to the dynamic transverse force (lift, transverse to the direction of the unperturbed flow); and in addition, even when the body has a good air- or hydrofoil profile, as a result of its finite length also an induced drag (opposing the direction of motion within the fluid, like every resistance force).

10.11 Applications of the Transverse Force

The transverse force which acts on rotating bodies (the lift force in Fig. 10.37) is used for example in sports. Example: A “cut” tennis ball, i.e. a ball hit with a glancing blow, flies further than a ball which is not rotating, because the transverse force compensates the weight of the ball. – For bodies which have the profile of a wing (airfoils, hydrofoils, sails), there are numerous applications of the transverse force: In airfoils, the component of the transverse force which is perpendicular to the direction of flight is used as lift;insails,the component which is in the direction of motion drives the ship. – Examples: 1. If the lift acting on an aircraft is equal and opposite to its weight, it will fly horizontally, obeying the scheme indicated in Fig. 10.33.Its forward velocity is usually maintained by an engine. The essentials were already summarized in Sect. 5.11 – If the engine is shut off, the induced drag (a result of the finite length of the wings) and the frictional losses in the boundary layer will consume its kinetic energy. ; r F = a . F even in sail glide ratio aces the wind, rising steeply higher, since it can use its , we see a ship sailing “close-hauled” (also called is held against the wind by a string from the ground; shows the ). Therefore, that quotient is called the Sailing “close- 10.39 Arrow 3 10.33 child’s kite of the wind, collectingloops kinetic around energy. and Near f tostored energy the to surface penetrate of thecreasing the layers the sea, lift of acting increasing it on wind itsit speed, wings. turns Once thereby again to it in- has follow the reached wind a in certain a altitude, downwards glide, and so forth. Examples A seagull suspended inbehind the air the stream stern that ofabove is a and moving ship; around upwards a a at tall bird an chimney. of angle prey circling in the rising, warm air Example The albatross glides along a sloping flight path downwards in the direction cf. Fig. direction of the sidewards drift. a direction which ising. not the In same Fig. as that in which the wind is blow- this permits the aira horizontal to direction. – flow The aroundimation profile to of it, a a good without kite airfoil, is carrying but a it it rather suffices. poor3. away approx- in A ship canhard move to readily move in it thetance sideways. direction to of Like motion its the can long kite-string,are hold axis, this surrounded the but by transverse ship it an resis- against is air the flow. This wind, makes so it that possible its to sails a) By gliding in a rising column of air (a ‘thermal’); Gliding is the mode of flight ofpilots sailplanes, and and it some is practiced birds. bygaining glider – altitude: In gliding flight, there are two methods of hauled”: “working to windward”); i.e. the wind is coming obliquely from the Figure 10.39 2. A b) by making use ofity. the vertical In gradient of the a(land boundary horizontal or wind layer veloc- sea), of the thealtitude. horizontal air wind above velocity the increases surface with of increasing the earth These losses must be compensatedi.e. from the the aircraft stored must slowly potential glide energy; its to flight the ground. path is The angle determined of by glide the of ratio of the lift to the drag ( Motions in Liquids and Gases 10 242 Part I 10.11 Applications of the Transverse Force 243 forward direction. The velocity of the ship must be subtracted vectorially from the velocity of the wind relative to the ground; as a result, the wind is flowing “flatly”, i.e. at a small angle of attack, around the Part I sails. The component Fv of the transverse force in the direction of travel drives the ship forward. The component perpendicular to the direction of travel leads to a small sidewards drift. 4. The rotor blade of a wind turbine as motor. Figure 10.40 shows the cross-section of a rotor blade. The wind is blowing perpendicular to the plane of the rotor, but hits the blades at a small angle of attack. The component Fv of the transverse force which is parallel to the plane of the rotor drives its rotation. The orbital velocity of the blades at some distance from the rotor hub may exceed the wind velocity by a large factor.

In order to make this intuitively clear, put a flat wedge on a smooth horizontal surface and press on its flank vertically with the point of a pencil. This will slide the wedge horizontally by a distance which is greater than the vertical motion of the pencil. As a toy, one can build a windmill with blades of a symmetric profile, for example a semi-cylindrical cross-section (as shown in Fig. 10.41). A starting push gives the blade (at the position of the cross-section shown in the figure) a velocity u relative to the ground. The direction of u determines whether the initial vortex is cast off the left side or the right side of the blade. This in turn determines the sense of rotation of the circulation and thus of the whole rotor. The blade is subject to an air flow under a small angle of attack. All else occurs as shown in Fig. 10.40.

Figure 10.40 The principles of operation of a wind-turbine blade. The component of the transverse force in the direction of arrow 3 puts a load on the vertical axis of the support structure.

Figure 10.41 A toy windmill with two blades which have a symmetric cross- section. They consist of half-cylindrical rods which can rotate around the A-A axis. When they are given a push in the direction indicated, the initial vortex is formed at the right and the windmill rotates clockwise as seen looking in the direction of the air ﬂow (wind). . 2 /s. u are 3 . 2 again % and 1cm C10.27 1 u D and 10.42 t 1 d % = ) V 10.5 of the petroleum ether. is ﬁlled with water up to % R and densities 2 R The online version of this chapter (doi: and The sketch shown in Fig. . 1 R and the density of the opening? (Sect.

C10.26 r backwards. Accelerating the airstream requires a force, The prin- ) . The water is flowing out through a circular opening in the h A cylindrical container of radius Two balls of radii 10.3 cating pistons, and this represents a considerable technical advance Of course, the air streama could be free-standing produced propeller. by an Modern enclosed turbojet fan engines instead of dispense with recipro- air stream The aircraft propeller How large is the radius sinking in petroleum ether with the constant velocities Determine the viscosity Figure 10.42 ciples of operation of an aircraft propeller blade. The propellers or screws of ships operate in analogous fashion adepth bottom of the cylinder, with a volume current of d (Sect. 10.2 represents the cross-section of athe propeller transverse blade. force perpendicular The to componentvides the of the plane driving of force theof or propeller thrust the pro- which aircraft. maintainsaircraft; This the he is flight or velocity seen she would differently say:an by The an propeller is observer a sitting fanand which in the is counter-force the blowing acts on the aircraft and drives it forward. Electronic supplementary material 10.1007/978-3-319-40046-4_10) contains supplementary material, whichable is to avail- authorized users. Exercises 10.1 5. The HAIN ttingen. .O explained at (see e.g. , in particular ANS V OHL , W.J. Boyne and OHL OHL Motions in Liquids and Gases bore into the fluid like 10 the Wikipedia article on R.W. P Jet Age D.S. Lopez, eds. Smithso- nian Institution, Washington, DC (1979), p. 25). See also this fact in mind in discussing the other examples in this section, as well! not a corkscrew. Its blades are simply rotating airfoils (hydrofoils).” One should keep this point: “The propeller of an aircraft or motor ship does Reference 17. toral student and assistant in the research group of R.W. P aircraft were carried out in 1935 by H (1911–1998) in Gö He was at that time a doc- C10.27. The first experiments with jet engines for (1930), P C10.26. In the first edition 244 Part I Waves and Vibrations II 245

Part II oee,frmn hscl ehia n uia upss n re- one purposes, quires musical and technical physical, many for However, envi- its and pendulum ronment. the mainly they between lost, friction which gradually unavoidable energy is the excitation” through the “pulsed and initial the time, from with obtained decrease amplitudes their i.e. iuswihhv enivne oacmls hscnb summa- be can this keywords accomplish the by to rized invented been have which con- niques lo the energy for the calls of vibrations replacement undamped tinuous such of production The time. h asv rieta lmn ftepnuu.Te are They to pendulum. directed the impulse of element an inertial by or produced massive the were pendulums simple these trdptnileeg fala fmass of load a of energy potential stored uu clock dulum vibration. of mo- direction The right momentary its the of at sense motion the its in ment accelerates which mechanism a activates 11PeiiayRemarks Preliminary 11.1 Vibrations .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 1 pendulum such Figs. simple of in scheme of presented The vibrations was law. sinusoidal vibrations force the linear only a under treated pendulums have we far, Thus Vibrations Undamped Producing 11.2 b and astonishingly pos- vibrations body an mechanical human over of The waves detector music. sensitive to extremely and con- an hearing close sesses of their sense in our origin its to nection has waves and vibrations of Knowledge h edlmt oto h tpiertto fteconweland wheel crown the permits of rotation escapement an stepwise the This with control to together “crown pendulum pendulum. the asymmetrically, a the cut by to are accomplished attached is which “escapement” transfer teeth energy with The wheel”, height. certain a cutc ssc sls mhszd h aeili hsadthe mind and in this this in with organized material and The chosen been emphasized. has chapter less while next is courses, introductory such in as priority acoustics given are waves and brations h nto rqec s1hertz 1 is frequency of unit The D lsia aaimfralcnrltruhfebc stepen- the is feedback through control all for paradigm classical 0H o2 0 Hz) 000 20 to Hz 20 undamped ,Fig. irtos hs mltd ean osatover constant remains amplitude whose vibrations, 11.1 trpae otkntceeg ytpigthe tapping by energy kinetic lost replaces It . feedback olsItouto oPhysics to Introduction Pohl’s 1 oa,gnrltpc ftepyiso vi- of physics the of topics general Today, . D 1s odfeunyr frequency road or 1 4.13 uocontrol auto abeitdHz) (abbreviated ssmnindaoe h tech- The above. mentioned sses and M 4.14 hthsbe itdto lifted been has that h edlmitself pendulum The : ne(fro ange h irtosof vibrations The . O 10.1007/978-3-319-40046-4_11 DOI , roughly m damped C11.1 . , amncoscillator harmonic the e.g. as implied, is erality gen- more when ‘oscillations’ terms, both use we Here, 2). Vol. (see ﬁelds magnetic and electric e.g. systems, non-material in phenomena periodic to refer also may ‘Oscillations’ chapter. this of subject the motions, acoustic and mechanical to applied usually is and general less is latter The motions. riodic pe- to refer both ‘vibrations’ and ‘oscillations’ terms The igwt Sect. with ning begin- chapter following in the treated be will sense ciﬁc spe- the in Acoustics C11.1. 11 12.24 . . 247

Part II will a , and thus ac- b . In the position M , so that the pendulum a ndulum via relative to each other (frictional and immediately thereafter, the other hook b , we see a gravity pendulum about the size of an average clock Auto-controlling of a gravity pe Auto-controlling of a gravity 11.2 In Fig. pendulum, in a side view. It is connected to a shaft of about 4 mm diameter Demonstration Figure 11.1 pendulum through feedback via anment escape- and crown wheel soon as the pendulum passes theslide midpoint off of the its hook swing, the tooth will shown in the figure,the a inner tooth flank of of thecelerates the crown the right wheel pendulum hook is in of the pressing the direction against escapement, of the arrow, to the left. As Figure 11.2 again engage another toothpressing of against the the upper crown flank wheel. of the This hook tooth is now the transfer of energy from the descending load friction with a rotating axle (frictionallength vibrations). of The the pendulum isis around 200 g. 30 cm, The and leatherrubbed its in with mass the resin, clamps like on a the violin axle bow. has to be is accelerated to the right, and soPositive forth. feedback ofa this great kind variety of canample, ways, one be often can by produce practically purely ato implemented mechanical periodic a means. connection in of source For a ofbodies ex- vibrating which energy system are by momentarily means at rest vibrations). of “sticking” or “hooking” of two Vibrations 11 248 Part II 11.2 Producing Undamped Vibrations 249

Figure 11.3 Pneumatic feedback control of a tuning fork (Video 11.1)

Figure 11.4 Auto-controlling of a tuning fork by pneumatic feedback (Video 11.1) Video 11.1: “Vibrations of a tuning Part II fork” http://tiny.cc/idgvjy The video shows the tuning fork as in the figure. Note the adjustment of the feedback by turning the knurled screw by two stuffed leather clamps. When the shaft is rotated, the pendulum is on the cylinder b in Fig. 11.4. pulled forwards with it. The clamps initially stick to the shaft (“sticking friction”). At a certain deflection of the pendulum, the torque resulting from the pendulum’s weight becomes too great and the friction of the clamps is overcome; they slip back along the shaft (“sliding friction”). The pendulum swings back, reverses, and then during its next swing forward, at a certain moment its velocity relative to the shaft is zero – the pendulum clamps and the shaft are moving synchronously. The clamps now again stick to the shaft, and the pendulum is again accelerated up to the point where it breaks free. Its next swing begins, with the same amplitude as the first, and so forth.

Feedback control via airﬂows (pneumatic control) is also widely used. Figure 11.3 shows an example, used to excite the vibrations of a tuning fork.

The essentials are shown in Fig. 11.4 as a cross-sectional view. A piston a is ﬁtted smoothly into the cylinder b, but not touching its walls. The cylinder is connected to a source of compressed air by a tube. The air pressure forces the piston out of its resting position in the cylinder, pushing the tine of the tuning fork to the right. When the piston is outside the cylinder, a ring-shaped gap opens between the piston and the wall of the cylinder, allowing the air to escape along closely compressed streamlines. From BERNOULLI’s equation (Eq. (10.13)), the resulting static pressure of the air is reduced and the piston is pulled back in. The natural frequency of the tuning fork controls this process through the mechanical coupling of the piston’s motion to the vibrations of the fork.

The use of electrical devices to control mechanical vibrations has long since gained importance. The oldest example is the electric doorbell, familiar to every school child today (Fig. 11.5). A pendulum with an iron rod (‘clapper’) vibrates in front of one pole of an electromagnet M. The rod also carries the contact spring of an interrupter switch S.

In descriptions of the operation of a doorbell, the decisive point is often misunderstood. When the interrupter contact is closed, the iron rod of the 1. ! ): The lamp ndulum allows 11.6 0, but also during the ndulum is reversing at 1mustbelessonaver- 2 Hz), and put an auxil- ! D ! 1, the acceleration is in the ( ition of the pe (distortion!). Elimination of ! 1. The time variation of the vibra- slowly . ! Every feedback mechanism perturbs 0. The current in the electromagnet must C11.2 ) ! accelerated towards the magnet. This accel- 1.9 ) which swings 11.5 1; but along the path 0 ! 0 must be greater than the energy loss along the path 0 shows a trace of the motion of the clapper of a door- (a “choke”) with a large inductance into the circuit (see Vol. 2, ! Auto-controlling L 11.7 iary coil Chap. 10). A small lamp under the rest pos us to follow the slow increasebegins of to the glow during current each clearly swing (Fig. its only maximum deﬂection when 1. the pe pendulum (the ‘clapper’) is eration occurs not onlyhalf-swing during 0 the half-swing 1 wrong direction. Itretarding is the opposite pendulum’s to motion theadditional and condition motion must reducing necessarily be of its fulﬁlled: the The energy. gainthe swinging in path As energy 1 pendulum, along a result,Only an the difference between thesedriving two the quantities vibrations of ofcurrent energy in is the the useful pendulum. electromagnet for age along than Practically, the that path this along 0 the meanstherefore path increase that 1 with time the from 0 Technically, this increase inof the the current electric is circuit. produced –(as by For shown the the in demonstration, Fig. self-induction we use a gravity pendulum tions in this casethe reveals curves clear seem deviations to from bethe pointed. a sinusoidal simple form sine ofthe curve; damping the is vibrations bought atvibrations. the price But of the abandoning distortion strictly can sinusoidal be reduced by careful design, to Figure 11.5 of a gravity pendulum byelectromagnet an Figure bell without the bell gong.a The slit clapper rod and was itspassing vibrating motion in through was front (cf. of recorded Fig. photographically using the light . 11.22 11.5 in Sect. Vibrations 11.23 11 C11.2. See also Figs. and 250 Part II 11.3 The Synthesis of Non-Sinusoidal Periodic Processes from Sine Curves 251

Figure 11.6 The current as a function of time in the feedback circuit of Fig. 11.5 (schematic of a doorbell) Part II

Figure 11.7 The time graph of the non-sinusoidal vibrations of a doorbell clapper a much greater extent than in this intentionally exaggerated demonstration experiment. Many modern electrical feedback mechanisms for mechanical vibrations make use of triodes (electron tubes or transistors). Examples can be found in Vol. 2, Chap. 11.

11.3 The Synthesis of Non-Sinusoidal Periodic Processes and Structures from Sine Curves

The deflection, velocity etc. of most periodic processes are not strictly sinusoidal. Likewise, most periodic structures do not have strictly sinusoidal profiles. Nevertheless, sine curves or sine functions play an important role in physics: It is possible to synthesize, i.e. both to generate and to represent non-sinusoidal, periodic curves by making use of simple sine functions. We demonstrate this first by using vibrations which we generate kinematically, referring to the well-known relation between a circular orbit and a sine curve (Sects. 1.7 and 4.3). We begin with the superposition of two sinusoidal vibrations of differing frequencies. We move a rod along a circular path in front of a slit and present the time sequence of the slit images as separate pictures spread out in space (using a polygon mirror in the optical path). We can see the rod and the slit in the upper part of Fig. 11.8. The rod is attached at its two ends to the circumferences of two disks I and II so that it can rotate with them; , . , 5 5 S ::: S and 3 11.8 S 1. The and , and 2 W 1 1 11.10 S S S , 1 S 5. For the ratio between the two W .–Nowforsome A 5 and whose amplitudes have a ra- phase difference W , we have chosen a value of about 3 5 A . W 1 C11.3 A of the two vibrations to be adjusted to the desired 1. This picture as well as the following figures to be established between the rotation rates of the , can be pulled out and turned by the desired angle before W F whose frequencies are in the ratios of whole numbers 3 S The superposition of two sinusoidal oscillations Demonstration , we see the traces of two sinusoidal oscillations 5 A 11.9 W are photographic images, made using the apparatus shown in Fig. 1 To adjust the phaseby difference, the the spring upper gearbeing on engaged the again right, to the which lower is gear. held A We use the sameexamples: indices for their amplitudes In Fig. whose frequencies are in the ratio 1 i.e. oscillations whose frequencies areof in the the amplitudes ratio 1 vibrations can be chosen. bottom trace shows their superposition: The trace of the combined tio Figure 11.8 apparatus for the superposition of two sinusoidal vibrations. The two shaftsand 1 2 are rotated viagears their by an electric motor which drives shaft 3 Figure 11.9 The slit can beamplitude slid horizontally ratio within thevalue. window. This allows the Oscillations they are driven by an electricfrequency motor. ratio The gears permitdisks, a and fixed, furthermore, integer any desired 11.11 will be denoted in the following by integer indices, i.e. Vibrations 11 “palpable” manner! however have an advantage in that their operation can be understood intuitively in a in a much simpler way by using electronic oscillators or frequency synthesizers. Mechanical devices may C11.3. Today, we can of course demonstrate the superposition of oscillations 252 Part II 11.3 The Synthesis of Non-Sinusoidal Periodic Processes from Sine Curves 253

S 10

S 9 Deflection S r T v v 1=1/ (10-9)=1/ 1 Time t

Figure 11.10 The superposition of two sinusoidal oscillations S10 and S9, i.e. two oscillations whose frequencies are in the ratio 10 W 9 and whose am- Part II plitudes are approximately equal. The resulting curve Sr exhibits beats.

oscillation Sr resembles a sine curve which was drawn by someone with a very shaky hand.

In Fig. 11.10, upper part, we show two sinusoidal oscillations S9 and S10 with the frequency ratio 9 : 10, of nearly equal amplitudes, A9 A10. Their superposition is shown as the bottom trace, Sr.It again resembles a sine curve, but with a periodically-variable amplitude. Oscillations of this form are called beats. The beat frequency, C11.4 often denoted as B , is equal to the difference of the two in- C11.4. Note that this beat dividual frequencies. In this example, the oscillation amplitude goes frequency is twice as large as to zero at each beat minimum. At the time of the minimum, the two the difference frequency de- equal amplitudes of the two sinusoidal oscillations are opposite; their rived from the trigonometric phase difference is 180ı. At the time of the beat maxima, in contrast, sum formula: the two amplitudes add with a phase difference of zero to give a value sin ˛ C sin ˇ D ˛ C ˇ ˛ ˇ twice that of the individual amplitudes. With two component oscilla- 2sin sin . tions of unequal amplitudes, the minima of the beat curve would not 2 2 go to zero.

In Fig. 11.11, upper part, we see the traces of two oscillations S1 and S2 with an amplitude ratio of A1 W A2 3 W 2. Their superposition gives a curve Sr which is symmetric around the time axis.

In Fig. 11.11, lower part, we see the same oscillations S1 and S2 as in the upper image, but now the oscillation S2 begins at a time t D 0 ı with a phase angle of 90 (its maximum), while oscillation S1 has a deflection of zero (phase angle 0ı). The resulting superposed oscillation Sr looks quite different, in spite of having the same amplitudes and frequencies. It is not symmetric around the time axis. In this example, we see clearly the influence of the phase difference on the form of the resulting curve. Summarizing the superposition of just two sine curves: In Figs. 11.9 through 11.11, we could “synthesize” some non-sinusoidal curves using two sine functions, i.e. we could generate the curves as well as describing them. In the non-sinusoidal oscillations in Figs. 11.9 through 11.11, a certain oscillation pattern repeats itself in every detail after a period T1. The reciprocal 1=T1 is called the fundamental frequency 1 of the shows r S whose frequen- . The approxi- 2 r S S The frequency of and . 1 S C11.5 er multiples of this fundamental s be integer multiples of the funda- ss. The frequencies of the two , a periodic sequence of “rectangles” to this beat curve 11.13 / 9 . We obtain the curve labelled . In the bottom curve, we have added a third 10 5 . 2. The comparison of the two resulting curves 10 S S W S D ::: ,and and 1 , top, we synthesized a beat curve from two component The superposition of two oscillations 3 9 S 9 S S S , , 1 7 S S 11.12 the influence of the phases on the form of the superposition curve. curves, mation can be improved as much as desired by adding additional sine cies are in the ratio 1 component the third component is thusfrequency) taken of to the be equal twomaxima to are other chosen the to frequencies. difference fall (beat together withaddition Furthermore, those of of its the such beat positive curve. a –curve, difference The which oscillation was transforms symmetrical around the therical original abscissa, into curve. beat an asymmet- The magnitudeway of on the the asymmetry amplitude of depends the in difference a oscillationFor clear-cut employed. our second example, we wishas to the synthesize upper the trace oscillation in shown Fig. non-sinusoidal oscillation proce component oscillations arefrequency. integ In a corresponding manner,sine functions, by we adding can more “synthesize”amplitudes ever and and more phases more complex of curves. component thesen. The components Their can frequencies be must alway appropriatelymental cho- frequency of the periodic curve– which Two we examples: want to synthesize. In Fig. Figure 11.11 or “boxes”, by superposing sine curves. Thiserate can degree be done of with a approximation mod- bycurves, graphically adding just three sine functions EUERMANN , 227 (1978)). 27 vibrations was de- Vibrations three 11 Praxis der Naturwis- of scribed by G. B C11.5. An apparatus for the mechanical superposition senschaften ( 254 Part II 11.3 The Synthesis of Non-Sinusoidal Periodic Processes from Sine Curves 255

S +S 10 9

S =S 1 (10–9) Deflection

S T v r 1=1/ 1 Time t

Figure 11.12 The asymmetric oscillations Sr resulting from the superposition of two sine functions S10 and S9 with their difference-frequency curve S1 D Part II S.109/. The frequency of this latter curve is thus equal to the difference of the other two frequencies.

Figure 11.13 Synthesis of a rectangular oscillation I by superposing three sine curves, S1, S3, and S5; Sr is the resulting 0 curve. The curve Sr shows curve I shifted upwards by A, so that it ‘rests on’ the axis of the abscissa (cf. Fig. 11.14).

The oscillation curve I sketched in Fig. 11.13 can thus be described approximately using the three sine curves shown below it. In analytic form, this description is as follows: Â Ã 4A 1 1 x D sin !t C sin 3!t C sin 5!t C (11.1) 3 5

.x D deﬂection, A D amplitude of the rectangular function, ! D 2=T/:

This example of “FOURIER analysis”, i.e. the representation of a complex periodic curve in terms of a sum of sine curves whose angular frequencies are integer multiples of the fundamental frequency !, can only hint at the great importance of this method. FOURIER analysis can of course also be applied to spatially-varying structures; we need only change the labels on the coordinate axes. The time coordinate t is replaced by a length coordinate, and ! D 2=T by k D 2=D,whereD, the length period, corresponds to the time period T and reflects the spatial periodicity of the structure. Fur- thermore, FOURIER analysis can also be applied to propagating or , 0 r ), / S kz ,which t 11.13 3 upper ,whichis .! ,thesim- by illation is il- 11.13 t illation curve ! (cf. Fig. u oscillations by their of its component oscil- line spectrum ); but we do not require . . u D 11.11 eriodic curve III; the sharp T .Itisa = 12. becomes smaller, the number of D = is an arbitrary number, e.g. 10 frequencies 1 T a D 11.13 ; D = / ); we need only replace , we see the spectrum belonging to the D T = 11.13 A spectrum contains no information about = 11.1 2 direction, with a velocity 11.14 z /=. from Fig. of the individual component spectral representation T 0 r S scillations. = illations, the result is the p 2 . , the frequency spectrum of this rectangular osc , we are dealing with a special case. There, we chose The frequency spectrum of the rectangular osc Knowledge of the phases is to be sure indispensable for D 2. If the ratio of of Complex Oscillatory Processes we have taken = k 11.15 1 amplitudes 11.13 != D 11.15 In Fig. lustrated with its first 20 spectralcomponent lines. osc If one adds up the first 10 of these T ‘zero’ would not occur. has been divided out ofthe the representation of component the frequencies oscillation assymmetric curve a I around common at the the factor. axis top – of of In Fig. the abscissa, the spectral line at the frequency A spectrum is plotted againstlations the as abscissa.cate The the ordinate values, calledlengths. spectral Thus, lines, in indi- Fig. oscillation curve dence of the oscillation: the phases. tracing the oscillation curve (e.g. Fig. plest representation of anless oscillatory information than process. the complete A representation spectrum of contains the time depen- Figure 11.14 shown at the bottom of Fig. 11.4 The Spectral Representation When the oscillatory processespenses are with very a complicated, representation of one theand often time uses dependence dis- instead of its the oscillation travelling waves. For example,right, a in box-shaped the wave travelling positive to the this knowledge for many physicallynon-sinusoidal important o tasks associated with In Fig. = component sine functionsFig. required increases. As an example, in can be described by Eq. ( where Vibrations 11 256 Part II 11.4 The Spectral Representation of Complex Oscillatory Processes 257

τ= 1 T 12 T=0.1s 4 b c b c I 2 h current)

Deflection a d ad (e.g. electric 0 0.05 0.1 s Time 0.6 A =2h sin(nπτ/T) 0.4 n π n II 0.2 oscillations component 0 5 10 15 20∙10 Hz Amplitude of the Frequency of the component oscillations

4 Part II 2 III

4 bc bc IV

deflection 2 Momentary 0 0.05 0.1s Time

Figure 11.15 Curve I: A rectangular oscillation or pulse train in which the time width of the pulses (e.g. electric current pulses) is much smaller than their repetition period T (i.e. the pulse duty factor is small). Curve II shows the first 20 spectral lines of the associated line spectrum. A spectral line at the frequency ‘zero’ means that there is a “constant” offset (e.g. a constant value of direct current). Curve III is the resultant superposition of the first 10 component oscillations (spectral lines), and curve IV shows the superposition of the first 20 components.

corners b and c are still missing. In curve IV, the next 10 spectral lines are added in. This has at least begun to reproduce the upper corners b and c. To reproduce the lower corners a and d, a still greater number of additional spectral lines must be included. The same holds quite generally for those parts of curves that contain straight segments that are steeply inclined to the time axis, e.g. the steeply falling segments on one side of a sawtooth proﬁle.

We show as examples two more spectra of important oscillation phenomena. 1. Frequency spectra of damped sinusoidal oscillations with periodic pulsed excitation. We take a numerical example for brevity: Some sort of oscillator is to be excited to undamped sinusoidal oscillations of frequency D 400 Hz. After its initial excitation, a sine curve of constant amplitude and unlimited length results. The spectrum consists of only a single spectral line at the frequency 400 Hz. Now, this oscillator is somehow damped. As a result, after a one- time pulsed excitation, it exhibits an oscillation of ﬁnite length whose amplitude decays with time (Fig. 11.16, part g). Above this curve, we see the oscillation of the same system with periodically-repeated pulsed excitation. In part e, a new excitation pulse is applied after each 8 oscillations; in part c, after each 5 oscillations; and in part a, after each 2 oscillations. ) of the damped illations, or pulse 200 400 600 800 200 400 600 800 200 400 600 800 200 400 600 800 Frequencies in Hz Frequencies in Hz Frequencies in Hz Frequencies in Hz cm∙s

2 1 3 2 1

–2 –2 –2

Amplitudes in cm in Amplitudes 1.5 0.5 integral

Amplitudes in cm in Amplitudes pulsed after each 2 oscil- 0.5 1.5 b cm in Amplitudes

d a f

2∙10 1∙10 3∙10 Fourier transform Fourier h OURIER corresponding frequency spectra illations, or pulse frequency 50 Hz, f only a single spectral line at the , d pulsed after each 5 osc , c b 0.02 0.03 s continuous spectrum (F pulsed after each 8 osc Time Time Time Time h e r The same damped sinusoidal oscillation as a function of time T transforms) of the oscillation curves to the left (note the scales r T r T 0 0.01 0 0.01 0.02 0.03 s 0 0.01 0.02 s 0.03 0 0.01 0.02 0.03 s

4 2 0 2 4

4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 cm cm cm Deflection Deflection Deflection cm Deflection only a single excitation pulse; OURIER g c a e g frequency 80 Hz, oscillation at the left, with only one excitation pulse of the ordinates); for pulsed excitation at different pulse frequencies; lations, or pulse frequency 200 Hz, (F Figure 11.16 Alongside each of these threespectrum. time functions, None we ﬁnd ofundamped the oscillation, them associated that is isfrequency similar with 400 Hz. to Along the with this simple original spectrum frequency of of 400 Hz, the we Vibrations 11 258 Part II 11.5 Elastic Transverse Vibrations of Linear Solid Bodies Under Tensile Stress 259 see a whole series of additional spectral lines. In each of the three spectra, the lowest frequency is that corresponding to the pulse rate, or for short, the pulse frequency. In the three spectra, from above, it is 200, 80 and 50 Hz. The pulse frequency is the fundamental frequency 1 of the three non-sinusoidal oscillations. All the other spectral frequencies must be integral multiples of the pulse frequency used for that particular spectrum. Therefore, the spectral lines occur only occasionally at the same frequencies in the spectra for different pulse frequencies. But, they are – and this is important – always in the same frequency range. It is called the formant range (the dashed

curves in Fig. 11.16b, d, and f). Part II With decreasing pulse frequency, the required number of frequency components or spectral lines in the spectral representation continues to increase. We require an increasingly large number of sinusoidal oscillations in order to represent the broad gap regions between the damped oscillations by mutual compensation of their amplitudes. In the limit of a single excitation pulse (zero pulse frequency), we arrive at 2. A continuous spectrum for a damped oscillation with a one-time pulsed excitation. In Fig. 11.16, part g, we see the time dependence of the damped, decaying oscillation after a single excitation pulse; and in part h, its frequency spectrum. The spectral lines now occur with an inﬁnite density; they ﬁll the region under the envelope curve (dashed in the upper images) continuously. The region under the curve is therefore shown as a black area in part h. Instead of a spectrum with individual, separate lines, we now have a continuous spectrum. These important relationships have been treated only descriptively here. Their analytic derivation is dealt with in detail in every rea- sonably complete mathematics course under the topic of FOURIER analysisC11.6. C11.6. For example in H.J. Pain, The Physics of Vibrations and Waves, John Wiley, New York, 4th ed. 11.5 Elastic Transverse Vibrations (1993), Chap. 9. of Linear Solid Bodies Under Tensile Stress

Systems capable of vibrations, or oscillators, have in our treatment up to now been reduced to a simple schematic form, consisting of an object with inertia to store kinetic energy, and an elastic object (spring) to store potential energy. The most clear-cut form of this scheme was a massive ball held between two extended helical springs (Fig. 4.13). We will henceforth refer to this ‘ball-and-spring pendulum’ as an elementary oscillator (A system which oscillates with a pure sine function is called a harmonic oscillator). This simple scheme was thus far sufficient to represent the majority of the oscillators which we have considered, although at times only by stretching . . 11.17 proper in phase coupled el- or .Hereagain, N ı three natural or ). three coupled elemen- 11.18 synchronously 11.19 . We will start by investigating ntally without difficulties. With normal modes. In the limit of tors, we thus find two transverse ; a vibration perpendicular to the N phase shifted by 180 linear objects. For such an object, we a higher frequency than in Fig. ,or ı illa- illa- . This system can vibrate in two ways: In 11.18 , two such elementary oscillators are con- . 2 continuous (often called ‘eigenfrequencies’), corresponding 11.18 coupled (normal modes) of arbitrary bodies. transverse vibration and and of vibration with the corresponding (proper or eigen-) 1 Transverse vibrations Transverse vibrations , two instantaneous (‘stopped-motion’ or ‘snapshot’) im- longitudinal vibration 11.17 in opposite directions , we arrive at 11.17 coupled elementary oscillators, we thus obtain N normal modes frequencies we show two stopped-motion images (Fig. For two coupled elementary oscilla The vibration frequencies are different in thewatch, two we cases. observe in Using Fig. a stop- tary oscillators. Now, threefound; different all transverse three vibration are modesThey are shown can as be stopped-motion demonstrated images experime three in the figure. can thus expect a practically unlimited number of transverse normal proper frequencies to three normal modes of vibration. We could continue indefinitelymore in coupled this oscillator way, toementary repeatedly our oscillators, adding chain. one we Forlarge obtain a chain of Figure 11.17 In an analogous manner, we see in Fig. is called a spring axis is a tors; the balls vibrate withphase a shift, 180 i.e. in oppositetions direc- (stopped-motion images) the point somewhat. It isall however by the no cases means which sufficient to occurthe describe in inertial nature. element Often, andall, a the every separate object elastic of localization element whatever of is formdaily can not experience. vibrate; reasonable. We we have know After thus thiselastic from arrived vibrations at our the problem of the A vibration of the elementary oscillator along the axis of its springs of two coupled elementary osc tors; both massive balls are moving in phase (stopped-motion images) Figure 11.18 of two coupled elementary osc transverse vibrations. In Figs. nected together or the first case, both massive balls vibrate In Fig. ages of these vibrations arevibrate drawn. In the second case, the two balls Vibrations 11 260 Part II 11.5 Elastic Transverse Vibrations of Linear Solid Bodies Under Tensile Stress 261

Figure 11.19 The three possible transverse vibrations of three coupled elementary oscillators (stopped-motion images)C11.7 C11.7. One must be careful with the nomenclature: The ﬁrst normal mode is also called the fundamental vibration, and the second is called the ﬁrst harmonic,etc.! Part II

Figure 11.20 Photographic (time exposure) images (side view) of the second to the fourth transverse normal modes of a stretched rubber elastic band; the band appears light in front of a dark background. Where it shows a gray color, the velocity of motion perpendicular to its long axis is maximal. To excite a normal mode of frequency , one end of the band is moved periodically by a motor, either transversally to its long axis at the frequency ,oralong its long axis at the frequency 2. In the latter case, the excitation is technically termed parametric, because the tension of the band (as a parameter) is periodically varied. (The tension attains its maximum value twice during one vibration period of the normal modes.) (Video 11.2, see also Video 1) Video 11.2: “Transverse normal modes of a stretched rubber band” modes. As a first example, we show the undamped transverse vibra- http://tiny.cc/negvjy tions of a horizontally-stretched rubber elastic band. Figure 11.20 The video shows some trans- shows a side view as time-exposure photographs of its second to verse normal modes as fourth normal mode vibrations. In each of these examples, we see standing waves, up to the three quantities periodically distributed along the band, namely the eighth normal mode. transverse deflection, the transverse velocity,andtheslope of the band relative to its rest position. All three quantities show nodes and Video 1: maxima. At their nodes, each of the three quantities remains zero at “R.W. POHL Lecturing” all times. At their maxima, the three quantities exhibit their largest http://tiny.cc/fpqujy values. The deflection maxima and the velocity maxima are found at the same positions; likewise the nodes of these quantities. The maxima of the slopes, however, are at the positions of the nodes of the deflection and the velocity; for example at the two ends of the band.

For a purely kinematic visualization of transverse normal modes, a wire bent to a sine-curve shape with a crank at one end is quite satisfactory (Fig. 11.21). The wire is positioned in front of the projector lamp and single .Such . C11.8 simultaneously , we see a string which is xposure photos as seen in (Video 11.3) 11.22 and imagine that the band which its midpoint, can be seen to move 11.20 shows some examples: A single point 11.22 11.23 Shadow projections of the vibration curves of one point on Visualization of . This primitive setup is rather useful. 11.20 rotated around its longvidual axis. instantaneous graphs Thephases” of for projected the short), image one vibration after then (oftencan another. readily shows called When observe the the the the crank indi- is “vibration transitionFig. turned to quickly, the we time-e a vibrating string, using a rotating lens wheel Figure 11.21 transverse normal modes or standing waves normal mode, and the two normala modes simultaneous occur occurrence of several normalstringed modes musical is instruments. often found In in Fig. Figure 11.22 We return once more tois Fig. vibrating in itspage. fourth Then normal the mode whole is band struck begins in to the vibrate plane in of addition the in its ﬁrst stretched horizontally. Its vibrations can be excited by aWith violin a bow. slitselect which a is single mounted “point”tographically. perpendicular along to Figure the the string string, and record we its can motion pho- ments exhibit a very complex vibration spectrum along the string, e.g. intransversally Fig. to the string in afunction. way which Instead, is by one no sees meansvibration a as curves. simple a sine They rule result ratherber from complex, of the non-sinusoidal normal superposition modes of (proper anormal vibrations). mode large The can num- appearance be of achieved a onlythen by very only selective approximately. bowing and even In general, the strings of musical instru- . Vibrations 12.86 11 http://tiny.cc/odgvjy The string is plucked or bowed. Video 11.3: “Transverse vibrations of astring” C11.8. See e.g. the frequency spectrum of a violin toneFig. in 262 Part II 11.6 Elastic Longitudinal and Torsional Vibrations of Stressed Linear Solid Bodies 263

Figure 11.23 The time curves of the deﬂection of one “point” on a violin string which is undergoing transverse vibrations at multiple frequencies (Video 11.3) Video 11.3: “Transverse vibrations of astring” http://tiny.cc/odgvjy Part II For demonstration experiments, instead of moving the projection lens along a horizontal straight line (as in Fig. 1.9), one can use the lens wheel shown in Fig. 11.22. When it is rotated, the individual lenses are brought into the optical path one after another. The wheel is rotated by hand using the knurled knob K. The abscissa of the projected images obtained (the time axis) is slightly curved (compare Fig. 11.23;thisisaharmlessﬂaw).

11.6 Elastic Longitudinal and Torsional Vibrations of Stressed Linear Solid Bodies

At the beginning of Sect. 11.5, we deﬁned the longitudinal vibrations of an elementary oscillator as a vibration of the massive component of the oscillator parallel to the long axis of its helical springs. In Fig. 11.24, we illustrate the two longitudinal vibration modes of two coupled elementary oscillators. At left, both massive balls are moving in the same direction, synchronized “in phase”. At the right, they are vibrating in opposite directions, “phase shifted by 180ı”. We continue adding additional elementary oscillators to make a chain, and ﬁnd N normal modes with N oscillators. Thus we again arrive in the limit of large N at a linear object with a practically unlimited number of longitudinal normal modes of vibration. As an exam-

Figure 11.24 Three stopped-motion images each of the longitudinal vibrations of two coupled ball-and-spring oscillators; top and bottom at the times of their maximum deflections, center image at passage through their rest positions. On the left, both massive balls are vibrating with the same phase, on the right with a phase shift of 180°. . : of C11.9 (11.2) along the velocities l shows time- 1 of the band for the 11.27 . They can be con- ) images of the first l show the variations graphic representations Spacing of the stripes D 11.26 as the quotient l l the author demonstrates this ex- Bottom: density of stripes N N , and the longitudinal of its long axis at the frequency time exposure in Fig. d longitudinal vibrations of linear in the along the length l Video 1 l N N l ). torsional oscillations deflections 11.25 in the direction The first and second longitudinal normal modes of Length shows photographic ( Top: stopped-motion images 11.25 Number of stripes in a segment of the density of stripes l D l N N and the second longitudinal normal modes.mediately In see both examples, two we quantities im- periodicallynamely distributed the along longitudinal the band, The maxima of the deflectionas and the do velocity their fall at nodes. the same The points, third quantity is the elastic deformation band. We define the density of stripes The two a rubber band with whiteexposure cross stripes photos. on Where a black thevibration background, white 1 velocity stripes m long; are along time- visible theof only the band distribution as of is a maximum deflection high. grayalong (and blur, the of the band. the longitudinal To velocities) excitewas the pulled normal periodically vibration modes, the left end of the band first longitudinal normal mode, nearly atdeflections. the phases The of the maxima maximum band. of They are these thus at variations the are pointshave where their at the nodes deflection the (Fig. and the ends velocity of the In addition tosolid the bodies, transverse there are an also veniently demonstrated using a braidedters rubber wide band which a is fewexposure stretched centime- images horizontally. for three Figure normal modes. stretched black rubber band with white cross stripes. Figure (stretching and compression), whichalong is the also band. periodically distributed Thecauses periodic a periodic distribution variation of the elastic deformation Figure 11.25 ple, we show the undamped longitudinal vibrations of a horizontally- periment himself.) the normal mode by a motor. (In Lecturing” ). OHL 11.20 Vibrations 11 http://tiny.cc/fpqujy case of transverse vibrations (Fig. C11.9. This deformation corresponds to the slope in the Video 1: “R.W. P 264 Part II 11.7 Elastic Vibrations in Columns of Liquids and Gases 265

Figure 11.26 Photographic high-speed images (ca. 105 s) of the first lon- Part II gitudinal normal mode of a striped rubber band nearly at the phases of its maximum longitudinal deflections (to be observed stroboscopically!). The maximum deflection in the center is ˙ 8 cm. The somewhat longer vertical streak allows us to discern the vibration phases. Due to overstretching the band, the images are only qualitatively accurateC11.10. C11.10. The connection between the variations Nl shown in Fig. 11.26 and the quantity Nl defined in Eq. (11.2)isgivenby Nl D Nl Nl,mean, where 1=Nl,mean is the mean spacing of the stripes, averaged over the total length of the band. Since the maxima Figure 11.27 The third to the fifth torsional oscillation modes of a 1 m long of Nl are at the ends of the and 3 cm wide stretched rubber band. The holder at one end is turned back band, we find for the nth nor- and forth by an eccentric on an axis parallel to the long axis of the band; mal mode that (nC1) maxima angles of a few degrees are sufficient. exist for Nl. Compare Fig. 11.31 in the following section. 11.7 Elastic Vibrations in Columns of Liquids and Gases

As before, we treat liquids and gases together (as ‘ﬂuids’). Our experiments will be carried out for the most part using air. In the interior of liquids and gases (their difference: formation of surfaces), no transverse or torsional vibrations are possible, but rather only longitudinal vibrations. This is a direct result of the free motions of all the liquid or gas particles past each other2. As in the case of solid bodies, we will ﬁrst treat linear arrangements of liquids and gases. Linearly-bounded liquid or gas columns can be constrained in tubes. Gas columns can easily be excited to normal-mode vibrations. For example, as a demonstration, one can close off one end of a cardboard tube around 1 m long and several centimeters in diameter with

2 ‘Particles’ in the sense of small volume elements, not individual molecules. Hz 4 10 is found. 3 dust particles 10.17 . Imagine that the 11.6 ). ). 11.30 11.35 ’s dust figures. ily audible, but rapidly decaying ). Two windows allow us to observe UNDT 11.28 when the column vibrates or is blown as an organ ’ flame tube (Fig. UBENS Aerodynamic detection of the ). Excitation is provided by a flue pipe which is blown directly 11.29 (Fig. in front of the open end of the tube (Fig. has to bethe equal tube. to – one When of the vibrating the gas natural is flowing frequencies back and of forth the in the gas tube, column in We show the normal-mode vibrations at the frequency of The flame tube is several metersupper long side, and is there filled is withtube. natural a One gas. series end On of of the its tube burnerbe is excited openings closed, the by running other some the carries means a length membrane to which of give can the undamped vibrations. Their frequency Example In the interior ofpith-balls on a thin tube threads with (Fig. the a balls square as a cross-section, projectedis we shadow first image. hang placed The perpendicular two to linealong small the connecting the the long tube two axis axis, balls ofThe the the streamlines streamline tube. are pattern pressed Then together shown between for inapproach the a two Fig. each balls; flow other the balls must pipe. This is indeed the case. back-and-forth air flows along theorgan axis pipe. of (The an balls couldbehind also the be other hung instead one ofthe side air by flow side. in both Then repulsion directions of produces the a balls.) mutual Figure 11.28 The nodes of this longitudinal flowpowder can on be the shown inside by bottom scattering ofto a rest the fine at tube. the The nodes powder and particles form come K The periodic distribution of the flowstrated velocity with maxima can R be demon- a rubber membrane. Byexcite plucking this or “air column” tapping to the loud,normal-mode membrane, read vibrations. we Or can else we couldend put and a pull solid a cover removable overgitudinal cover one rapidly vibrations off take the place other in end.of a an These manner elastic lon- rubber basically band similar as to described those in Sect. plitudes”). – For demonstrationthe experiments back-and-forth with air a flows largebe at audience, shown the using maxima aerodynamic ofdirection of forces the motion. flow which velocity are can independent of the air column is divided upplace of into one thin stripe layers, on where the rubber each band. Between layer takes the the nodesforth. of the Their motion vibrations,suspended can in these the be air layers made column. flowand These visible thus can with back used be to observed small measure and microscopically the maximal deflections to both sides (“am- Vibrations 11 266 Part II 11.7 Elastic Vibrations in Columns of Liquids and Gases 267

Figure 11.29 KUNDT’s dust figures. During the vibrations, the dust forms a fine fog which hangs perpendicular to the axis of the tube like curtains. They move slowly along the axis. They demonstrate that the flow within the longitudinally-vibrating gas column is associated with complex side effects (“second-order effects”). These come about through the formation of a boundary layer between the walls of the tube and the flowing parts of the gas column. Part II

C11.11. H. Rubens and O. Krigar-Menzel, Ann. der Physik 17, 149 (1905). These Figure 11.30 RUBENS’ flame tube shows the distribution of the flow ve- authors also report that this locities in a longitudinally-vibrating gas column. The heights of the flames effect is observed only at are constant over time and their maxima lie above the maxima of the flow low sound amplitudes. At velocityC11.11. (Video 1 shows W. SPERBER igniting the RUBENS’flame higher amplitudes, the max- tube. http://tiny.cc/fpqujy) imal flame heights appear at the nodes of the motion. These maxima then a boundary layer forms along its walls; cf. Sect. 10.2. In this layer, the vibrate at the frequency of static pressure above the velocity maxima increases by an amount which the standing sound wave 3 is constant over time, and above the nodes, it correspondingly decreases . in the tube. RUBENS and This causes the periodic distribution of the flame heights, which is constant KRIGAR-MENZEL explain over time. An application of these phenomena is shown in Fig. 12.46. the spatial vibrations at low In the case of longitudinal vibrations of an elastic rubber band, the sound amplitudes by effects stripe density was distributed periodically along the length of the in the boundary layer, al- band. The maxima of the stripe density were located where the axial though not in as much detail motion exhibits nodes (compare Fig. 11.26). Precisely the same is as given here. – In a sim- true for a longitudinally-vibrating gas or liquid column; but now, in- ilar experiment at the 3rd Physics Institute in Göttin- stead of the stripe density, we observe the number density NV of the molecules (Sect. 13.1). For a longitudinally-vibrating column of gas, gen(Dr.D.RONNEBERGER), we thus obtain the distribution sketched in Fig. 11.31. Three phases standing sound waves (at are shown for the fourth normal mode in a tube which is closed a frequency of 2800 Hz) in at both ends. Gray shading indicates the normal number density, air in a plexiglass tube are a lighter shading is a reduced density, and a darker shading is an in- shown. The tube contains a small amount of water along its bottom. In this 3 Rotations of the gas within the boundary layer (Sect. 10.2 and Eq. (10.18)) cause experiment, pressure ampli- the formation of two stationary vortex rings between each velocity maximum and 2 the two neighboring nodes. The symmetry axis of these rings is oriented along tudes of the order of 10 N/m the long axis of the tube. The sense of rotation in two neighboring vortex rings ( 1mmH2O) are observed is opposite. At the walls of the tube, the stationary flows from the vortices ap- at the nodes, corresponding proach each other in those sections of the tube where the velocity maxima of the to the case of high sound longitudinally-vibrating column of gas are located. There, the static pressure in amplitudes in RUBENS and the boundary layer is higher. In those sections of the tube where the nodes are KRIGAR-MENZEL’s experi- located, the stationary flows oppose each other. There, the static pressure in the boundary layer is lower. ments. , , in V 11.35 N 11.31 schlieren show a still . . Hz (correspond- C11.12 4 11.34 10 and ultrasonic . It shows longitudinal ) into individual vortices. or ’s schlieren method (Vol. 2, 11.32 11.33 labium OEPLER , i.e. a half wavelength ultrasound 8 mm). Sound waves with frequencies 11.26 in Fig. a Hz are called 4 and The periodic distribution of the number density Three snapshot images of the distribution of the number den- 10 b of the section shown in the top and bottom images corresponds l along a closed gas column vibrating at its fourth natural frequency. V N in Fig. 21.26, was 4.8 m). a longitudinally-vibrating gas column excited byphotographed a flue in pipe as a in Fig. dark field with T ingtoawavelengthof above ca. 2 rather simple (nearly sinusoidal) pipe vibration and its line spectrum. Technically, normal modes of gas columnsthe play construction an of pipes, important whistles role andThe in wind usual instruments of types all of kinds. Their design function is are rather – complicated in atitatively detail least and elucidated. has superficially been In only – qual- decay the well case of known. of the the air flue flow pipe, past there the is lip a ( periodic The air flow isthe controlled pipe. by A the similar vibrations situationa of holds reed the for pipe. the column reed This ofform and mechanism air in the causes in air the deviations column from pipe in the vibrations. sinusoidal Figures Sect. 21.11, Fig. 21.26). Intance the between figure, the above and condenser belowb and are the reversed (The wire-shaped dis- slit in the exit diaphragm, sity can be demonstrated experimentally, most effectively with a method. We give an example in Fig. Figure 11.32 creased density. This distribution, shown schematically in Fig. Figure 11.31 Top and bottom, at theplitudes; times center, at when a the time in density betweenThe variation when has the length density its distribution greatest is uniform. am- to the images pipe. The frequency is in this case about 4 normal-mode vibrations of an air column excited by a small flue ), V N 11.26 1). In a tube ). C , the number of n vibrates (and with l N V N vibrates (cf. Fig. Vibrations -th normal mode for l n N 11 to the longitudinal vibrations of a rubber band, in which C11.12. In complete analogy of the tube, and maximathe of flow velocity there (see Exercise 11.2 ural frequencies, there are nodes of the number density (and the pressure) at the ends maxima is ( which is open at bothin ends, which the gas columnvibrating is at one of its nat- in a gas column, thedensity number Note that correspondingly, in the it also the local staticsure) pres- around a mean value. and for 268 Part II 11.8 Normal Modes of Stiff Linear Bodies 269

Figure 11.33 A nearly sinusoidal vibration in a pipe (image by F. T RENDELENBURG)

Figure 11.34 The spectrum of the pipe vibration shown in Fig. 11.33 Part II

Figure 11.35 A ﬂue pipe for frequencies from about 104 to 6 104 Hz. The lip slit and the lip are designed to be rotationally symmetric. The actual volume of the pipe represents only a very rough approximation to a linear column of air.

Flue pipes which produce high-frequency tones are an important tool for physics (Fig. 11.35). The right-hand image in the ﬁgure shows construction details of the pipe (essentially a whistle operated with compressed air).

11.8 Normal Modes of Stiff Linear Bodies

In considering the normal modes of solid, linear (one-dimensional) bodies, we have thus far treated only limp objects which had to be stretched by external forces, for example a rubber band. Demonstra- tion of the normal modes of stiff linear objects, such as rods made of glass or metal, is more difficult. Such rods have to be held on knife- edge supports under two of the nodes, or hung there from thin cords. . (11.5) (11.3) (11.4) C11.13 vibrations longitudinal bending vibrations ; % E : : s % % E 2 h l l D r s vibrations for such a case, using 2 l l l = 1 1 2 2 c stretched rope or a rod whose ends 453 : a rod resting on both its ends D D D 0 1 1 D C11.14 1 transverse shows a demonstration of of a short, cylindrical steel rod. They are excited by illustrates Longitudinal vibrations of Bending vibrations of a flat steel bar (87 cm long). The exci- 11.37 11.36 sound velocity in the rod) D c Figure (Video 11.4) for comparison: a stretched rope For longitudinal vibrations are either fixed or both free a rod hanging from a25 cord cm), (rod fundamental length frequency ( striking one end of thevibration. rod. This impulse The excitation yields cross-sectionnodes, a but of damped at the the roda maxima, contraction. an remains We expansion can unchanged hear alternates a at periodically tone with the which decaysThe within a frequencies few of seconds. thefollowing first relations: normal modes of these bodies obey the For transverse vibrations tation was provided bywhich a carried small an electromagnet alternatingmade current under visible at the by a left scattering frequency end sand of on of 252 Hz. the the rod. The rod, nodes are Figure a flat steel bar. These oscillations are called Figure 11.36 Figure 11.37 . 12.17 . Bend- 2 l ) and their nat- , John Wiley, N.Y., Vibrations 11.36 11 open at both ends (organ pipes!). For longitudinal vibrations, see also Sect. rod, the node is atThink its of center. the vibrations of a column of air inare pipes closed that at both ends or fixed at both ends, theof nodes the motion are held atends. the With a free-standing C11.14. In the case of a stretched rope and the rod proportional to nusoidal with nodes at the ends. But note that invibration this mode, the first natural frequency is inversely two ends (but not clamped there) are the same asof those a stretched rope, i.e. si- C11.13. The modes of vibration of a rod resting on its jected shadow image. This experiment was included in the book up to thetion. 11th edi- modes or standing waves on a small helical spring; they are readily seen as a pro- http://tiny.cc/ydgvjy The video provides a supplement to this section, in the form of longitudinal normal Video 11.4: “Longitudinal vibrations of a helical spring” Vibration Problems in Engi- neering 4th ed. (1974), Chap. 5. ural frequencies are rather complicated. For details, see e.g. S. Timoshenko, D.H. Young, and W. Weaver, ing vibrations (see also Fig. 270 Part II 11.9 Normal Modes of 2-Dimensional and 3-Dimensional Bodies. Thermal Vibrations 271

For torsional oscillations held ﬁxed at both ends (no rotation) or a free cylindrical rodC11.15 C11.15. When the ends of s the rod are ﬁxed, the nodes of 1 G the motion are held there; for D : (11.6) 1 2l % a free-standing rod, the node is at its center. (l D length, % D density, D tensile stress, h D thickness of the rod, E D elastic modulus, G D shear modulus, cf. Table 8.1.)

The vibrations of short crystal rods, for example quartz crystals, have

acquired considerable technological importance in recent years; in Part II particular in the broad area of communications technology and for watches and clocks, they serve as a “microscopic pendulum”.

The longitudinal vibrations of crystal rods are also suitable for generating standing waves in columns of liquid. These vibrations are excited electrically; to make the wave crests and troughs visible, a schlieren image with bright-ﬁeld illumination can be employed, as shown e.g. in Fig. 11.32.

11.9 Normal Modes of 2-Dimensional and 3-Dimensional Bodies. Thermal Vibrations

We can be rather brief here. In 2 and 3 dimensions, we can also trace the occurrence of normal modes of vibration to the coupling of many elementary oscillators. But this would be, apart from a few exceptions, an exceedingly challenging business mathematicallyC11.16. In C11.16. It can be solved the majority of all practically important cases, one remains dependent numerically today using the on experimental observations. To detect the nodal lines, usually the methods of so-called Finite- accumulation of dust or sand scattered on the surface is employed. Element Analysis (carried out Figure 11.38 shows the nodal lines of a square and a circular metal by fast computers). plate in various modes of vibration.

If, instead of scattered sand, one makes use of optical methods (refraction of polarized light), then considerably more complex normal-mode vibrations can be observed (see Vol. 2, Sect. 24.9, ‘Strain Birefringence’). Figure 11.39 gives two examples for short glass cylinders with circular cross-sections.

Bowing of the plates leads to a bell shape. The vibrations of these relatively simple forms are already unpleasantly complicated. In the simplest case, a glass as seen from above vibrates according to the scheme shown in Fig. 11.40.AtK, we see the crossover points of four nodal lines which take the form of “meridians”. We can imagine that the simplest vibrations of our skull capsules, which harbor our hearing organs in their walls, must take a roughly similar form. In the range of extremely high frequencies up to the order of 1013 Hz, all solid bodies, regardless of their form, possess an enormous number of elastic normal modes. Oscillations at such frequencies make C11.17 ) in solid bodies or 16 prisms. Their diameters are 30 (right). ICOL 11.24 and sound patterns (positive photographic images) 11.18 C11.18 HLADNI ) Glass cylinders vibrating at high frequency, observed with Simple vibrations of a wineglass, C ERGMANN shaped vessels with short, open necks are especially worth mention- crystals. At theatoms highest or end molecules of in this theized crystal frequency roughly lattice range, in vibrate Figs. the in the individual manner visual- In considering the normalfrequencies modes and of normal gas-filled modes volumes, of the air-containing natural spherical or bottle- and 44 mmL. and B the excitation frequency is 1.54 MHz (images taken by linearly-polarized light between two N up the disordered “thermal motions” (Chap. Figure 11.39 Figure 11.40 Figure 11.38 seen from above (schematic drawing) ”, Weid- ”, Ver- Vibrations 11 Der Ultraschall Entdeckungen über die “ lag S. Hirzel, Stuttgart, 6th ed. (1954), pp. 636ff. C11.18. L. Bergmann, Theorie des Klanges manns Erben und Reich, Leipzig (1787)). drawings of over 100 normal- mode vibrations of round and square plates (E.F.F. Chladni, “ C11.17. The first publication of this research contained 272 Part II 11.10 Forced Oscillations 273 ing. These are “HELMHOLTZ resonators”, which are important in C11.19. See e.g. T.B. Greenslade, metrology. They represent sound sources of a convenient form with “Experiments with Helmholtz well-defined fundamental frequenciesC11.19 (Video 11.5). Resonators”, Physics Teach., Vol. 34 (1996), p. 228. For architects, the normal modes of large dwelling and meeting rooms are important. They are excited when people speak, sing or play music. Video 11.5: “HELMHOLTZ resonators” http://tiny.cc/0cgvjy The video was filmed in 11.10 Forced Oscillations the historic collection of the 1st Physics Institute in Part II Following a pulsed excitation, every oscillator vibrates at one or more Göttingen. of its natural frequencies. But one can also cause every oscillator to vibrate at other, arbitrary frequencies which are not equal to any of its natural frequencies. In this case, the oscillator is undergoing forced oscillations. Such forced oscillations play an extraordinarily important role in many areas of physicsC11.20. C11.20. The torsional pendulum described in this section To treat forced oscillations, we must first deal with the concept and used for demonstrating of damping of an oscillator more precisely than we have done so forced oscillations, known far. Due to unavoidable losses of energy or intentional energy out- as “POHL’s pendulum”, can puts, the amplitude of every oscillator decays with time following be found today in every col- an excitation pulse. The graph of the oscillation as a function of lection of demonstration time is given by curves of the type illustrated in Fig. 11.42a(de- experiments and in every el- flection ˛). In many cases, these curves follow a simple law for ementary physics lab course. sinusoidally oscillating systems: The ratio of two maximum deflec- See also Vol. 2, Sects. 11.7 tions or amplitudes on the same side of the curve remains constant and 26.2. A detailed math- all along the time curve. It is termed the damping ratio K.Its ematical treatment can be C11.21 natural logarithm is called the logarithmic decrement . The found in H.J. Pain, “The damping constants and the logarithmic decrements are given in Physics of Vibrations and Fig. 11.42a. Waves”, John Wiley, New York, 5th ed. (1999), Chap. 2. The reciprocal of the logarithmic decrement is equal to the number of oscillations after which the amplitude of the deflection has decreased to 1=e D 37 % of its original value. C11.21. This means that the decrease of the amplitude ˛ Keeping these definitions in mind, we now want to elucidate the es- is exponential: ˛. / ˛ ıt sentials of forced oscillation using a demonstration experiment which t D 0 e , provides clear and detailed examples. This experiment makes use of with the logarithmic decre- torsional oscillations at a rather low frequency: as usual, the slower ment ,definedby the sequence of events, the clearer our observations. Figure 11.41 ıT D D ln K shows a suitable torsional oscillator (torsional pendulum). Its iner- (T D oscillation period). This tial element is a flat copper wheel with three spokes, whose hub is follows mathematically from mounted on an axle D.Aspiral spring is also attached to the axle, the fundamental equation for with its other end mounted on a lever at A; the lever can rotate back rotational motion (equation and forth around D. Via the connecting rod S and a motor-eccentric of motion, Eq. (6.7)inTa- arrangement, the lever carrying the end A of the spring can be moved ble 6.1 for a frictional torque sinusoidally at a chosen frequency (the motor is geared down to a low proportional to the angular rotational speed and can be further controlled by a rheostat) and vari- velocity (see Exercise 11.5). able amplitude. In this way, a sinusoidal torque of constant amplitude 3 for : and 3 D 0 e himself ˛ D T illations, The de- = ,waitfor 1 .

OHL D P e , C11.22 D damping ratio K Video 1 eces. The current in the in ; , the maximum amplitudes ough the magnetic field pro- and the . To illustrate this, the sequence e K is an arrangement which allows 42 Hz. The spiral spring has several : 0 M 3 N/rad, moment of inertia : (Video 11.6 D 2 e 39 s; thus the eigenfrequency but only then D : 2 b for four different damping ratios. The D D . Corresponding pairs of values of 0 e T ˛ eigenfrequency 11.42 are somewhat asymmetric Gauss curves. These C A torsional pendulum for demonstrating forced osc ,and a, and their endpoints are connected with a drawn-in line. B , and eigenfrequency 2 , A 11.42 42 Hz. The ratio of two consecutive amplitudes is found to be nearly con- kg m : 3 in the frequency range near the eigenfrequency of the pendulum The oscillation period 0 stant at 1.285; thisof is the amplitudes damping to ratio theFig. left and the right at intervals of 1.2 s are graphed in The damping mechanism operates bywheel. inducing eddy A currents small in electromagnet theof has copper the its wheel. pole The pieces oscillating on wheelduced moves each by thr side the of magnet, the withoutmagnet rim touching coils the can pole be pi varied to control the damping. See Vol. 2, Chap. 8. Numerical example are listed in Fig. 0 ˛ are the ‘deflection curves for forced oscillations’. When the damping is weak, flection of the torsionaloutside pendulum can the be wheel. read Below off left the at angular scale demonstrates his torsional pendulum.) curves turns (not shown in the drawing). the oscillator. For both measurements, thelever wheel in is its deflected rest with position the are and registered. its reversal points (maximum deflection) with a torsion constant of are especially large compared to the amplitudes at other frequencies. the damping of the wheel’s motion to be adjusted within wide limits. 10 Now for the actual experiment:the We set lever the attached “exciter” to in thethe motion spring – stationary – here and state record ofthe its oscillation frequency forced to amplitude be established, and then record Before beginning the actualnatural frequency demonstration, or we first determine the Figure 11.41 and variable frequency can be applied to the axle is , / / t is t ˛ // t D ˇ. ˛. 2 ’s pen- 0, i.e. the . ˇ D is the momen- D sin / D t Lecturing” 0 OHL ˛ / ˇ t ˇ .ˇ. , as described in : ˛. D A OHL D M ). We find D ˇ D Vibrations 6.7 C is the angular elastic / is the moment of inertia t acting on the axle P . ! 11 D

“Free and forced oscillations of a torsional pendulum (P Video 11.6: M “R.W. P dulum)” http://tiny.cc/uegvjy Video 1: equation which describes the motion of a free torsional pendulum (without damping). acting at the text. For rest position of the lever, we obtain the differential On the right-hand side is the additional sinusoidal torque generated by the lever of the pendulum wheel). http://tiny.cc/fpqujy ( (Eq. the torsional pendulum, this is the torque which enters into the equation of motion constant (torsion constant) of the spiral spring). Inmathematical the treatment of C11.22. By way of explanation: The total torque M the angle that describes the motion of the lever around the axle, and pendulum wheel. We then find for tary angular deflection of the ( deflection of the spiral spring. This deflection is where determined by the angular 274 Part II 11.10 Forced Oscillations 275

a K=1.29 degree A A Λ=0.25 100

B K=1.60 Λ=0.47 80

of free oscillations K=2.90 α C Λ 60 =1.06

D of the forced oscillations Part II 0 Deflection K≈50 α B 40 246 8s Time C b Amplitude 20

degree A B c C D 150 120 between

Δϕ 90 60 30

0 0.2 0.4 0.6 0.8 Hz excitation and resonator Frequency v of the excitation Phase difference 0 0.5 1.0 1.5 2.0 Frequency v of the excitation Eigenfrequency ve of the undamped resonator

Figure 11.42 a The time dependence of free oscillations of the torsional pendulum in Fig. 11.41 with various values of the damping; curve D was recorded with a camera. The small increase in the period with increasing damping (at D 1 only 4 %) is negligible compared to the precision of the present measurements. This allows us to refer to only one eigenfrequency in the text, e D !e=2, independently of . b The deflection amplitudes of forced oscillations of the torsional pendulum at a constant excitation amplitude, as a function of the excitation frequency and the damping of the resonator. With increasing damping, the maximum value of ˛0 at the resonance frequency e decreases visibly. At an excitation frequency of zero, ˛0 is simply determined by the two endpoints of the end of the spring (A in Fig. 11.41). c The influence of the excitation frequency and the damping on the phase difference ' between the excitation and the resonator. The excitation is always ahead in phase. The measured points were taken from ‘stopped-motion’ photographic images. At the resonance frequency e,the phase difference ' D 90ı, independently of the damping. Note the optical illusion at the crossover point of the curves (see also Sect. 1.1). (Video 11.6) Video 11.6: “Free and forced oscillations of a torsional pendulum (POHL’s pendulum)” http://tiny.cc/uegvjy Z .At move (11.7) in our A A by which . cshowsthe all along the ; 11.42 e phase shift This torque reaches its . and that of the pointer . Figure . Referring to this term, any and the excitation A C11.23 resonance curves Z , it attains a value of 12.5. This ), peculiar features are seen: The at its right-hand extreme value) of the amplitude of the torsional . This means that A D 11.41 are ı A C , , parts b and c; the lower axis is inde- resonance B ahead , A 11.42 ;forcurve )istermed e Amplitude at the eigenfrequency D Amplitude at excitation frequency zero enhancement is just moving out of its rest position to the right; the ex- , and the curves D A V At large values of the damping (curve maximum of the resonance curvelower is frequencies. barely visible, and it is shifted towards in the demonstration). We thus have to record simultaneously both resonator demonstration) precedes the amplitudeZ of the resonator (thethe pointer position of the end of the spring at for the torsional pendulum in Fig. when the torsion wheel is just passing through its rest position. The the amplitude of the excitation (the end of the spring at results. At very low frequencies, the pointer path of the pendulum,that it the always exciter accelerates tensesthe the the left-hand motion maximum spring of rotation the in ofspring torsional such the at pendulum torsion a wheel, way the end of the citer thus produces a torque to the right together, with zero phase shift,same and time. both reverse As their thethe motions excitation excitation at begins frequency to the increases, move the amplitude of pendulum (the resonator) more and more. At resonance, the phase shift is 90 maximum value (end of the spring oscillator operated in thea mode of forced oscillations may be called is called the special case ( The resonance curves whicha we particular have example – observedgenerally. experimentally our That for torsional is pendulum the reason –the why frequency hold axis we in in have Fig. included fact two quite captions on The ratio pendent of the particularapparatus. numbers It obtained gives with the frequency our ofeigenfrequency demonstration the of excitation the as undamped a resonator, fractionbe. whatever of its the nature Thus, may theacoustic curves forced oscillations, apply but not alsosystems only to with those to vastly of differing any electrical resonance and kindSects. frequencies optical 11.7 (see of and also mechanical 26.2). Vol. 2, or Given the universal applicabilitytions of of these the curves mostpressures, for electric diverse forced currents, voltages, systems oscilla- field strengths, and etc.),keep amplitudes one in should (lengths, mind angles, athis simple end, general we picture consider of anotherthe how experimental influence they observation: of come the It about. excitation concerns frequency on To the Vibrations ; see Comment C11.22. ˇ 11 C11.23. This is the torque D 276 Part II 11.10 Forced Oscillations 277 Part II

Figure 11.43 The resonance curve of the angular velocity of the torsion wheel as it passes its rest position (˛ D 0, its maximum angular velocity !0), for curve C from Fig. 11.42b. At resonance, the maximum value of the angular velocity is !0 D 71 deg/s D 1.23/s. This resonance curve has its maximum at e, independently of the damping. This is also true of the resonance curve of the energy, since the latter is proportional to the square of the velocity. torque is zero (the end of the spring again at its rest position) just at the moment when the wheel is reversing at its right-hand maximum rotation. For the oscillation of the torsion wheel from the right to the C11.24. Here, it is impor- left, we have the same situation with a reversed sign. At resonance, tant to distinguish between the external torque, which leads the deflection ˛ by a phase angle the time-dependent angu- ı d˛ of 90 , is thus feeding energy into the torsional pendulum along its lar velocity ! D and whole back-and-forth path. Without the losses due to damping, the dt the constant angular fre- oscillation amplitude at resonance would increase without limit (“res- quency 2 with which the onance catastrophe”). resonator is being excited Figure 11.43 gives the maximum angular velocity !0 of the torsion (i.e. the frequency of the ex- wheel for curve C in Fig. 11.42b, i.e. the amplitude of the angular ternal sinusoidal force). In velocity with which the wheel passes through its rest position. It the stationary state, for the is related to the maximum deflection ˛0, i.e. the amplitude of the deflection we have C11.24 deflection, by the simple equation ˛.t/ D ˛0 sin.2t/ and for the time-dependent ! ˛ 0 D 0 2 (11.8) angular velocity d˛ . D excitation frequency/: D ˛ .2/cos.2t/, dt 0 whose amplitude is denoted The pendulum goes through its rest position with the maximum value by !0 in Eq. (11.8). of its kinetic energy E0.Itisgivenby 1 !2 E0 D 0 (6.5) C11.25. Away from the rest 2 position, the total energy of . is the moment of inertia of the torsion wheel/: the pendulum is given by the sum of its kinetic and poten- The dependence of the energy amplitude E0 on the excitation fre- tial energies. The resonance quency is shown in Fig. 11.44. A graph of this kind is called the curve shown in Fig. 11.44 C11.25 energy resonance curve . The kinetic energy E0 vanishes at an it thus identical to the curve excitation frequency of zero, in contrast to the maximum angular defor the total energy of the flection ˛0, i.e. a pendulum at rest stores no kinetic energy. resonator. initially , to be equal to C11.26 ition. If the excitation . To a good approximation, e . max . It plays an important role in E quantities are being plotted in the . of the resonance curve is the frequency H C11.27 a. The half-width The resonance curve of the kinetic energy of the pendulum by Resonance . The energy of the resonator is increased from its starting . ı stimulated energy transfer 11.42 = 90 e D ' D of Fig. the eigenfrequency of thecitation resonator (resonance then condition). leads the Theof deflection ex- of the resonator with a phase shift set at zero. Then we adjust the excitation frequency Now we come to90° the to 270° new without effect: interrupting theexcitation. We operation of increase As the a the pendulum result, or the phase the motionforth shift of path the from resonator is along continually its retarded, back-and- the and excitation system, its until stored the energy pendulumis comes is to released called rest. to This process optics and in electrodynamics H when the torsion wheelis is turned passing off, through this its energyC rest will pos be dissipated within about 8 s, as seen in curve one should keep in mind justcorresponding which resonance curves. 11.11 Energy Transfer Stimulated In the previous section, wecitation demonstrated was forced provided oscillations. by The a ex- electric lever which motor was equipped moved with periodically an byical eccentric. an phase – shifter Now, between we the put motor a and mechan- the eccentric Figure 11.44 In considering the various, diverse applications of forced oscillations, value of zero to its maximum value range at whose limits the amplitudeas of its the kinetic maximum energy is at only half the as (resonant) great frequency Optik und refers here to , Oxford Uni- , Springer Ver- OHL Vibrations 11 the phase shifter described in earlier editions (see image). mon Hooker and Colin Webb, Laser Physics versity Press (2010), Chap. 2. e.g. R. W. Pohl, Atomphysik lag, Berlin, 13th ed. (1976), Chap. 14, Sect. 15; or Si- the LASER (Light Am- plification by Stimulated Emission of Radiation); see the other. C11.27. For example in maximum deflection of the pendulum on the one sideits to maximum deflection on simply produced by shut- ting off the motion of the eccentric for half an oscillation period, e.g. from the The phase shift of 180°quired re- here could also be C11.26. P 278 Part II 11.12 Resonance for the Detection of Pure Sinusoidal Oscillations. Spectral Apparatus 279 11.12 The Importance of Resonance for the Detection of Pure Sinusoidal Oscillations. Spectral Apparatus

According to the considerations in Sect. 11.10, the forced oscillations of an oscillator or resonator can attain very large amplitudes even when the periodic forces acting on them are quite small. This requires that the oscillator be only weakly damped, and its eigenfre-

quency must be as close as possible to the frequency of the excitation Part II force. There are many demonstration experiments which show how amazingly large amplitudes can be produced in this way (see also Video 11.7). We will give another example here, using the forced Video 11.7: bending oscillations of a leaf spring (Fig. 11.45). We have already “Forced oscillations with employed its forced oscillations in Sect. 1.8,Fig.1.8 to explain stro- a pocket watch” boscopic time measurements. The excitation was produced by an http://tiny.cc/zegvjy asymmetric axle passing perpendicularly through the holder of the This experiment was included spring. by POHL up to the 11th edition (1947). The video shows A series of such leaf springs or reeds mounted on a common base can how a pocket watch pendu- function as a useful measuring instrument, a vibrating-reed frequency lum (with an eigenfrequency meter (Figs. 11.46 and 11.47). The oscillations being investigated are of 1.95 Hz) is excited to either coupled mechanically to the base (for example by attaching it forced oscillations by its own to the frame of a vibrating machine) or more conveniently by using escapement, which oscillates an electromagnet. Frequency meters of this kind are typical of spec- at a frequency of 2.00 Hz tral apparatus. They decompose an arbitrarily complex oscillation when the watch is at rest. without regard to its phases into a spectrum of pure sine-wave com- The swinging of the watch ponents. From this spectrum, one can read off the frequencies of the reduces the frequency of the individual sinusoidal components and their amplitudes. This can be escapement to 1.95 Hz. Thus, most simply demonstrated using an electrical system. We offer two the swinging watch produces examples: a periodic excitation at its 1. We refer to Fig. 11.13 and pass a rectangular or chopped direct own resonance frequency, current through the electromagnet of the vibrating-reed frequency and this generates a large meter. Figure 11.48a shows the arrangement, which needs no fur- amplitude. ther explanation. The rotating commutator revolves once in the time T D .1=20/ s, and the current is switched on for a period of (1/40) s

Figure 11.45 A leaf spring or reed which is being excited to forced vibrations (compare Fig. 1.13), time exposure 40 Hz each D 4 . Immedi- , the difference 11.12 4 70 Hz and and in Fig. 7 D ndividual component can r 7 S are small integers). b and a 60 Hz. The next-higher frequency ( are also present. They can be computed 2 and replace the direct-current source of D b ˙ 3 cillations. Every i 1 11.12 Production of beats with two sine-wave alternating- a 30 Hz is also indicated. b D c D 4 Production of a rectangular direct-current waveform using ). Section of the scale of a Sketch of a vibrating-reed 20 Hz and 7 b). The spectral apparatus indicates these two frequencies. a (storage batteries) by two alternating-current generators combination frequencies D 11.47 1 100 Hz can just be discerned in the spectrum. 11.48 11.48 Difference oscillations or differencesome frequencies sort of occur “nonlinear element” in iscillations; general involved that in is, when the when transmission of for the exampleor os- a (as force and with the the resultingother. deformation, rectifier) A voltage difference and frequencycalled current occurs only are rarely not alone. proportionalfrom Usually, to the other scheme so- D 5 (Fig. frequency excite the leaf springs of the vibrating-reed frequency meter to forced a rotating commutator; ately, in addition to the two frequencies one side, so that it resembles the curve – Then we add a rectifier to the circuit and cut off the beat curve on a technical vibrating-reed frequency meter which shows the frequencyhousehold of alternating current (50 Hz) frequency meter. The leaf springssuitably graded have resonance frequencieseach and has a white, square(see knob Fig. at its tip Figure 11.46 2. We now refer to Fig. Fig. connected in series, with the frequencies current signals, and cutting off the beat waveform with a rectifier Experiments of this kind aresinusoidal quite oscillation important; process they behaves showindividual like that component a a os physical non- mixture of its Figure 11.48 during each revolution. The spectralcies apparatus indicates the frequen- Figure 11.47 Vibrations 11 280 Part II 11.13 The Importance of Forced Oscillations for Distortion-Free Recording 281 vibrations, independently of all the others. This physical independence of the individual component oscillations plays a large role in all applications of forced oscillations. The next section gives an important example.

11.13 The Importance of Forced Oscillations for Distortion-Free Recording of Non-Sinusoidal Part II Oscillations

In the majority of cases, our sensory organs suffice for simply detecting mechanical vibrations. Our bodies can for example register vibrations of the ground ( ca. 10 Hz) already at a horizontal amplitude of only 3 m. Our fingertips can feel vibration amplitudes of around 0.5 m(at D 50 Hz) with a gentle touch. We will give numerical details of the enormous sensitivity of our ears later in Sects. 12.28–12.30. In general, however, one is not satisfied simply with detecting vibrations; rather, a true-to-form or distortion-free recording of their time dependence is often desirable. In every recording method, the vibrations in question produce motions in some sort of probe instrument (levers, mirrors, microphones, sensory organs). These motions are recorded continuously, for example (after electronic amplification) by an oscilloscope. The probe instrument is excited to forced oscillations; it is a resonator with the eigenfrequency e: For a flawless recording which correctly reproduces the time dependence of the process, two error sources must be avoided: First, the recording system must not give preference to certain frequency components of the oscillation due to their frequencies . Second, the phases of the individual components must not be shifted relative to one another. – Both requirements are fulfilled as long as is not greater than ca. 0:7 e and the damping ratio K of the recording sys- tem4 is 50. Justification: From curve D in Fig. 11.42b, for the same excitation amplitudes, the amplitudes of the forced oscillations up to 0:7 e are independent of the frequency .–AccordingtocurveD in part c

4 This statement does not hold for seismographs is 50. In their case, the eigenfrequency of the instrument (the “resonator”) must be low compared to the frequencies of the seismic waves. This is for two reasons: First, the ground serves as an accelerated frame of reference; it generates inertial forces; these cause the ground to serve as the excitation source for forced oscillations. Second, the scale of the instrument (the resonator) is also part of the motions of the excitation source. As a result, at the same excitation amplitudes, the deflections shown by the instrument (the resonator) are small at low excitation frequencies and large at higher frequencies, since the resonator can follow the excitation source (the ground) less and less as the frequency increases, in contrast to curve D in Fig. 11.42b. . by OHL ,but labile const. 11.49 ,P D t up to controlled Video 1 t 2 electric current D T = t const and thus phase shifts relative to one 2 D t D tion curves is rather challenging, ). without ' . This water stream is 12.28 . Then, 2 water to of the oscillations plays an important role: The the input of energy from an external source. This ). Tiny motions of the nozzle cause changes in the , the phase angle A stream of water as an acoustic amplifier. (In 10.6 control , i.e. they are recorded t proportional amplification 11.42 is time e 7 : This means that the components atsame all frequencies are delayed by the acoustic/musical applications, for example for thesical production recordings of and mu- their playback,less the stringent requirements (compare are Sect. fortunately another. The recording of undistorted oscilla but it is a task which is essential for many purposes in science. For throb from the membrane. Similarly,fork the held vibrations against of the a water smallclearly nozzle, tuning or audible the all ticking over of a a large watch, auditorium. become A jet ofa water strongly from damped, a stretchedforms glass membrane (e.g. a nozzle a smooth flows tamborine). filament; nearlyobject such The (Fig. horizontally jet a onto filamentary streamcross-section downstream, is or a interruptions very ing of impulse the of flow. the The streamlatter chang- of to water rapidly hitting decayingtance the away. vibrations membrane Thus, excites a which light the tap can on be the nozzle heard will be some amplified to dis- a loud instead from a currentthe of vibrations which aresuccessfully to with be the amplified. relatively This primitive can setup be shown demonstrated in Fig. demonstrates the amplification, althoughhe silently, presses using against the a water tuning nozzle.) fork which Figure 11.49 ertheless, it is usefulwith to a comprehend clear-cut, theamplification, purely essentials the mechanical of energy example. amplification is not For taken this from mechanical an 11.14 The Amplification of Oscillations For recording oscillations and numerous othernology, tasks of vibration tech- essential characteristic of amplification isthemselves always that the oscillations is accomplished today almost exclusively using electronics. Nev- of Fig. 0 Vibrations 11 282 Part II 11.15 Two Coupled Oscillators and Their Forced Oscillations 283

Figure 11.50 A water ampliﬁer regulated by auto control of undamped vibrations (feedback). (This experiment is also demonstrated by the author in Video 1 Video 1: (again silently).) “R.W. POHL Lecturing” http://tiny.cc/fpqujy Part II

Electronic amplifiers are used technologically on a large scale to generate undamped electrical oscillations. Our mechanical amplifier can accomplish the same task by generating undamped mechanical vibrations. One need only set up a feedback loop between the resonator – here the membrane – and the glass nozzle. The vibrations of the membrane must be mechanically transmitted back to the nozzle; then the membrane controls the decay of the water stream with the rhythm of its own eigenfrequency. It suffices to lay a metal rod on the membrane and the nozzle, as in Fig. 11.50. Immediately, loud-sounding undamped vibrations are heard. Their frequency can be varied as desired, by adjusting the tension of the membrane and thereby changing its eigenfrequency.

11.15 Two Coupled Oscillators and Their Forced Oscillations

We now wish to investigate some of the properties of coupled pendulums, for simplicity considering only two pendulums, each of which has the same natural frequency. We distinguish three different types of coupling mechanismsC11.28: C11.28. A mathematical treatment of the examples 1. Acceleration coupling (Fig. 11.51a). One pendulum is hanging discussed here is given in from the other. It is located in the accelerated frame of reference of Müller-Pouillet’s “Lehrbuch the ﬁrst pendulum and is thus subject to inertial forces. der Physik”, 11th edition, 2. Force coupling (Fig. 11.51b). The two pendulums are connected Vieweg und Sohn, Braun- to each other by an elastic element (spring). schweig (1929), Vol. 1, p. 55.

Video 11.8: “Coupled pendulums: Force coupling” http://tiny.cc/pfgvjy The video shows the normal mode vibrations of two coupled gravity pendulums Figure 11.51 a Acceleration coupling; b Force coupling; c Frictional cou- (Fig. 11.51b) as well as their pling. In the case of frictional coupling, no beats are observed. The first beats; and also the beating of pendulum excites the second so that it begins to swing, and from then on, a spring-torsional pendulum both swing together with the same amplitudes and phases (Video 11.8) (see and a double gravity pendu- Exercise 11.7). lum (Fig. 11.53). D of 11.5 Sec- . one actio C11.29 2 ). corresponds to 2 and 1 , rubs against a part of the . Exercise 11.6 a b . Pendulum No. 1, which was applies to synchronous swing- , with its smaller mass, exhibits 1 to the initially immobile pendulum c). A part of one pendulum, e.g. the ). 11.51 11.24 ,and ). Something surprising happens – pendulum No. 1 which can rotate about S 11.18 , Two coupled gravity , two bifilar gravity pendulums with the same vibration 11.52 (Video 11.9) of the two superposed eigenfrequencies 11.17 forced vibrations at resonance 11.53 . We are dealing here with forced vibrations with a strong . It accelerates No. 2 along its whole swing with the correct ı beats reactio negative feedback from the resonator to the excitation source. We offer two more examples of coupled oscillations: In Fig. period are hung from one another.mass The than upper ball the has lower anoticeable one. much push, greater the If lower pendulum thebeats upper of one a is large givencan amplitude. a barely be small, The seen. hardly beats of the larger-mass pendulum swinging in opposite directions (analogously toin the Figs. vibrations shown the two pendulums (No.lease 1) it away (Fig. from its rest position and then re- initially released at itssource maximum for deflection, pendulum No. serves 2 as asof the resonator, and 90 excitation leads it by a phase angle ond, as sign. It is itself braked by the counter force according to ing of the two pendulums, and the higher frequency pendulums Figure 11.52 This process of energy transferas can be described in two ways: First, Now we come to a new observation: Initially, we move only gradually passes all of itsNo. energy 2 and excites itcomes to a to large rest vibration during amplitude. thisthe Pendulum process. 1 roles of itself Then the the two cycle pendulums reversed begins ( anew, with 3. Frictional coupling (Fig. connecting rod other pendulum, e.g. the rotatable collar In the following, we consideration only coupling the and first force two coupling. cases,the After that the two is pendulums eigenfrequencies are acceler- coupled, whichare we observed. The have lower already frequency seen in Sect. )as ). Two 11.52 Exercise 11.6 Vibrations 11 served ( The two individual vibrations of two coupled gravity pendulums (Fig. well as their beats are ob- “Acceleration coupling and chaotic oscillations” http://tiny.cc/8cgvjy Video 11.9: an Introduction for Applied Scientists and Engineers”, Francis C. Moon, John Wiley, New York (1987)). exhibit a new form oftions vibra- at large amplitudes, so-called chaotic vibrations (see “Chaotic Vibrations, similar gravity pendulums coupled in a different manner can be described in termstheir of normal modes (eigenfrequencies). C11.29. The effect seen in this example holds quite generally for coupled pendulums: All the vibrations 284 Part II 11.15 Two Coupled Oscillators and Their Forced Oscillations 285

Figure 11.53 Two coupled gravity pendulums, hung in a biﬁlar arrangement, with inertial elements of very different mass. They can swing in a plane perpendicular to the page. (Video 11.8) Video 11.8: “Coupled pendulums, coupled by force” http://tiny.cc/pfgvjy Part II

Figure 11.54 A tuning fork with an attached strongly damped leaf spring (MAX WIEN’s demonstration, Ann. d. Phys. 61, 151 (1897)) (Video 11.10) Video 11.10: “Coupled oscillations with damping” http://tiny.cc/5egvjy

C11.30. Another example of such an application are the In the following examples, one of the two pendulum vibrations can be “vibration dampers” in tall damped. A strongly-damped leaf spring (reed) is attached as a little buildings. These are large ‘rider’ to a tuning fork. The arrangement is shown in Fig. 11.54. pendulums mounted within The damping of the reed is produced in the usual way by its rubber the buildings, whose vibra- mounting. The reed and the tuning fork each have the same natural tion frequency is the same frequency. as the eigenfrequency of First, we hold the reed with a fingertip to keep it from vibrating. Then the structure, thus forming the tuning fork’s tone gradually dies away after it is struck once, over a system of coupled pen- a time of around a minute. We can make its vibration readily visible dulums. When the building from some distance away by using the mirror M. We then repeat the is excited to vibration by experiment, leaving the reed free to vibrate. After being struck, the the wind, i.e. it begins to tuning fork stops vibrating very quickly, after only a few seconds. sway, energy can be trans- Its energy of vibration, which is transferred to the coupled reed, has ferred to the pendulum and been dissipated as heat in the reed’s rubber holder. Instead of the damped there by friction. See long beats heard with an undamped oscillator, now we can register Ch. Ucke and H.-J. Schlicht- only a few. When the spring is properly dimensioned, the energy will ing, “Schwingende Puppen be completely absorbed by the end of the first beat. und Wolkenkratzer”, Physik in Unserer Zeit 3/2008(39), So much for the free oscillations of two coupled pendulums. In technical p. 139; or B. Breukelman applications, the forced oscillations of two coupled pendulums play a sig- and T. Haskett, “Good Vi- nificant role. We limit our considerations to a single example, the reduction brations”, Civil Engineering, of rolling motions of ships in heavy seasC11.30. ASCE 71/2001(12), p. 55. .By ı (left), we see of this type, ir frequency on axis). Even a su- 11.56 A . When this valve is – H A ), and thereby excites the as a steamship, and the leaf ˛ antirolling tank s (simple oscillators) exhibit 11.54 ttle valve pendence of the hange of kinetic and potential en- illate. The board, that is the model ple the “wobble oscillations” which shows an These phenomena are given preferen- 11.55 illations (rolls) after being tipped to 40 . The two arms of the U-tube are connected A after only two or three oscillations, and returns to its rest ı Model of an antirolling tank Oscillations opening the valve, weprovide can damping release of the its watertippedto40 oscillations. column and Now, at the the model same stops time rolling when position. We can think of thespring tuning as fork a in strongly Fig. more, damped we pendulum imagine the built pulsed into excitationaction the of of the ship’s the tuning hull. waves fork to onple. Further- be the the The ship; periodic strongly-damped then pendulum we iscolumn implemented can in technically already a as large recognize a U-tube. the water The princi- model illustrated in Fig. attached as a pendulumcan to a rotate board (roll) withabove the around profile through of an aclosed, air ship’s hull, the pipe which water with columnship, a performs cannot thro around osc 20 osc a wooden column which is standing withbaseplate. its A two small linear “feet” impulse on in afoot, the flat tipping direction the of column the to arrow the lifts left the (angle right tial treatment in every physicsA textbook, owing number to of their oscillation simplicity.into or – this vibration simple scheme, phenomena for exam howeverare do frequently not observed in fit everyday life. In Fig. wobble oscillations, a periodic exc ergy (the latter alternatingeither between two to metastable the positions, left tipped or to the right around the (Video 11.11) Figure 11.55 ideal limit ofnitely vanishing and losses, sinusoidally this atamplitude exchange a of can the frequency oscillations. continue which is indefi- independent of the 11.16 Damped and Undamped Wobble Spring pendulums and gravity pendulum a clear-cut, unified scheme for the occurrence ofThere their is oscillations: a periodic exchange of potential and kinetic energy. In the wobble oscillations, namely the de perficial observation allows us to grasp the defining characteristic of controls the frequency Vibrations H 11 Sinusoidal oscillations: ship’s hull. The throttle provides the damping. ing of the oscillating water column so that it canto be the adjusted roll frequency of the “Antirolling tank” http://tiny.cc/jfgvjy Varying the throttle open- Video 11.11: in every physics textbook, owing to their simplicity”. “These phenomena are given preferential treatment 286 Part II 11.16 Damped and Undamped Wobble Oscillations 287 Part II

Figure 11.56 Left: Demonstration of wobble oscillations using a rectangular wooden column on a steel plate. The column is about 30 cm high and stands on two long “feet”. Right: After an initial push, the oscillations of the column were recorded photographically using a light beam. (Video 11.12) – Forced Video 11.12: wobble oscillations can be readily excited by employing inertial forces: the “Wobble oscillations” baseplate is periodically moved back and forth in the direction of the double http://tiny.cc/qcgvjy arrow P by a motor with an eccentric. In the video, besides the wooden column that tips back and forth around its center- ˛ their amplitude. The smaller the angular amplitude 0, the higher the line A-A, a metal disk is also oscillation frequency becomes (compare Fig. 11.56, right). shown (covered with plastic to enhance the visual effect). Wobble oscillations can also be maintained with a constant amplitude, that The disk is held at an angle is without damping. A first example of external control, i.e. forced oscillations, is described in the caption of Fig. 11.56. A second possibility is on a flat glass plate and given briefly illustrated in Fig. 11.57: A and B are two index fingers which are a spin around its axis. It rolls placed on the edge of a table T at a certain distance. A metal bar S is rest- around its circumference on ingonthem.ThenA and B are moved up and down periodically, but with the glass, at a continually in- opposite phases as “exciters”. creasing frequency, for about Externally forced controlled, i.e. wobble oscillations exhibit a notable pe- a minute, until finally coming culiarity: Each excitation frequency corresponds to a certain amplitude, both of the resonator and also of the exciter. Therefore, there is no dan- to rest on the plate. ger that the column in Fig. 11.56 will tip over, as long as the excitation frequency does not fall below a certain lower limiting value5. Wobble oscillations with auto feedback can also be illustrated using the example in Fig. 11.57.Now,A and B denote two sheets of lead attached

5 In the St. Gumbertus church in Ansbach, Germany, the church towers are not attached rigidly to the nave. When their bells are rung, they are excited to wobble oscillations with a predetermined amplitude of 20 cm. This presented no danger to the rest of the structure (E. MOLLWO). Nevertheless, this example of forced wobble oscillations was unfortunately eliminated by applying a rather simple structural change: The vertical plane in which the bell swings was rotated by 90ı.An- other example of wobble oscillations (or “rattling”) is the “Rattleback” or “Celtic wobble stone” (see: https://en.m.wikipedia.org/wiki/Rattleback). This is a small, slightly asymmetric, boat-shaped piece of glass, plastic, stone or wood. When given a spin around its vertical center axis, it stops spinning after some time, and its kinetic energy is transferred to a different mode: it begins to wobble or “rattle” around a body axis. After some time, the energy is transferred back to a rotational mode and it again starts to spin, but in the reverse direction. This may be repeated one or two times. For a video demonstration, see e.g. https://m.youtube. com/watch?v=69Xm762qE8o. ’s ’s rocker) . A steady b REVELYAN metal bar, made and REVELYAN a 1 Hz, are the spark : hot 0 tions occur whenever is a S the group of relaxation oscilla- eached. Then the lamp flashes on ever, cannot occur at all without from the continuously-available ndefinitely after an initial pulse illations of constant amplitude, either nguished from ordinary oscillations . itial push, the bar touches first the left and then C11.31 ons”. Relaxation oscilla shows an analogous setup in the form of an electrical circuit. shows a simple example. A liquid container with the Producing wobble osc Oscillations 11.59 11.58 rocker” or thermophone) discharges which can be observedmachine) with equipped an with influence machine a (Wimshurst Leyden jar. Figure A condenser is slowlyresistor with charged a from high alamp value, in current until parallel the source with ignition throughand the voltage discharges of a condenser the a is control condenser glow-discharge r trical quickly. relaxation oscillations at Another, a similar low frequency, example of elec- to a metal block (ate.g. a spacing of of copper. ca. 6 After mm),the an and right in lead sheet, each timewhich causing accelerates it the to rod’s bulge motion more with deeply the downwards, correct phase (“T with external excitation or by using thermal feedback (T or vibrations (periodic inter-conversion ofergy) by potential two and essential kinetic characteristics: en- stream of water fillsthe the left. container At a and certain shiftsthe point, left, the its ‘seesaw’ emptying center becomes the unstable contents of and ofto gravity tips the the to to container, right. which The then cycle tips begins back anew. Relaxation oscillations are disti excitation, at least inimportant the group ideal of oscillations, limit how ofa negligible continuous input losses. of – energy:tions This Another or “toggle is oscillati profile of a non-equilateralrock triangle back is and mounted as forth a between seesaw. two resting It points can 11.17 Relaxation (or Toggle) Wobble oscillations can continue i Figure 11.57 there is a delay timetial (relaxation energy and time) its between conversion thethe into end storage kinetic of of energy this poten- delay inreleased time, an energy the oscillator. is stored At not energy used isa released; again however, to new the refill quantity the of storageexternal element; source. energy rather, is taken Figure , , for The Phys. Teacher Vibrations 11 a video demonstration. Rocker?”, Vol. 12 (1974), p. 382; and https://www.youtube.com/ watch?v=U23iwbVX-Dk the same volume in frontpage of 429. See also I.M.man, Free- “What is Trevelyans C11.31. A. Trevelyan, Trans. Roy. Soc. Edinburgh Vol. 12 (1834), p. 137.illustration An can be seen in 288 Part II Exercises 289

Figure 11.58 Mechanical relaxation oscillations. a and b are resting points for the “seesaw” container.

Figure 11.59 Electrical relaxation oscillations Part II

1. Their amplitudes cannot be changed without modifying the construction of the oscillator (e.g. the container in Fig. 11.58); only their frequencies can be varied (in Fig. 11.58 by opening or closing the water faucet, in Fig. 11.59 by adjusting the variable control resistor). 2. Relaxation oscillations can be readily controlled by an auxiliary oscillation of small amplitude. As a result, they can easily be synchronized. In everyday life, relaxation or toggle oscillations are extremely important. We can observe them for example in the creaking of a door or the screeching of chalk on a blackboard, and in the operation of a pneumatic hammer. In particular, relaxation oscillations play an especially important role in the lives of many organisms. The excitation of nerve cells and muscular activity are examples. The functioning of the human heart can be elucidated down to the ﬁnest details in terms of three coupled relaxation oscillations. Unfortunately, the mathematical treatment of relaxation or toggle oscillations is complex and difﬁcult; it is therefore not dealt with as extensively as it deserves in the literature of physics. However, the electrical demonstrations mentioned above have contributed notably to the understanding of relaxation oscillations.

Exercises

11.1 Sound waves with the frequencies 1 D 225 Hz and 2 D 336 Hz, each containing their ﬁrst two overtones at 2 and 3,are sounded together. Show that two of these overtones lead to beats of frequency B D 3 Hz. (Sects. 11.3 and 11.7) ). D ) l and of the K 11.7 11.44 2 in the rod. and l 1 c and spring con- (see also Com- m ) 11.10 b (note that here again, 340 m/s). (Sect. , which represents a sit- e free oscillations of the D D 11.15 , the damping ratio of the torsional pendulum, c 11.51

e of one of the pendulums when 0 0 D avity pendulums (“acceleration The online version of this chapter (doi: a). Find from this result the value of the shortest capillary tube whose air d ; cf. C11.21.) of the exponential decay 11.42 ) ) shows the longitudinal vibrations of two cou- 11.15 11.7 of the resonance curve of the energy (Fig. ’s dust figures are produced by a whistle; their spacing H 11.24 ’s pendulum)” begins with th ) ) OHL UNDT . a) Calculate the frequency K Figure Video 11.6, “Free and forced oscillations of a torsional pen- The longitudinal fundamental vibration of a rod of length In Video 11.9, “Acceleration coupling and chaotic vibra- Determine the length 11.10 11.8 D (i.e. the time constant 1 4 m is 800 Hz. Evaluate the velocity of sound : stants pled ball-and-spring pendulums with massive balls vibrations of the coupled pendulums.the b) two Replace balls the by spring a between weaker spring, i.e. the other is being held fixed, and also the frequencies (Sect. the half-width ment C11.4). (Sect. 11.7 and derive the logarithmic decrement torsional pendulum (Fig. ı uation very similar to that shown in Fig. one should observe the vibrations ofother one is of being the pendulums heldlongitudinal while vibrations the fixed). in this Determine case. the (Sect. three frequencies of the 11.5 dulum (P (Sect. is found to be 1.2of cm. the sound. Determine (Sect. the wavelength and the frequency 11.4 2 11.6 tions”: Forcoupling”), the use a twosymmetric stopwatch and coupled to the gr the determine antisymmetric beat the normal frequency. frequencies modes Thenthree of investigate of frequencies, the the vibration which relationship between is and these hinted at in Fig. column can be excited to normal-modea vibrations frequency by of a 520 Hz, tuning if forkthe at tube a) is both closed ends (sound are velocity open; in and air: b) if one end of Electronic supplementary material 10.1007/978-3-319-40046-4_11) contains supplementary material, whichable is to avail- authorized users. 11.2 11.3 of the amplitude, and the frequency Vibrations 11 290 Part II sasao mg.I smvn ihtevelocity the with moving is It image. shadow a as eouin ftehlx on olwdb h y,eg h crossing the e.g. eye, the by followed point a helix, the of revolutions (Fig. window the behind shape has which helical wire a a into put We between bent motion: been relation circular the and demonstrates oscillation it sinusoidal a since better, pulling is velocity the by arrangement at experimentally ent wire of wave made travelling curve sine a a such moving represent is can We crest wave or point crossing the a along such which with velocity ti oae rudisln xs(ytrigtecak,wt h fre- the with crank), the turning (by axis long quency its around rotated is it points crossing the e.g. wave, z the along correspond- points two between ing distance The troughs. wave and crests wave Figure n Radiation and Waves Travelling .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 12.2 Figure 12.1 Figure a represents thus image shadow This tion: Fig. In Waves Travelling 12.1 rudisln axis long its around rotated be can and helix a into bent been has which wave travelling sinusoidal a of (‘snapshot’) image neous xs rtoscesv aecet,i aldthe called is crests, wave successive two or axis, 12.1 12.1 z D xsi aldthe called is axis Awire Instanta- eaeloigtruhawno oseasn curve sine a see to window a through looking are we , hw saso’o h ae edsigihbetween distinguish We wave. the of ‘snapshot’ a shows N = t ,thatis olsItouto oPhysics to Introduction Pohl’s N hs eoiyc velocity phase eouin ihnatime a within revolutions c rvligsnsia wave sinusoidal travelling attewno.Adiffer- A window. the past ftewave. the of O 10.1007/978-3-319-40046-4_12 DOI , wavelength c ˛ nthe in and t 12.2 .During ˇ .Then ). z nthe on direc- .The N . 12 291

Part II . axis z (12.2) ), they (12.1) . All the after each vibrations vibrations z .Atapar- 12.2 is the cir- T at the phase .Its depend only on 2 ˛ N phase velocity t D ! position D ! , we consider the vi- z s Á 2 z . Its neighbor to the right is just touching the wave T ˛ D = t 2 t ! ct cT 2 , it has reached a deflection of t t direction. Nevertheless, no point and die phase sin : ! z further to the right. Along its path 2 D x 0 x z , and thus the name D ' D . Therefore, instead of Eq. ( c D sin t ˛ (small wooden beads on a thread). Each T 0 x x phase or 2 t D 12.3 x D = , moves along a distance nusoidal oscillation with the period ' N , the deflection D 12.1 t 0. Then after a time axis. This can best be seen by imagining that the wire = z s , it has a phase of D t Beads arranged along a he- t D c . in Fig. . The deflection is repeated at the same vibrations T is the maximum possible deflection (amplitude), ˛ 0 time x ( cular frequency). (Video 12.1) are delayed by a phase angle of brations of a point atupwards, a it distance crosses the abscissa later than the point This is a fundamentalDuring relationship the for rotation of every the waveweaving helical along phenomenon. wire, like every a observer – sees snake the in wave the the period In order to arrive at a description of the are described by: relative to those of the point Figure 12.3 helical form, as in Fig. point undergoes a si ticular time at the time on the wire ispoints on actually the moving wire in aredicular moving the to circularly the in direction planes of whichhas are travel been perpen- dissolved into a series of individual points which follow the For the point lix has the same phaseright along somewhat the later: wave is its what isFor in a quantitative fact description, moving we firstof to need a to the single examine point. the Assume that the point velocity Travelling Waves and Radiation 12 ing the crank (clockwise or counter-clockwise). This can be clearly seen in the video. wound (right- or left-handed), and on the sense inthe which experimenter is turn- The direction in which a wave is travelling as given in the text depends on the sense in which the helix is “Model of a travelling wave” http://tiny.cc/2fgvjy Video 12.1: 292 Part II 12.2 The DOPPLER Effect 293

Figure 12.4 A ripple tank for observing surface wave fields. At the end of a lever is an immersion contactor C which can be moved up and down by means of a vibrator (after THOMAS YOUNG, see Comment C12.3). For circular waves, the contactor is a short, pointed rod; for linear waves, it is a horizontal bar (Video 12.2) Video 12.2: “Experiments with water waves” Part II or, with D c= and 2 D !, http://tiny.cc/tfgvjy Á z x D x sin ! t : (12.3) 0 c This is the specification of a travelling wave in terms of an equation. For the wave, the deflection x and the phase !.t z=c/ depend not only on the time t, but also on the position z. The deflection repeats itself at the same position after each period T and at the same time at positions separated by z D . Any state which travels at a finite velocity can form waves. To observe waves, it is expedient to begin with states which travel at small velocities c. Among these, small deformations of liquid surfaces occupy a prominent position; they form surface wave fields. These can be observed in a ripple tank. In Fig. 12.4, a ripple tank is sketched in cross-sectionC12.1. For pro- C12.1. Many of the ex- ducing the waves, it contains an immersion contactor which vibrates periments described in the sinusoidally up and down at the surface of the water and serves as following were filmed using a wave source. Its frequency is chosen to lie between 10 and 20 Hz; this ripple tank (Video 12.2). then we obtain wavelengths between 2.5 and 1.2 cm. Pointed con- In order to aid in finding the tactors give circular waves, which travel out as concentric circles individual experiments, the (Fig. 12.5). Linear contactors (bars) give wave crests and troughs times (in minutes) at which in the form of straight lines (linear waves), as seen in Fig. 12.11. they appear in the video are noted in the margin of the The edges of the tank must be flat with a gentle slope, so that the waves page (the complete video are damped and no disturbing reflections occur there. – Illumination from runs for somewhat more than below can be used to project an image of the waves in the tank onto the 6 minutes). ceiling or, using a mirror to redirect the image, onto a wall screen. Strobo- scopic illumination allows a particular phase, e.g. a certain wave crest, to be easily followed in the projected image.

12.2 The DOPPLER Effect

In Fig. 12.5, we can imagine that somewhere, there is a receiver E which responds to the oncoming waves and can count them; it could be for example the eyes of an observer. If the source and the receiver of the waves are at rest relative to the wave-carrying medium, waves to (12.4) , and the phase u mechanical Á ) separate from that of C 12.6 c u ˙ 1 effect is observed: During the prop- D , the receiver observes a wave frequency c , u

. Then the following observations can be c 12.5 1 OPPLER D . 0 individual waves (i.e. one crest and one trough) are t , they are compressed; therefore, the receiver measures / C12.2 ut (Video 12.2) D 0 N denotes a “rect , A two-dimensional Similar to Fig. t E emitted from a moving sourcetheir path which ( is approaching the receiver. Along In a time Receiver at rest and aInstead moving of source the source frequency of: but with the wave sourceto moving the right. ceiver” which is at rest.10 cm A long scale is shown. a moving receiver agation of the waves,increases a the decrease frequency observed inin by the their the spacing spacing receiver, of reduces whilesion, the the an one frequency. wave increase must crests – bekeep For careful the a when case quantitative dealing of discus- with a moving source (Fig. Figure 12.5 Figure 12.6 Let the relative velocity of source and receiver be here the surface offrequency as the was water, emitted then atare the the moving, source. then receiver the measures If D the the source same or the receiver wave ﬁeld on the watersnapshot surface, (exposure time 0.002 s). The wave crests and troughsconcentric are circles made: velocity of the waves be , rel- 12.20 , .) 12.19 12.23 ,and motion is of importance Travelling Waves and Radiation 12 12.22 to demonstrate refraction of waves in Figs. velocity, which is changed when the depth of thedecreases. water (This effect is used Circular waves (3:00). Note that the wavelength depends not only on the excitationquency, fre- but also on the phase waves” http://tiny.cc/tfgvjy http://tiny.cc/tfgvjy Video 12.2: “Experiments with water ative (see Vol. 2, Sect. 23.4). C12.2. In contrast to light waves, for which only the 294 Part II 12.3 Interference 295

a wavelength of 0 D .c u/t=N0 D .c u/=. Inserting 0 D c=0 yields Eq. (12.4).

Moving receiver and a source at rest

Á u 00 D 1 ˙ (12.5) c

If t individual waves would arrive within the time t at a receiver which is at rest, then for a receiver which is moving towards the source by a dis- ut t ut= tance during the time , an additional individual waves arrive. The Part II moving receiver will thus detect N00 D .t C ut=) individual waves in the time t. It observes a frequency of 00 D N00=t, giving with D c= Eq. (12.5).

In Eqns. (12.4)and(12.5), the upper sign applies when the source and the receiver are approaching each other.

C12.3. THOMAS YOUNG, physician (1773–1829). An 12.3 Interference appraisal of his scientific achievements can be found in R.W. Pohl, Physikalische Using a ripple tank, THOMAS YOUNGC12.3 discovered and named Blätter 5, 208 (1961). a fundamental effect in 1802; it is basic to the understanding of all wave phenomena. This is interference, which occurs when two waves are superposed. – In order to demonstrate this effect, we use two Video 12.2: points which are rigidly connected together as excitation sources in “Experiments with water the ripple tank. The result is shown as a snapshot in Fig. 12.7:The waves” two waves add to each other, and the resulting wave field is chopped http://tiny.cc/tfgvjy up by interference maxima and minima. YOUNG’s experiment (3:30) is easily reproduced, in Along the symmetry direction 00, there are travelling waves. On particular with long wave- both sides of this symmetry axis 00, one can observe alternating nar- lengths. One can clearly row, dark regions with no waves and broad regions containing waves. discern two directions of neg- In the wave-free regions – called for short interference fringes –for ative interference. When the every point along their center lines, the path difference, i.e. the dif- frequency is increased (short- ference of the distances to the two wave sources, is constant and is ening the wavelength), the an odd multiple of =2. As a result, the two waves cancel each other. number of such directions increases; however, the images also become more complex. Figure 12.7 Snapshot of The reason for this is that the a two-dimensional wave field showing interference be- waves excited by an object tween two waves (exposure which oscillates harmoni- time 0.002 s). This image cally at the water surface are and all those on through by no means simple sinu- Fig. 12.19 are photographic soidal waves with only one positives (Video 12.2). wavelength, as one can see in many of the experiments in this video. Their interference patterns are correspondingly complex. is m (12.6) (12.7) times within ), where n ). The interfer- ˛ D : t ), we find = 2 at a sufficiently large dis- n , as shown by the numbers ; /= index D . Then the interference fringes 1 12.8 B D D or m C m D C ). More details are given in Vol. 2, 2 . ˛ D sin and . – The curves of equal path difference, ˛ , the two exciter points are no longer rigidly sin 12.7 vice versa ordinal number 12.7 ) can be read off Fig. 12.7 00 . In a periodic sequence, they arrive at the same posi- The derivation of Different Source Frequencies )and( , or at the “beat frequency” t 12.6 wander instead of trough and Fig. 12.35. and for the minima The amplitudes of neighboring wave trains have opposite signs (crest Eqns. ( Figure 12.8 ence fringes exhibit fixed positions only in stopped-motion images. 12.4 Interference with Two Slightly Suppose that in Fig. connected to each other, butmotors, instead at are the frequencies driven independently by two tance from the sources. For the maxima (at angle In contrast, thefied. waves Along the between center lines the ofcoming the interference from wave-containing regions, the the fringes waves two sources are addpath with ampli- differences the are same integral phase, multiples and of thus their will regions, are hyperbolas. The anglesymmetry between axis their asymptotes and the a whole number (an in the margin ofi.e. Fig. the center lines of both the wave-free and the wave-containing tion previously occupied by the neighboring fringe ( a time Travelling Waves and Radiation 12 296 Part II 12.5 Standing Waves 297 Part II

Figure 12.9 Two two-dimensional wave fields showing interference, in which the second wave was produced by reflection of the first from a wall; left: an instantaneous image, right: a time exposure, with a larger spacing of the two wave sources. The left-hand image corresponds to the left half of Fig. 12.7. The right margin of the picture corresponds to the surface of the reflecting wall, and to the line 0–0 in Fig. 12.7.Thearrows in the right-hand image show the direction of propagation of the travelling waves in the interference field. Furthermore, the wall at the right and the wave nearest to it, with a path difference of 0, are not shown.

In many cases, it is impossible to arrive at strict synchrony between two sources and thus to avoid frequency differences between them. Then, one can resort to a trick: The second source is replaced by a mirror image of the first, guaranteeing synchrony. That is, we allow a wave to reflect from a smooth wall and observe the superposition of the reflected and the incident waves. Figure 12.9 (on the left) shows an example.

12.5 Standing Waves

In Fig. 12.9, at the left we see the interference of two waves of the same frequency as an instantaneous image. We now show a similar interference pattern as a time exposure (Fig. 12.9, right). Here, the second wave source was also produced as a mirror image of the ﬁrst. In this time exposure, we see nothing more of the progressive sequence of wave crests and troughs; we see only the hyperbolas of equal path differences. Between the dark interference fringes, the waves travel in the directions shown by the short arrows; above the line ZZ upwards, and below the line ZZ downwards. Along the line ZZ (and practically also in its immediate neighborhood), we observe standing waves with nodes and maxima: At the maxima, crests and troughs follow each other in a periodic sequence. At the nodes, the medium (water surface) is at rest1 Along the connecting line Z-Z

1 At the nodes, a stick oriented perpendicular to the water surface would change only its angle to the vertical over time; its lower end would not move up and down. ) 12.3 (12.9) (12.8) ( ). In (12.12) (12.11) (12.10) 11.20 , and obtain ˇ . D C12.4 2. Standing waves Á ˇ z = c direction, we have ˇ/ : 2 direction. : z ! : : . t z ˛ z c sin Á ! Á C ! z c z c t C cos ! sin ˛ C cos ˇ z t upwards t ). t sin 2 C ! . ! and 0 ! ˛ x at the frequency sin 12.10 ˛ 1 mm on water surfaces sin sin cos 2 0 : D x 0 0 0 0 l D x 2 x x 2sin x 2 2) are formed by the Á D D < D D C z c r = D l r x x x ˇ x ! x D periodically D sin t x ! C ˛ changes periodically along the A time exposure of linear standing (Video 12.2) sin z cos 2 is equal to half thethis excitation way, frequency one (cf. can the observe caption of Fig. Standing waves on liquid surfaces can be “parametrically”ing excited by the caus- container to oscillate in the vertical direction. The wave frequency 0 x We abbreviate Therefore, they appear of the waves. or The wave travelling to the left is described by This is the equation2 of a sinusoidal oscillation whose amplitude waves in front of abright wall regions which (spacing is atmaxima the right. when the The water surface is bowed for the resulting wavetravelling function, waves resulting from the two oppositely- to obtain Figure 12.10 We then make use of the trigonometric identity between the two wavesitely sources, towards one the another. In two a strict waveswaves sense, only move one in can exactly speak this of oppo- situation, standing sinceterference only minima then at is its the smallest spacing value, of namely two in- can be produced in aface; one simple need manner only to experimentally cause onincidence linear a from waves to a water reflect flat sur- with wall perpendicular (Fig. The equation for the standingthe waves wave travelling can to be the derived right as in follows: the For positive , 655 51 Travelling Waves and Radiation 12 vidually and observe the footnote 1. Standing waves form by reflection from a wall at perpendicular incidence (1:00). Examine the images indi- “Experiments with water waves” http://tiny.cc/tfgvjy Video 12.2: C12.4. See W. Eisenmenger, Physikalische Blätter (1995). 298 Part II 12.6 The Propagation of Travelling Waves 299 12.6 The Propagation of Travelling Waves

How do travelling waves propagate? Why do we speak of the emission of waves? – Again, we use the ripple tank to help us answer these questions. We set obstacles (pieces of wood or metal) in the path of the waves. First of all, we let linear waves with a broad wavefront fall onto a long

wall at perpendicular incidence, as in Fig. 12.11, in order to block off Part II “half” of the waves. From symmetry arguments, we would expect the dashed line in the ﬁgure to be the boundary of the waves in this “semi- plane”. Beyond this line, the “shadow” of the wall should begin. But in fact we observe something quite different. The waves pass over this geometric boundary and move along large arcs into the shadow region. (For waves, their wavelength plays the role of a characteristic dimension. Here, “large” thus means “large compared to ”.) This path of the waves is described in words, strangely enough, in the pas- sive form; we say that the waves “have been diffracted”. The waves which appear beyond the shadow boundary are diffracted waves. In Fig. 12.12,wehavemadeaslitintheobstacletowavepropaga- tion by using two semi-planes, above and below, with a gap between them. Again, the dashed geometric boundaries are clearly crossed over by the waves. The connection to Fig. 12.11 is immediately evident. – In Fig. 12.13, the slit (gap) has been replaced by a rectangular obstacle of the same width. Here, the diffracted waves appear even

Figure 12.11 Blocking linear waves by a semi-plane. This and the following Figs. 12.12–12.20 and 12.22–12.24 are instantaneous images (‘snapshots’, with ca. 0.002 s exposure time). (Video 12.2) Video 12.2: “Experiments with water waves” http://tiny.cc/tfgvjy Diffraction (1:40). Behind a semi-plane, the waves propagate within the opti- Figure 12.12 “Cutting out” linear cal shadow region. waves by a slit C12.5 show the corresponding experiments for 12.15 and . In this case, the geometric ray-tracing construction The shadow of an Cutting The very incom- 12.14 off linear waves by a slit; secondary maxima in the diffraction region (Video 12.2) plete formation of a shadow behind a small obstacle inear a wave lin- ﬁeld Figure 12.13 Figure 12.15 minima can be seen on bothonly sides. indistinctly Behind visible the even obstacle, the at shadoware short is hardly distances; the weaker diffracted than waves on both sides in the free wave ﬁeld. Figure 12.14 more clearly: The diffracted waves comingedges from the of upper the and obstacle lower the interfere with obstacle each becomes other, increasinglyWaves and continually propagate washed the along “shadow” out the of axis at of greater the shadow! distances. Figures smaller dimensions. The width ofonly the about slit 3 and offails the even obstacle as is a now roughfans out approximation. broadly, and Behind in the the slit, diffraction the region, clear-cut wave maxima train and obstacle produced by linear waves spot” (Vol. 2, Travelling Waves and Radiation OISSON 12 http://tiny.cc/tfgvjy Diffraction by a slit (2:00). Video 12.2: “Experiments with water waves” Sect. 21.1). a“P C12.5. In optics, this is called 300 Part II 12.6 The Propagation of Travelling Waves 301

Figure 12.16 Elementary waves behind a very small opening on which linear waves are incident Part II

Figure 12.17 Elementary waves which are produced by scattering from a small obstacle (Video 12.2) Video 12.2: “Experiments with water waves” http://tiny.cc/tfgvjy Note here the short wave train which was produced by dipping the excitation bar only once into the water surface (2:30). According to FOURIER, such a pulse consists of waves from a broad In Fig. 12.16, the width of the slit is now smaller than the wave- frequency spectrum (see length of the waves; the incident waves which pass through it form Fig. 11.16). The waves of essentially semicircular wavefronts behind the slit. – In Fig. 12.17, shorter wavelength travel an obstacle of the same width was used. The incident waves take so faster (compare Fig. 12.74; little notice of this obstacle that we can barely see its effects against capillary waves), and thus the background of continuous waves. Therefore, in Fig. 12.17 we let a characteristic wave field is a wave train of limited length (produced by a brief, pulsed excitation formed in which the shorter of the wave source) pass by this small obstacle. In the stopped- waves move ahead of the motion image, this short wave train has already passed the obstacle. longer ones (“spectral anal- We see the result: The obstacle has produced a new, circular wave. ysis by dispersion”). This – The observations in Figs. 12.16 and 12.17 can be summarized as phenomenon can be readily follows: The slit in Fig. 12.16 and the obstacle in Fig. 12.17 have observed when swimming in become source points for new waves. Behind the slit, these propa- a calm lake! gate as semicircular wave fronts, and around the obstacle, they are complete circular waves. Both are limiting cases of diffraction. In this limiting case, they are termed “scattered” waves, or “waves produced by scattering”. The name elementary waves is also often used. Their amplitude decreases with decreasing width of the slit or obstacle. However, they are present even at arbitrarily small geometric dimensions. When the amplitude of the incident waves is sufficiently large, they can always be detected. Even the smallest objects make their existence known through scattering of waves (“ultramicroscopic detection”). The results of our experiments thus far are: We can describe the propagation of waves with geometric boundaries by using simple rays or bundles of rays. However, this works only when the geometric di- . ! A (12.14) (12.13) (12.15) refraction : (above). Linear waves : B A ! A pass over this boundary. An n : for the transition from law of refraction ˛ B B B A D ! A ! ! A c A A n n n D const D B . B D c in a ripple tank separates a shallow-water ˛ ˇ (“geometric optics”). 0-0 sin sin is governed by the B (Video 12.2) ). This fact can be used to demonstrate , divergent waves are incident on a smooth, planar ob- , the line Reflection of divergent (width of the slit or obstacle) are large compared to the B 12.36 (below) from a deep-water region to region 12.18 12.19 Using it, we find for the wavelengths B A . (perpendicular to the boundary). The transition of the waves from C12.6 and for the velocities of the waves: It defines the index of refraction B waves by a mirror atdence an of angle 45°. of The inci- smoothmirror appears surface to of be the distorted dueripples to on the the water Figure 12.18 In Fig. mensions available wavelength 12.7 Reflection and Refraction region stacle at an angle. Theare imperfections in small its compared surface to (scratches, bumps) theirif wavelengths. by The a waves are mirrorversed reflected (specular by as the reflection). mirrorcircles are The representing sketched rays the in. which incidentthe They have and rays are been the drawn, perpendicular re- the reflected toequals law wave the the of fronts. angle reflection of reflection. holds: For sition Above and The the interference mirror, angle we of of see the theright, incidence incident we superpo- and can the discern reflected the waves.out shadow To of owing the the to mirror diffraction. with its edges washed In shallow water, waves(see propagate Eq. more ( slowly thanIn Fig. in deep water incident from the upper left atincident an angle and a refractedNN ray are sketchedregion in, also the “normal axis” .If B 12.11 n = A n D B ! A n (see also Sect. Travelling Waves and Radiation 12 Reflection of plane waves by a wall at an incidenceof angle 45° (1:15). Video 12.2: “Experiments with water waves” http://tiny.cc/tfgvjy vacuum and, in Vol. 2, Sect. 16.3.) defined for all other regions. This is done in opticsting by the set- index of refraction equal to 1 for the “medium” we take a fixed value forof one the regions, then theof value the index of refraction is indices of refraction, each of which applies to oneticular par- region or transparent medium: C12.6. This is the “relative” index of refraction. We can write it as the ratio of two 302 Part II 12.8 Image Formation 303

Figure 12.19 Refraction of linear waves on passing into a region of lower wave velocity: snapshot. (The waves which are reﬂected at the boundary 0-0 upwards and to the right are not visible; they are too weak.) (Video 12.2). Video 12.2: “Experiments with water waves” http://tiny.cc/tfgvjy

Refraction (4:20). In the Part II 12.8 Image Formation video, the depth of the water in region A is 8 mm, and in B, it is 2 mm. The incident Figure 12.20 shows a “shallow-water lens”: We place a flat, transpar- waves are refracted just like ent, lens-shaped object into the ripple tank. Between its upper surface light waves which are inci- and that of the water, the remaining depth is only about 2 mm. This dent on a glass plate (towards “lens” is mounted on both sides in a frame which blocks the waves. the normal to the boundary), The divergent waves are delayed most when they pass through the since their phase velocity in thick center portion of the lens, and the least at its edges, correspond- region B is smaller than in A. ing to the decreasing thickness of the lens there. As a result of this delay, the curvature of the waves changes its sign. The waves con- verge behind the lens towards the “image point” and again diverge after passing through it. – Reflection from concave mirrors can show the same effect. We illustrate this in Fig. 12.21. This time, the wave source (the “object point”) is on the left at “infinity”, i.e. we use linear waves. In this case, the image point is termed the “focal point”. Fig- ures 12.20 and 12.21 are quite instructive: Image points are in reality not the convergence points of the rays, but rather extended diffraction patterns of the lens or mirror mount. Their diameter depends on the wavelength of the waves in question and on the diameter of the lens or mirror. The greater this diameter, the smaller the diffraction pattern, the true image point. To represent a wave, a single line is often sufficient, namely the normal to the wavefront; it is also called the principal ray. Referring to this popular method of representation, one frequently calls a wave field which is bounded by parallel lines (i.e. a wave bundle) a beam. One thus speaks of e.g. beams of sound and of light beams.

Figure 12.20 Awave source point as object point Video 12.2: is imaged by a shallow- water lens at an “image “Experiments with water point”. The object point waves” is intentionally not on http://tiny.cc/tfgvjy the symmetry axis Z In the video, the image point (Video 12.2) of an “infinitely” distant object is shown (i.e. linear wavefronts) (4:50). ). 12.17 (12.17) (12.16) 90°), that D ˇ : ); cf. Eq. ( . Experimentally, 1( 1 ˇ D 0-0 < B ˇ critical angle for total B ! , no refracted wave can 1 12.48 A ! T 1 . n A n D D ed, no waves at all should pen- A A ˛>' ! B ! , they are already refracted by 90°, B n T is called the n ' T D ,definedbysin ' D T , see also Vol. 2, Sects. 16.7 and 18.2). Time total reflection ' T ' const where the wave velocity is greater; instead, with its higher wave velocity (due to the fact sin than the angle of refraction A D A ˛ ˇ focal length sin sin smaller shows refraction for the case that the waves are incident The refraction Prominence of the focal point in the interference field of short is the is f ˛ ; . At angles of incidence for total reflection is exceed r T 12.22 ' has a maximum value of is exposure, mirror diameter 14 cm. Compare Fig. cannot be greater than one. ˛ i.e. they are propagating parallel to the boundary that at an angle of incidence of surface waves on watercurvature in front of a cylindrical concave mirror (radius of enter into the region the incident waves undergo reflection of linear waves on passing into a region with higher wave velocity Figure 12.22 incidence we find Figure 12.21 12.9 Total Reflection Figure from below right upon the boundary surface. In this case, the angle of etrate into the region angle According to this formal geometrical consideration, when the critical Travelling Waves and Radiation 12 304 Part II 12.9 Total Reflection 305

a λ A d =17.8 mm

0 0 λ B s =14.4 mm nB→A=0.81

b deep β water =90°

0 0 Part II shallow α =54° water T

0 0 α=63°

0´ 0´ 1 λ Video 12.2: d= s 0 0 4 “Experiments with water α=63° waves” http://tiny.cc/tfgvjy Total reflection (5:30). The e waves are produced in the shallow-water region (water 0´ 0´ d=λ depth 2 mm), and are incident s 00on the boundary line to the deep-water region (8 mm) at α=63° an angle which is greater than the critical angle for total reflection. Besides the reflected 10 cm wave,weseeatransversally- Figure 12.23 Demonstration of the total reflection of water waves. The wave damped wave, which travels propagates from below right onto the boundary 0-0. During total reflec- along the boundary. Its depth tion (images b und c), below 0-0, sinusoidally modulated waves travel from of penetration into the deep- right to left, i.e. the waves are split up by horizontal interference minima water region is of the order (Video 12.2). of one wavelength. When the width of the deep-water region (channel) is reduced We now should consider what in fact happens in total reflection. To (5:30), one can also observe this end, we use the same setup as in Fig. 12.22; we again allow a wave on the other side awaveat the angle of incidence ˛ (here around 40°) to arrive at the of the channel; this is the boundary from below right, above which the region A with a higher “tunnel effect” (see the end wave velocity begins. In Fig. 12.23a, we see both refraction and re- of this section and Vol. 2, flection. The amplitudes of the reflected waves are much smaller Sect. 25.9). 0 b, The 0-0 . ) is not 12.23 0 C12.7 0-0 d, the deep- 12.23 e, the distance c. The result: The 12.23 12.23 4 mm/17.8 mm = 0.81. At has been further increased : , a region of shallow water has been narrowed down to 0 ˛ 14 -0 is equal to a quarter of a wave- 0 D 0-0 0 0 bove the boundary (they are called d 0-0 = s , total reflection begins. In Fig. ı D . 54 A by about 1 mm. Their propagation direction is ! these waves are strongly damped in a direction for the appearance of total reflection B 0-0 D n 0-0 ). In the figure, the white wave crests are overstep- ˛ . c, the angle of incidence C12.8 81 or necessary : 0 . The refracted waves are perpendicular at the boundary and 12.23 ı D 90 ˛ (Their continuation as curved, diffractedcan waves even is distract very our clearlyfigure. attention seen. as But They a diffraction disturbance alsointo belongs from a beam, inseparably the etc.) to essentials any of bundling the of waves D again follows it. The distance next two experiments demonstrate this. In Fig. small relative to the wavelength.dex Otherwise, of the region refraction of does smaller notwaves. in- They represent are an able to impenetrable pass obstacle throughweakened, the to as ‘forbidden’ the region, though although a pathtunnel had effect opened through a tunnel: this is the sin extend upwards as curvedreflected “diffracted” waves waves. is The now comparable amplitude to of that the ofIn the Fig. incident waves. parallel to the boundary.quickly The in amplitude an of upwards these direction,tion waves i.e. decays direction. perpendicular very Thus, to theirperpendicular propaga- to their direction of propagation. length. The deep-water regionof is the therefore transversally-damped narrower waves than in the Fig. width water region above the boundary a small strip or channel. Above ˇ has been extended upnow to offers one sufficient wavelength.form. space The This for also deep-water restores channel the the transversally-damped total reflection. wavesSummary: to Total reflection can occurgion only with when a the smaller width index of of the refraction re- (in our example The transversally-damped waves incording the to second formal-geometric region considerations should A, beare which free in ac- of fact waves, than those of theindex incident of waves. refraction In this example, we have for the to 63°. We are thus intion, the and middle there of we the observe angular the range for following effects: total reflec- Some waves are still propagating a evanescent waves ping the boundary reflection is no longer total;across the some boundary waves clearly propagate upwards And finally, the counter-experiment: In Fig. of the d is a “for- A d; that is, the 12.23 Travelling Waves and Radiation as an “evanescent wave” in 12 C12.8. This is the name of the effect in wave mechanics. A matter wave, e.g. antron elec- wave, can also penetrate as a damped (evanescent) wave into a forbidden region, i.e. for example into a region where its potential energy would be greater than its quantum energy. This can also occur at normal incidence. If the width C12.7. This can be heuris- tically justified by the following argument: How do the incident waves “know” that the region bidden zone” if they cannot penetrate it at all? Some (small) portion of the incident wave must penetrate to a limited distance (ca. one wavelength) into the region A order to impart this information, so that the major portion of the incident wave will be (“totally”) reflected at the boundary. forbidden region is chosen to be sufficiently small, the matter wave can pass throughregion the and can be observed on its other side, into analogy Fig. particles corresponding to the matter wave can “tunnel through” the potential barrier with a certain probability. 306 Part II 12.10 Shockwaves when the Wave Velocity Is Exceeded 307 12.10 Shockwaves when the Wave Velocity Is Exceeded

Suppose that an object dips into the water surface in the ripple tank, and moves horizontally at a constant velocity u. Let this velocity be greater than the phase velocity c for propagation of surface waves on the water. Then a shockwave is formed, similar to the bow waves from a ship with which we are all familiar. Spatially, it corresponds to a conical wave, as produced for example in Fig. 12.87 by a bullet. Part II Such shockwaves can also be readily produced as a periodic sequence. Figure 12.24 shows a suitable setup. It contains a channel with boundaries 0-0 and 0 0-0 0, bordered on both sides by regions of shallow water. Beyond the left margin of the picture, an oscillating excitation source produces periodic waves in the channel: Each wave crest which propagates along the channel acts like a moving object, and it produces a shockwave in the shallow-water regions on either side of the channel. The same phenomenon can be described in quite a different manner, namely as a limiting case of refraction at an angle of incidence of ˛ D 90ı. This is illustrated in Fig. 12.25, for the lower boundary 0-0, to facilitate the comparison with Fig. 12.192. The location of the normal N-N to the boundary was arbitrarily chosen, since the incident ray is propagating parallel to the boundary and not, as in Fig. 12.19,

B shallow water 0´ 0´Waves in the A deep water channel 0 0 B shallow water

Figure 12.24 Production of a periodic sequence of shockwaves

Figure 12.25 A periodic se- N quence of shockwaves can be A treated as a limiting case of re- deep water α fraction (a representation which is popular in geophysics) 0 Outgoing beam0 β B β shallow water N

Wave front

2 In Fig. 12.19, the wave was incident from the upper left at the angle ˛.

the boundary the A refraction

Normal to Normal Angle of Angle to the

H β I that the α β B A ı ); it is then 90 Angle of D Region Region is thus the same incidence F 12.22 ˛ G α ˇ (wave front) of the is different, in a cor- . at a boundary which II E B .Itshowsthepathof c ! be a reflective boundary 00 A which is characteristic of n = B 12.28 0-0 c 1 D refraction ,let ˇ ), it follows for . 12.26 D ’ ’ Principle . Each point can then be considered to 12.13 B 12.11 c = A . and , the critical angle for total reflection. More c B (a wave front). This crest moves to the right T ! ’princi- D ' A T FH - n UYGHENS is given by sin T EG D = . Its ends, which abut the walls of the channel A UYGHENS c FH and its caption explain The origin of UYGHENS const The origin of re- . To do so, we refer to Fig. . These elementary waves are indicated by short circular ’ principle. In Fig. one after another D is a wave crest of a linear wave incident from the upper I ˇ 12.6 is shown as a dashed line; all such paths are traversed in the 12.25 12.27 sin II angle of incidence are traversed in the same times. ˛= UYGHENS angle of refraction sin details will be given in Sect. Figure separates two regions where the waveresponding velocity manner. Finally, we consider thein limiting Fig. case of refraction which is treated same times. ple. The borders of theare wave represented beam as rays. fraction according to H principle. The ray paths EG Their ratio is the same asof the the ratio velocities of the wavestwo in regions, the i.e. as when the directionthe of the rays is reversed (Fig. according to H specular reflection from a plane, Figure 12.27 Figure 12.26 12.11 H An explanation forH refraction and reflection can be provided by at an angle to it. From Eq. ( surface; left. It strikes theary points which surface are arbitrarily markedbe on the the source bound- of anin elementary Sect. wave of the type that we encountered asinglewavecrest at the velocity arcs in the figure. Theirreflected tangent wave. is a One wave ofcrest crest the paths which lead from the crest are the source points oflar elementary waves, waves however which at propagate the as lower circu- velocity Travelling Waves and Radiation 12 308 Part II 12.12 Model Experiments on Wave Propagation 309

Figure 12.28 The origin T´ of “MACH’s angle” shallowB t c B water =c B s B χ T deep A s =c t c water A A A 00T χ shallow s B water B c B T´ Part II the shallow-water regions B. The common tangent of all these elementary waves represents the new linear wave crest T-T0.Fromthe sketch, one can read off the relation c sin D B ; (12.18) cA and here, the angle is called Mach’s angle. Applying this result to the question raised at the beginning of Sect. 12.10 (conical shockwaves), we obtain for MACH’s angle

Phase velocity c of the waves sin D : Velocity u of the object

12.12 Model Experiments on Wave Propagation

In Figs. 12.26 and 12.27, neither the transverse width of the incident wave nor a structure of the boundary 0-0 was taken into account. If this simplification is not permissible, then the common tangent (II in Fig. 12.26, G-H in Fig. 12.27) of the elementary waves is no longer sufficient. One then has to consider the interference due to superposition of the elementary waves. This interference can be most clearly illustrated by model experiments. Initially, we will deal with the case shown in Fig. 12.12 in this way; that is, ‘cutting out’ linear waves by passing them through a broad slit. In Fig. 12.29, the double arrows indicate a wave crest which has arrived at the opening, and its length also shows the width B of the opening. Furthermore, the system of concentric circles indicates a single elementary wave train, which originates from a single point within the opening. – This wave pattern can be transferred to a glass slide and projected onto a screen; the double arrow is drawn on the screen. Then we imagine that using additional projectors, a continuous series of similar glass slides are projected side by side on the ). 12.30 ) along its 12.31 which one can imagine . 3 . Along the beam axis, P (left) and quickly move and , if we leave the upper edge . 12.12 2 1 12.29 P P , 1 12.29 P zone construction . , right. It shows the structure of the wave 12.11 . The wave field takes on a simple form only P 12.29 Cutting out a portion of a linear wave train by using a broad ’s case of observation). At the left, the wave pattern has been Model ex- as space points for the glass-slide image from Fig. RESNEL 12.14 one when the distance from thefor slit example becomes to large the compared right to of its the width, arrow In the model experiment of Fig. transferred to a glassnot plate. sinusoidal, The but profileprocess. rather of rectangular, the Look to waves at haslong avoid the been axis; losing right-hand chosen the detail image to arrows in (and be indicate the later points printing also Fig. to lie on theSect. symmetry axis of the wave field. These points will be used in slit (F periment for diffraction by a semi-plane Figure 12.30 screen. In practice, one uses athe more clever arrangement: We use only the waves are initially interrupteddicated by by nearly the wave-free arrows stripes, as in- Figure 12.29 its wave center upusing and some down sort in oftographic the a film direction mechanical can of setup.rapidly the separate in double these space Neither arrow, images and theall in which eye the time; follow nor elementary they each wave pho- registeris trains. other only reproduced in the This Fig. superposition produces of field the more wave clearly pattern than that the earlier Fig. of the slitdownwards, as then we is, arrive at but diffraction byThis move a corresponds semi-plane to the (Fig. Fig. lower edge a considerable distance Travelling Waves and Radiation 12 310 Part II 12.13 Quantitative Results for Diffraction by a Slit 311 Part II

Figure 12.31 A model experiment demonstrating FRAUNHOFER diffraction by a broad slit, and showing the formation of an “image point”, here a “focal point” F. In its neighborhood, the waves are linear. Compare the neighborhood of the focal points in Figs. 12.21 and 12.48.

In Fig. 12.29, the wave train after passing the slit is divergent. This case is called Fresnel diffraction. By adding a converging lens, one can convert the divergent waves resulting from the diffraction into convergent waves. Then one refers for short to Fraunhofer diffraction: The waves propagate more slowly within the lens than in its surrounding medium. As a result, the center of the beam is delayed relative to its outer parts. The wave front becomes concave; thus, the straight double arrow in Fig. 12.29 should be replaced by a circular arc. Everything else proceeds just as before. We move the wave source point (using some sort of mechanical setup) back and forth along the curved double arrow. The result is shown as a photograph in Fig. 12.31.

FRAUNHOFER’s observation method yields a diffraction pattern in the focal plane of a convergent lens which is as simple as that obtained only at very large distances from the slit in the FRESNEL case. For this reason, the FRAUNHOFER case is usually preferred.

Finally, we show two model experiments on FRESNEL diffraction from narrow slits (Fig. 12.32). Both of the diffraction patterns already exhibit a simple structure, even close to the slit; this is obtained for broad slits only at a considerable distance.

12.13 Quantitative Results for Diffraction by a Slit

To begin with, we use Fig. 12.33 to understand how the minima that can be seen on either side of the central diffraction maximum in Fig. 12.32 come about. To this end, we suppose that the observa- . P ele- N of regions, for N 2. This means that = , between the ﬁrst and s . Then the path difference ), there is no wave amplitude ˛ , left) to the end of a vibrating reed ) into a number . For the demonstration, we attach the 12.29 B B are practically parallel. Furthermore, P . The essential aspect of this addition is the P 12, each of which is the same size, and number them Calculating the diffraction pattern from a slit Two model experiments showing the passage of waves through is very far away from the slit, so that the two rays leading D P N which can be made tothat vibrate of back a and doorbell. forth by an electromagnetic drive like glass plate with the half-waves (Fig. Figure 12.33 Figure 12.32 tion point from the edges of the slit to narrow slits with different widths we divide up the slit (of width as 1, 2, 3of etc. origin of Each an of elementary these wave with regions is the same considered number. to be All the point the twelfth elementary wave, is equal to example mentary waves meet and areThen superposed the at amplitudes the of the observationamplitude point elementary at waves the add point todifferent give paths the taken overall by the individual elementary waves. Suppose that the maximum path difference in the corresponding direction (angle between the ﬁrst and the sixth,elementary or waves, between etc., the is second in andthe each the case amplitudes seventh equal of to each of these pairs cancel each other; as a result, Travelling Waves and Radiation 12 312 Part II 12.13 Quantitative Results for Diffraction by a Slit 313 at all at P, and we have a minimum; from Fig. 12.33, this angle ˛ is given by

sin ˛ D : (12.19) B

This equation agrees with the observations. It allows us to calculate if we know the slit width B and measure the direction of the ﬁrst

minimum. – Along other directions, we can carry out the addition of Part II the elementary waves graphically. This yields the “amplitude landscape” or diffraction pattern which plays an important role for waves of all types (see Fig. 12.35). It gives the distribution of wave amplitudes for the various observation directions at some distance behind a slit of width B.

The path difference between any two of the N neighboring elementary waves is s sin ˛ D D B : (12.20) N N

For the point P0 on the slit’s symmetry axis 0-0,weﬁnd

s D 0;˛D 0; sin ˛ D 0;D 0 :

Thus, all 12 amplitudes in Fig. 12.34 add here without a phase difference, as shown in the auxiliary ﬁgure 0. Their sum is drawn in as a heavy arrow R0 below the image and as the height R0 in Fig. 12.35 above the point sin ˛ D 0 on the abscissa. For the next point P , we choose s D ;thenwehavesin˛ D ,and 1 3 3B 1 the path difference of two neighboring elementary waves is D , 12 3 1 or, in angular units, ' D 120ı D 10ı. 12 The amplitudes of the 12 elementary waves add as shown in the auxiliary ﬁgure 1. As their sum, we obtain the arrow R1.ItisisenteredinFig.12.35 as the result of the graphical addition above the point sin ˛ D on the 3B abscissa.

Figure 12.34 Auxiliary ﬁgures for the graphical construction of Fig. 12.35 on B contains at the : : = ı ı 2 : 20 45 Fraunhofer 12.35 12.34 3 . D ı 2 D D ˛ R 30 D ). 2 , we choose 2 P lit. Figure ; ' ; ' 12.3 3 2 litudes. Thus, for a com- 4 2 3 3 2 ;' R ), the ordinate values of this 1 1 ΒΒΒ Β 12 12 12 D D D 2 λλλλ λ R Β 12.62 1 3 1 R 0). 0 =0 , keeping in mind its symmetry on both ; ; ; α R

D B B B ˛ 3 2 3 2 sin 12.35 D D D ˛ again lies on the axis. This should be sufﬁcient. , both the amplitudes of the elementary waves 1 ˛ ˛ ı , we have treated the limiting case of 0). 60 D on the abscissa, we have put the corresponding point i.e. sin 12.35 D 12.33 i.e. sin i.e. sin B

' = and ; diffraction pattern diffraction

; ;

D Amplitude of the waves in the the in waves the of Amplitude D The “amplitude landscape” (diffraction pattern) resulting from ,or 2 2 3 3 . The incident wave crests are practically straight lines. The s ˛ in the observation plane are “infinitely” distant to the right from . 2 12.14 D D 4 P s s R D s sides of the center point (at sin the abscissa in Fig. We can readily complete Fig. In Figs. the slit, or they lie in the focal plane of a lens. diffraction points Auxiliary figure 2 yields the sumFor as the the next length point, of the we arrow choose The graphical addition is shown in auxiliary figurefirst 4. 8 The amplitudes elementary of the waves formto a closed the octagon; 12th theirarrow sum amplitudes is yield zero. The a 9th half-octagon and thus correspond to the For We continue in an analogous manner. For the point The amplitudes of the 12to give elementary a waves closed add polygon; in their the sum auxiliary is figure zero. 3 Therefore, in Fig. on the axis (ordinate value sin to 6 and those of 7 to 12 give zero, so that the point at sin Finally, we take diffraction pattern must be squared.taken into Here, account the signs (see of the the note amplitudes at the are end not of Sect. the auxiliary figures needed for theergy construction. current, The radiation i.e. power (or thewaves en- is energy proportional transported to by the the squares waves per of unit their time) amp of the parison with measurements (e.g. in Fig. the passage of a plane wave through a rectangular s Figure 12.35 Travelling Waves and Radiation 12 314 Part II 12.14 FRESNEL’s Zone Construction 315

12.14 FRESNEL’s Zone Construction

In Sect. 12.13, we treated a special case of a general procedure which we will now deal with; it is known as FRESNEL’s zone construction. In Fig. 12.36,letS be the wave source point and P the “observation point” (or “receiving point”). Using P as their center, we draw a system of spherical waves with the wavelength of the radiation used (wave crests are black and wave troughs are white). Furthermore, centered on the source point S, we imagine a spherical surface of radius a. It intersects the wavefronts from P, deﬁning ring-shaped, Part II alternately white and black zones. Looking from the observation point P, we see a spherical surface sector containing a system of concentric rings, similar to that shown below in Fig. 12.38.Forthe radius rm of the mth zone on the spherical sector, we ﬁnd the simple geometric relation ab r2 D m (12.21) m a C b .for the derivation, see Fig. 12.36/:

The path of the waves via the mth zone is longer by D m=2 than along the line directly connecting the source point S and the observation point P, whose length is .a C b/. All the zones have approximately the same areas, namely

ab A D .r2 r2 / D : (12.22) mC1 m a C b

Now, we add to Fig. 12.36 the object, either a circular hole in an opaque screen, or a circular disk; the double arrow indicates the diameter of this object. Then only a portion of the zones remains visible from the observation point P. Looking from P,weseethe(spheri- cally convex) zone areas as in Fig. 12.37. The number of “surviving”

P´ a rm d x S a b P

Figure 12.36 The FRESNEL zone construction. m is the index of the black 2 2 . and white zones, which are numbered sequentially. We have rm D a a /2 2 2 . /2 = x , rm D d b C x ,andd D b C m 2. From these three equations, we 2 2= 2 compute rm by neglecting small terms containing 4andx . , . ), is is in al- a C12.9 three B 1and 12.29 right 12.29 D m , then at the number. odd .Furthermore,we combine. Whether b obstacle circular aperture circular aperture m and a . Here, the opening . The right-hand image can be 2 P after forming pairs remains at ; cf. also Vol. 2, Sect. 21.1 ). 12.36 gives the amplitude at that point. . Here, the aperture leaves the 1, 2 and 3); thus an P 3 12.15 ). P D left over and m !1 Video 12.4 itional rings outwards, with decreasing line thick- 12.13 number. In the zone construction, we must keep ), and by a circular opaque disk of the same size ( left even The zones which are not covered by an opaque screen with , point 5 (and ) is very large ( 12.20 2); thus an 12.21 . Each pair of black and white zones practically cancels out . The effect of the zone D right, for the observation point in mind that in thisEq. case ( of an incident plane wave,2. the quantity The number of zones allowed to pass by a odd full strength. The observationaxis point which lies along contains a waves. region of – the This beam can be seen e.g. in Fig. 3. An experimental investigation ofbut with the a examples point-like mentioned wave source above, Sect. (sound waves), will be described in lows only the two innermost zones to pass through (with m a circular opening ( reduced to two-thirds their sizethought in of as Fig. containing add nesses. tary waves. These interfereelementary with waves each that other. arrive TheExamples: at resultant of all the 1. The number of zones allowed to pass by a Figure 12.37 zones changes depending on theconsider distances each of the remaining zones to be the source of new elemen- even (but not completely!). Thefree observation region point along lies the inon beam a axis. the nearly This right wave- can at be the seen observation e.g. point in Fig. the “shadow” in Figs. innermost zones free (with 4. If we replace the circularobservation aperture by point, a all circular the zones ofthere higher index is one moreelementary waves or at less the observation issame point value; unimportant. has waves are practically always always The present the there, resultant e.g. of on the all center the axis of , Hirzel . Principles of Op- , Springer-Verlag, Travelling Waves and Radiation , Max Born and Emil 12 https://archive.org/details/ PrinciplesOfOptics English: tics Wolf, Pergamon Press, 4th ed. (1970), available at Optik 3rd ed. (1933), reissued in 1972, pp. 144–147. for example in P. Drude, Lehrbuch der Optik Publishers, 2nd ed. (1906), p. 155; or also in M. Born, C12.9. Understanding this fact is by no meansA trivial. derivation can be found 316 Part II 12.15 Narrowing of the Interference Fringesby a Lattice Arrangement of the Wave Sources 317

Figure 12.38 A zone plate for light waves (red ﬁlter light) and an observation point P at a distance of 2.7 m (actual size). The plate acts as a lens with more than one focal length. The longest is f D 2:7 m; about ten of the shorter ones can be readily observed.

5. The zone construction can also be applied to observation points that are not on the axis of symmetry. Imagine the zone surface to be mounted on an arm which can be swung around (a Cb in Fig. 12.36). Its pivot point is at the source point of the waves, and its free end Part II at the observation point. Then a sideways motion of the observation point from P to P0 moves the whole zone surface at once; thus the zones that are free to pass through the aperture or alongside the disk (ﬁxed double arrow in Fig. 12.36!) are different from before. The resultant of all their elementary waves yields the maxima and minima outside the center of the image. 6. At large values of a, the zone surface becomes nearly planar. Then the zone image of a circular aperture can be transferred to a glass plate (e.g. photographically) without serious errors. The black rings are made opaque and the white rings transparent. Such a zone plate can be used to form images. As an example, in Fig. 12.38, a zone plate for light waves ( D 0:6 m) and an observation point at a distance of 2.7 m are shown in original size. Their focal length b is given by Eq. (12.21). Additional focal points are to be found at b=3, b=5, etc. (see also Vol. 2, Sect. 21.8).

12.15 Narrowing of the Interference Fringes by a Lattice Arrangement oftheWaveSources

Figure 12.7 shows the experiment that THOMAS YOUNG used to demonstrate interference. In Fig. 12.39a, we repeat it as a model experiment by means of the superposition of two transparent wave images. This time, we do not place the two wave sources next to each other, but rather one above the other. A continuation of this model experiment makes use of a lattice of N wave source points, arranged equidistantly along a line. In Fig. 12.39b, there are three sources; in Fig. 12.39c, there are four, and so on. – This model experiment clearly demonstrates two fundamental facts which are basic to all interference phenomena3: 1. With an increasing number N of wave source points, the maxima seen already with two interfering wave trains remain, but they are

3 Both of them can also be derived graphically using the same scheme as in Sect. 12.13. 1 0 1 1 0 1 1 0 1 2 2 2 2

4 3 2

N= N=

x N= Number of wave source points source wave of Number a c b . m A model ) are pro- 12.29 jected one above the other. The numbers refer to the order indices experiment to show the interference of two, three and four wave trains from equidistant sources (marked with points at left). Two, three or four glass-plate images (cf. Fig. Figure 12.39 Travelling Waves and Radiation 12 318 Part II 12.15 Narrowing of the Interference Fringesby a Lattice Arrangement of the Wave Sources 319

Figure 12.40 The interference max- 3 ima from a linear lattice; here as X 2 a schematic illustration. The numbers α refer to the order indices m. 3 1 α 1 0 Z

3 Part II compressed into a more closely-spaced angular region: The interference fringes are narrowed. 2. Between each two neighboring maxima, .N 2/ sub-maxima appear, i.e. one in Fig. 12.39b,twoinFig.12.39c, etc. At large values of N, the sub-maxima form a nearly continuous background. The resulting scheme is sketched in Fig. 12.40. These facts, found here with the help of model experiments, play an important role in the precise measurement of wavelengths, especially in all the spectral regions of optics. We will therefore treat them in some detail experimentally in Sect. 12.20 (points 6–8). We start by using narrow slits, spaced equidistantly, as wave source points, and we arrange for the waves to be incident in the z direction, as in Fig. 12.40. The waves which emerge from the slits are spread out over a large angular range due to diffraction (cf. Fig. 12.32, right), so that they superpose nearly as well as elementary waves and exhibit interference. This experimental trick, basically a minor point, has led to the terms diffraction grating or optical lattice. For the angular dependence of the maxima of mth order, i.e. the interference maxima with a path difference of D m, when the waves are incident perpendicular to the lattice plane, we ﬁnd

m sin ˛ D (12.6) m D

(m is the order index, and D is the distance between neighboring wave source points, called the lattice constant).

As a second possibility, we will use the mirror image of a ﬁrst source as a second source, as described in the following: Imagine that in Fig. 12.39, the wave source is one point along the x axis and the second, third, fourth . . . wave source points are its mirror images. Thus, in Fig. 12.41a, one wave source S is replaced by two mirror images S0 and S00 produced by two levels of transmitting and reﬂecting planes. The waves reach the receiver along two paths which subtend the small angle 2ı. Their path difference isshowninFig.12.41a. γ γ γ d d sin sin sin d d d d d d d δ =2 =2 =2 2 β β β Mirror images To the receiver To cos cos cos A A d d d d Source 2 =2 =2 =2 β β β β Δ Δ Δ Δ δ 2 γ γ γ Receiver S S S S´ S´ S´ S´ S´´ S´´ S´´ S´´ S´´´ S´´´ S´´´´ S´´´´ a b c d , 00 ; S , 0 S RAGG plane- B d c – ; (divergent light b , Point 8. ( OHL P 12.20 ABRY Interference arrange- a AIDINGER H and F b of one source serve as wave EROT ;::: P 000 S beams); parallel light beams). Afollows demonstration in Sect. Figure 12.41 ments in which mirror images d source points. Travelling Waves and Radiation 12 320 Part II 12.16 Interference of Wave Trains of Limited Length 321

In Fig. 12.41b, the receiver has been placed at a great distance and therefore the angle 2ı is practically zero. The waves reach the two re- ﬂecting surfaces along the same path. Maxima of the reﬂected waves, or minima of the transmitted waves, occur when the path difference obeys D 2d cos ˇ D 2d sin D m (12.23)

(m is an integer, is often called the Bragg angle).

In Fig. 12.41c, four transparent, reﬂecting surfaces are placed one

behind the other at equal spacings d.InFig.12.41d, between two Part II strongly reflecting but still somewhat transparent surfaces, multiple reflections occur. In both cases, the number N of wave source points is increased (S0, S00, S000;:::) and thus the condition is fulfilled which leads to narrowing of the interference maxima. If we for example rotate the stacked transparent plates in Fig. 12.41c around an axis perpendicular to the plane of the page at the point A,thensharp,in- tense maxima appear in the directions of the arrows one after another, separated by broad, flat minima.

12.16 Interference of Wave Trains of Limited Length

Up to now, in treating the propagation of waves, we have tacitly made two assumptions: 1. The wave trains are excited with constant amplitudes and have unlimited lengths; and 2. the wave sources are point-like, i.e. the diameter of a wave source is presumed to be very small compared to the wavelength of the waves it emits. – If these two conditions are not fulﬁlled, then special features occur. They are particularly important for light waves. It is therefore expedient to discuss these effects in the section on optics (Vol. 2, Chap. 20).

12.17 The Production of Longitudinal Waves and Their Velocities

The knowledge that we have gained in this chapter will now be applied to the propagation of 3-dimensional waves. For this purpose, high-frequency longitudinal waves in air are very well suited; these are short-wavelength sound waves. First, some remarks about the formation of longitudinal waves. – A state can propagate as a wave if and only if it travels at a ﬁnite velocity. For the transverse surface waves on water, we have treated this fact for the time being as experimentally given. Its detailed discussion will follow in Sect. 12.21. – Longitudinal waves are formed , t t to the (12.27) (12.26) (12.24) (12.25) z /: (12.28) 8.3 /: by t , upper image in z during the time . This segment has t c ; t 12.42 (usually called the sound D t l : , so that within the time % t c AE A A F : = l z l l F t % E ) yields 1 E D s c ’s law, this is D D t z t D z c D D z 12.26 c m m thus moves in the time % OOKE is the cross-sectional area of the rod l tA A )to( act on the rod in Fig. c t 12.24 has two effects on the segment of the rod: First, F t . It produces an elastic disturbance. This disturbance F ,isgivenby F The is the density, .% .AccordingtoH is Young’s modulus for the material of the rod; see Sect. C12.10 E . 12.42 val. Combining Eqns. ( right; see the lowersponding process image. repeats itself In within the the next position time and shown length there, inter- the corre- and it follows fromnal this elastic disturbance that moves, the velocity with which the longitudi- The segment of length velocity) Figure 12.42 Second, it gives the segment a momentum directed to the right: when elastic disturbances propagatewe at begin ﬁnite immediately velocities. with a In quantitative this treatment. case, We let an impulse with the force propagates to the right with theit velocity affects a segment ofamassof the rod of length The impulse it compresses the segment byFig. the small amount calculation of the longitudinal sound velocity in a rod . ) / )), 2 :In ), the

1 11.6 12.28 rods. With 8.1 ). Thus, the 1 /. thin 8.3 C 1 , . % % G E , Sect. 0.27, Table s s

5.6 km/s. Travelling Waves and Radiation D . ı D D D numerical example 12 l t l

90 which one can represent as the superposition of two linearly-polarized transverse waves with a phase shift of longitudinal sound velocity is c In an unbounded medium, we have c For transverse waves, this complication does not apply. equal to the velocity of a torsional wave in aresponding rod (cor- to Eq. ( ratio occur due to the changesthe in cross section of the rod (determined by the Poisson other geometries, one cannot neglect the forces which C12.10. Equation ( holds only for ( A an unbounded steel body c longitudinal sound velocity in an unbounded medium is 322 Part II 12.18 High-Frequency Longitudinal Waves in Air. The Acoustic Replica Method 323

Numerical example For steel, E D 2:0 1011N=m2, % D 7850 kg=m3. Then the longitudinal velocity in a steel rod is r kg kg km c D 2:0 1011 =7:85 103 D 5:05 : ms2 m3 s

12.18 High-Frequency Longitudinal Waves in Air. The Acoustic Replica MethodC12.11 C12.11. Sections 12.18 Part II and 12.19 were the topic of POHL’s farewell lecture, As the source for high-frequency sound waves, we use the ﬂue pipe given on July 31st, 1952. An which we encountered in Fig. 11.35. It emits longitudinal waves audio recording of the lecture with spherical symmetry. Figure 12.43 can serve to illustrate this. is provided in the biographi- It shows a section of a meridional plane as a stopped-motion image. cal ﬁlm “Einfachheit ist das We see a periodic distribution of air pressure and density. In the dark Zeichen des Wahren”(“Sim- sketched wave crests, the pressure and density are greater, while in plicity is the Mark of Truth”) the light sketched wave troughs, they are less than in undisturbed (Vol. 2). air. The sinusoidal line added at an angle below represents the same phenomenon. The straight line indicates the normal air pressure p, while the sine curve shows its deviations p to higher and lower pressures. The absolute values of the amplitudes p0 will be given later in Sect. 12.24. The whole distribution which is visualized as an instantaneous image in Fig. 12.43 moves with spherical symmetry outwards, at a velocity of around 340 m/s. The changes in air density can be made directly visible by converting the travelling sound waves into standing waves. This is illustrated in Fig. 12.44 (following the scheme shown in Fig. 12.9, right). The

Figure 12.43 The propagation of spherical travelling sound waves in air (stopped-motion image). Medium gray indicates the normal density of air.

Figure 12.44 Standing sound waves in air in the interference field in front of a wall at R ( D 8 mm, time exposure using the schlieren method (see Fig. 11.32 and Vol. 2, Sect. 21.11); L: supply hose for compressed air). In this image, up and down are reversed. :It 4 and the following 12.5 ). This should be distinguished , this is not necessary. . They can be observed best ). At the right end, the beam is 11.30 12.47 12.46 (for example as seen in Fig. 12.44 flame tube (Fig. , i.e. a “mirror” for sound waves. The reflected R and acoustic replica method UBENS 11.35 surface waves A sound source in The acoustic replica method for demonstrating standing waves ) passes at a grazing angle over the surface of a liquid ; it is a flue pipe like that 12.45 12.46 As a result of the pressure distribution in the standing wave, similar to that ob- exhibits striations, as seen inat Fig. a grazingcan angle. be shown For asa demonstrations transparent schlieren bottom to and in placing a a a small bright large light source field, audience, underneath it. by they using a basin with in a free sound field in air Figure 12.45 the position in which it isFig. used in shown in Figs. from transverse figures). Figure 12.46 4 served with the R In the(Fig. acoustic replica(water or method, petroleum ether; a Fig. collimatedwaves beam are of superposedwaves on in waves the the air in incident front(and of waves, the density) mirror. producing Directly maxima, beneath standing their the pressure liquid surface is slightly deformed regions of constant density andriodically, those i.e. where the the density nodespattern changes with pe- and a maxima, dark-field method. are observedUnderstanding as a a dark-field image schlieren requires anof elementary geometric knowledge optics; we havethe to eyes. For understand the the role of the pupils of reflected by a panel Travelling Waves and Radiation 12 324 Part II 12.18 High-Frequency Longitudinal Waves in Air. The Acoustic Replica Method 325 Part II

Figure 12.47 The interference field of acoustic plane waves in front of a flat mirror (reflector); standing waves (time exposure using the acoustic replica method; D 1:15 cm, D 3 104 Hz). A moving mirror (u ¤ 0) would make the interference fringes drift, due to the DOPPLER effect.

If we move the reflector in the direction of the incident waves, or opposite to them, at a velocity u ( c), then the interference field also moves. The nodes of its standing waves pass by a given observation point (e.g. the point marked with the arrow a in Fig. 12.47)atthe “beat frequency” B D 2u=. This is a result of the DOPPLER effect which acts twice (on the incident and on the reflected waves)C12.12. C12.12. This is a particularly impressive and perhaps sur- If a wave of frequency strikes the moving reflector, the latter receives the prising demonstration of the 00 . = / wave with the frequency D 1 ˙ u c . The reflected wave, which is DOPPLER effect (Sect. 12.2)! “emitted” by the reflector, has the frequency 0 D 00.1 ˙ u=c/ D .1 ˙ u=c/2,andforu c,wehave0 D .1 ˙ 2u=c/. Thus, two oppositely- travelling waves with a frequency difference of 0 D D 2u= interfere with each other. This frequency difference causes the interference fringes to wander (Sect. 12.4); here the nodes of the standing wave. They pass the observation point, e.g. a, at the beat frequency B D D 2u=. It follows from this that B=2 D u; or, in words: Using the readily measurable beat frequency B, (for example with electromagnetic waves of short wavelength), the velocity u of the reflector (e.g. a moving automobile!) can be determined (Exercise 12.7).

In the acoustic replica method, the form of the reflector R can be varied in many ways. Figure 12.48 shows some examples (time ex- posures!). Figure 12.49 is also very instructive. There, the reflector for the acoustic replica method is a hand. In the course of the demonstration, its shape is changed. In the language of optics, we say that the hand is a “non-luminous source”; we see the “secondary radiation” which it “emits” by reflection. Acoustically, this means that we “see” the high-frequency acoustic radiation with which bats can recognize obstacles and find their prey in complete darkness or without using their eyes. This is the ancient acoustic archetype of radar technology “This is the ancient acoustic for localizing aircraft or other objects. archetype of radar technology for localizing aircraft.” ). ). (or right ). – At 12.24 and 12.21 acoustic 12.18 of the sound waves, ), and a 90° angular mirror ( waves not only oscillates at their can be used. We place it in front left radiation pressure 12.50 ound waves in front of a hand which is reflecting is particularly useful. This instrument is based on ound waves. The acoustic replica method makes visible the interference Sound Radiometers The interference fields of short-wavelength acoustic waves in radiometer front of a concave cylindrical mirror ( field of high-frequency s them (a ‘snapshot’, i.e. the hand did not move during the exposure time) analogous to the radiation pressure ofpressure light. should This not constant, be one-sided confusedsure with of the sinusoidally the varying sound pres- waves themselves (see Sects. For the quantitative investigationsound) of sound fields,a little-known the but significantsound fact: waves experiences Every a surface one-sided which pressuresound in is waves. the struck It direction by is of called the the Figure 12.49 12.19 The Radiation Pressure of Sound. Figure 12.48 – At the left,the the right, focal the pointterference acoustic is pattern. replica clearly method It visiblehigh-frequency s shows is (compare an suitable Fig. especially for pronounced convenient and in- low-cost detection of of its blades. The higher the acoustic radiation power of the sound (A thin membrane struck byfrequency, but sound also bulges to oneof side the in sound the waves!) direction of propagation For a qualitative demonstration“pinwheel” of sketched the in radiation Fig. of pressure, a the concave small mirror so that the focus point of the mirror is on one Travelling Waves and Radiation 12 326 Part II 12.19 The Radiation Pressure of Sound. Sound Radiometers 327

Figure 12.50 A “pinwheel” as an indicator for short- wavelength sound waves Part II Figure 12.51 An acoustic radiometer. At the right, we see the circular rotor behind an inclined glass window, and behind it on the housing, the edge of the entrance aperture. For receiving a collimated beam of sound waves, one puts this aperture at the focus point of a concave mirror which concentrates the waves (Video 12.3). Video 12.3: “The acoustic radiometer” http://tiny.cc/gggvjy

waves, the faster the rotation frequency of the wheel. It represents a practical receiver for high-frequency sound waves. If we use only one blade, and replace the bearing made of a glass thimble and a sharp pin by a ﬁne metal band which is under tension, then we have a sound radiometer, a quantitative measuring instrument. Today, one can purchase small, easy-to-use models with magnetic damping and a short, aperiodic reaction time (ca. 2 s). Fig- ure 12.51 shows an instrument of this type. Its scale deﬂection can be read off using a mirror and a light-beam pointer.

We will explain the origin of the radiation pressure of sound waves and its magnitude by referring to Fig. 12.52. The two straight lines indicate the boundaries of a collimated beam of travelling sound waves. Within it, the air particles flow sinusoidally back and forth in the direction of the double arrows with a maximum velocity u0. This decreases the pressure p 1 % 2 within the beam according to BERNOULLI’s equation by an amount 2 u0 (% is the density of the air). As a result, air flows inwards from outside the beam. If the beam now strikes an absorbing wall at the right with perpendicular incidence, the velocity of the air particles at the wall is zero; thus the pressure there increases by an amount equal to the stagnation pressure 1 % 2 1 % 2 pS D 2 u0. This is the radiation pressure. – The quantity 2 u0 is however . I (12.29) )towardsthe 12.45 reflecting : b c , usually in the form of of the sound field V ). As receiver, we use a sound of the acoustic energy (the derivation ı 12.45 Sound velocity of the sound field Radiation strength V D ı D ). ). Then for the radiation pressure of a sound wave, Vo l u m e S p 12.24 12.51 . Often, one can even use the same accessories for : We direct the sound source (Fig. Diffraction and Interference of 3-Dimensional Waves The origin of acoustic Kinetic energy within the volume C12.13 high-frequency sound waves in air that is, the spatial (volume) density is given in Sect. we have Shadowing of sound wavesvincingly can, without any by instrumentation the atof way, all. be your Rub the demonstrated right thumb hand quite and together forefinger con- at about 20 cm from your right ear. You will at the same time given by the quotient It is doubled when the irradiated surface is completely Shadowing parallel-collimated beams (Fig. Figure 12.52 radiation pressure 12.20 Reflection, Refraction, Three-dimensional waves, whoseof wavelengths 1 cm, areing are the of especially most theuse important suitable of order fundamentals for of experimentally wave theory. demonstrat- We make Most of these experiments couldmagnetic also waves be demonstrated using electro- both types of waves.source by We one need of onlyelectric the or replace small, electromagnetic the commercially-available waves flue (e.g. transmittersradiometer pipe by microwaves), for and a as small the wave receiver sound antenna and detector. Out of the largewe number can present of only impressive a demonstration small experiments, selection1. in the following sections. receiver and place ansound-wave beam. obstacle, for example a human body, in the radiometer (Fig. )) is complex. 12.29 Journal of the , 1673 (1982). Travelling Waves and Radiation 72 12 E. Apfel, Acoustical Society of Amer- ica andradiationenergyden- sity (Eq. ( A detailed treatment is given by Boa-Teh Chu and Robert C12.13. The relation between radiation pressure 328 Part II 12.20 Reflection, Refraction, Diffraction and Interference of 3-Dimensional Waves 329

hear a high tone, not dissimilar from that of our ﬂue pipe. Then use your left hand to block your right ear. You will no longer hear anything at all, since the left ear lies completely within the sound shadow of your head.

2. Reflection, lattice planes as a mirror: The reflection of sound waves has already been demonstrated in some of our earlier experiments (e.g. Figs. 12.44–12.49). Here, a few supplementary remarks will suffice: The reflecting surfaces need not be smooth; they can in fact consist of lattice planes. Two examples are illustrated by Fig. 12.53. There, the distance between the balls or the apertures from their neighbors must be of the order of the wavelength of the Part II sound waves. Then, one can readily demonstrate their specular reflection from these lattice planes using the setup sketched in Fig. 12.54.

In this setup, the angle of incidence ˇ2 can be varied by moving the sound source which is attached to a swivel-mounted arm. A small auxiliary apparatus (a pantograph) moves the reflecting surface R at thesametimebyanangleofˇ2. Then the reflected beam maintains a fixed direction. Therefore, we can use a fixed receiver. – We find a strong reflectivity from the lattice planes (Fig. 12.53) at every arbitrary angle of incidence. The reflection of sound waves from the interface between warm and cool air is also quite impressive. Figure 12.56 shows a suitable setup. Using a comb-shaped gas burner (Fig. 12.55), we prepare a nearly planar vertical wall of warm air, with a low density. It clearly reflects

Figure 12.53 Lattice planes which act as mirrors for sound waves

Figure 12.54 Demonstrating the law of reflection for a fixed direction of the mirrored beam; the sound source is the same as in Fig. 12.45. The receiver is a concave mirror at whose focus the input aperture of the sound radiometer (or a microphone) is located. ). 12.57 can be rotated (see : are sketched for the Pf 78 ˇ : 0 D 5 64 : . From this, we find a relative : ı 0 0 8 : D 39 ı sharply collimated beam of sound ı 8 D : r reach the receiver. Nothing more ˇ 12.56 is 30° for the prism. From the sketch, we sin 30 sin 39 ˛ D and the angle of refraction air , and thus ˛ ı ! is the acute angle of the prism, and at the same time 2 C ˛ CO ˛ n D gas. : For the demonstration of refraction of sound waves, ˇ : We point the beam of sound waves from the source ). The refraction of a beam of acoustic plane waves by a prism 2 The gas burner used to prepare Spec- C12.14 . ı 8 : 12.60 9 D Scattering Refraction index of refraction of The angle of incidence second interface. The angle can read off ı ular reflection of a beam of acoustic plane waves by a hot air layer a vertical layer of warm air in Fig. filled with CO Figure 12.55 the angle of incidencereceiver are at fixed, the whilealso second the Fig. face prism of and the the prism. flue pipe The scale and the Figure 12.57 4. the collimated beam froma wooden the or source, metal mirror. although not3. as precisely as we use a prism whichIts transparent is walls filled are with best made carbontic of dioxide silk are gas (paper, nearly cellophane, (Fig. or opaque plas- with to carbon the sound dioxide, waves). weto observe After an the prism angle is of filled refraction amounting Figure 12.56 can be discerned of the originally directly towards the receiverfrom and a then gas interpose burner which theor is hot we swung allow flame back the gases beam andpoured to forth pass through from through the a a beam; watering stream ofrandomly can. carbon in dioxide In all gas both directions cases,flection”), by the so reflection waves that and are they refraction scattered no (“diffuse longe re- Travelling Waves and Radiation 12 further”. air interface and conducted along the surface as indimensional a speaking two- tube, so that the sound “carries for example the voices of people at water level, are reflected at the cool/warm C12.14: Air at the surfacea of lake is often coolerair than higher the up. Then sounds, 330 Part II 12.20 Reflection, Refraction, Diffraction and Interference of 3-Dimensional Waves 331

Figure 12.58 Masking out FRESNEL zones for short-wavelength sound waves (Video 12.4) Video 12.4:

“Fresnel’s zones” Part II http://tiny.cc/fggvjy waves; it has been completely destroyed by schlieren in the air or the A setup similar to that in turbid medium. Fig. 12.58 with a variable opening 2r (iris diaphragm) 5. Fresnel zones. We refer to Sect. 12.14.–InFig.12.58,letS and a microphone as sound be a small ﬂue pipe as wave source and the observation point P the detector at P is used to show entrance aperture of the receiver (acoustic radiometer). In the center the variations of the ampli- between the two is a large iris diaphragm, framed by an opaque plate. tude (not the radiation power) The distances a and b areadjustedto50cm,andthewavelengthof on masking different numbers the sound emitted by the source is chosen to be 1 cm. Then from of zones. Eq. (12.21), we obtain

for the ﬁrst second third zone etc.

the diameter 2rm D 10 cm 14:1cm 17:3cmetc.

The indications of the radiometer are proportional to the radiation power which passes from the iris to the observation point P.Iffor example we choose 2r1 D 10 cm, then we observe a signal of ˛ D 16 scale divisions. When we open the iris from 10 cm to 14 cm, the power arriving at P is reduced; we ﬁnd only ˛2 D 3 divisions. A further opening to 17 cm again increases the power to ca. 15 divisions, and so forth. Figure 12.59 shows a complete series of measurements. 6. Diffraction by a slit: The phenomena that we encountered in Figs. 12.14 and 12.32 are demonstrated. The setup is sketched in Fig. 12.60, and Fig. 12.61 shows an image of the slit, while Fig. 12.62 gives the results of the measurements. Compare with Fig. 12.35: The values indicated by the sound radiometer are proportional to the squares of the amplitudes, so that the submaxima alongside the principal maximum as seen in Fig. 12.62 are relatively much smaller than in Fig. 12.35.

Figure 12.63 shows the same measurements as Fig. 12.62, but now represented in polar coordinates. Here, the radius r represents the signal indicated by the radiometer, or the radiation power, which is proportional to it. – Polar coordinates are preferred for technical purposes (“directional characteristic”).

7. Narrowing of interference fringes by a lattice array of slits serving as wave sources: Interference fringes become more sharp and narrow when the wave source points are arranged in a regular lattice. This was derived using model experiments in Sect. 12.15. Experi- , 5 in . P 12.15 grating ’s interfer- or OUNG lattice ; its of the iris diaphragm ). The principal maxima r 12.60 12.64 ) is replaced first by two and then 12.61 slits (Fig. . Concerning this name, we refer to Sect. narrow (see Fig. 5cm : The radiation power arriving at the observation point The diffraction slit used in Fig. Passage of acoustic plane waves through a slit 11 12.60 diffraction grating ), but with five interfering wave trains, they are consider- for various settings of the diameter 2 D B 12.65 12.58 Also called ence experiment to aslit lattice in of Fig. wave sources.by five To equidistant this end,of the the wide interference pattern(Fig. keep their positionsably during stronger and this narrower than process withlie two. between The them small form submaxima that aready nearly continuous demonstrate background. the Five characteristics slits of al- a typical width is 5 mentally, lattice arrangements of waveeither source in points the can form of bestart apertures realized with (usually slits) the or first asNo. method; mirror the 8. images. second We will We first be discussed demonstrate under the item transition from Y Figure 12.60 Figure 12.61 Figure 12.59 Fig. Travelling Waves and Radiation 12 332 Part II 12.20 Reflection, Refraction, Diffraction and Interference of 3-Dimensional Waves 333 Part II

Figure 12.62 The diffraction pattern of the slit shown in Fig. 12.61 at a wavelength of 1.45 cm. The shaded area B marks the geometric limits of the beam (geometric shadow).

Figure 12.63 The diffraction pattern from Fig. 12.62, shown in polar coordinates

as used in optics for spectral investigations. Under the topic ‘optics’ (Vol. 2, Chaps. 21 and 22), we will treat the properties of optical gratings in more detail. , in 12.66 consist A ,left.We OUNG Y 4) (Fig. 12.66 c and make use D m HOMAS 12.41 . The mirrors c), we could clearly rec- 3and D 12.54 ’s double-slit experiment to a lat- 12.39 m OUNG : We refer to Fig. D )). It is still in use today for many measure- ), i.e. a lattice-like arrangement of slits which serve : The archetype of all interferometers is the 12.64 2 submaxima between the principal maxima. They 12.64 D / The interference The transition from Y 2 aand interferometer N . 12.39 Narrowing of interference fringes by a lattice array of mirror im- 1807 for light waves (two(Figs. wave trains and two slits as source points tice or grating (Fig. as wave source points.by This superposition results of in more a than narrowing two wave of trains. the interference fringes will show here only one which is particulary important in physics, ments in laboratories and in technology. The radiation powersmall. transmitted by Therefore, theit later two possible on narrow to other slitscross-section. observe designs is interference were rather All from foundone collimated of which beam beams them make of of use waves large reflection into two or beams. refraction to Out separate of the many designs, we Figure 12.64 Figure 12.65 8. ages serving as wave sources of two and five wave trainsequidistant from slits; the spacing of neighboring apertures (center to center) is called the lattice constant of the experimental setup shown in Fig. right). In the model experiment (Fig. of four lattice planes withthus dimensions as use given only in four Fig. mirrordemonstration images experiment as shows wave two sources. ratherima sharp Nevertheless, the (“spectral interference max- lines” with the indices ognize are not visible in this demonstration experiment. 9. The interference setup which was described by T Travelling Waves and Radiation 12 334 Part II 12.20 Reflection, Refraction, Diffraction and Interference of 3-Dimensional Waves 335 Part II

Figure 12.66 The reﬂection of sound waves ( D 1:03 cm) from four equidistant lattice planes. According to Fig. 12.41c, these produce four mirror images as wave sources, arranged in a lattice. For the angles ˇ at which the reﬂected waves exhibit maxima, while the transmitted waves are minimal, cf. Eq. (12.23). is often called the Bragg angle.

Figure 12.67 The interferometer setup of A.A. MICHELSON

the MICHELSON interferometer (Fig. 12.67). In this design, the two reflecting surfaces are placed perpendicular to one another; T is a “beam splitter”, i.e. a surface which reflects about half of the incident waves and transmits the other half. The phase difference of the resulting two wave trains is determined by their path difference s. ). These photographic 10.5 ). A movable bar 10 shows an instantaneous . This is a time exposure 12.68 12.69 . In this section, we will treat the illumination) 5 cm). It is half-filled with water. As 12.17 30 The profile of a water wave of small amplitude. The wave Surfaces Streamlines in a travelling water surface wave ( positive exposure with bright-field crests do not “tip over”and no foamy breakers are formed. Figure 12.68 Figure 12.69 serves to initiate theby wave a motion; motor. When it the can wavea travels be along streamline the displaced channel, pattern we up as can and observe shown down in Fig. waves are produced information Sect. of surface waves. Thewhich results are will significant lead for us waves of to all some types. insights The waves on liquidture of surfaces sinusoidal can oscillations only beamplitudes. in represented the In by limiting general, casethe a the of crests simple wave very are troughs small narrow pic- and areimage high. broad Figure of and a flat, watermation while wave of which such is aa travelling wave long, to can narrow be the channel observedits right. made with sides of (around a – sheet 150 metal wave Thewe tank. with for- have glass seen This windows before, in is aluminumcan flakes be are observed mixed as into suspended the particles water (Chap. and 12.21 The Origin of Waves on Liquid We have discussed theagation most of important waves facts ortransverse concerning radiation surface the using waves prop- longitudinal on waves water. in air We and showed how longitudinal of 0.04 s duration. Thebe streamline seen pattern shown by in an thethe observer figure distribution who would of is directions at of rest the velocities in ofIn the the lecture a water room. particles. wave, the Itthe paths shows motion followed in of the the courseare of liquid by time is no by individual means not water particles identical stationary. to the As streamlines a (cf. result, Sect. Travelling Waves and Radiation 12 336 Part II 12.21 The Origin of Waves on Liquid Surfaces 337

Figure 12.70 The circular motion of individual liquid ‘volume elements’ (the orbital motion) in a travelling water wave (photographic negative with dark-ﬁeld illumination). The upper edge of the picture does not show the outline of a wave, but rather is due to the random distribution of the aluminum ﬂakes. Part II

paths look quite different. For moderate wave amplitudes, they are to a good approximation circular. We can observe these circular orbits both at the surface and also at considerable depths in the water. However, the diameter of the circular orbits is greatest for the water particles in the uppermost water layers, near the surface. To demonstrate these circular orbits of the individual water particles (the “orbital motion”), we add a small amount of suspended aluminum ﬂakes to the water. Furthermore, we take the period of the wave motion to be the exposure time for our photographs. In this way, we can obtain images such as the one shown in Fig. 12.70. Based on our experimental observations, we arrive at the scheme whichissketchedinFig.12.71. It shows the circular orbits of several liquid particles which are near the surface. Their diameter 2r is equal to the difference in height between the tips of the wave crests and the bottoms of the wave troughs. We will call the orbital velocity on the circular paths w,i.e.

2r w D : T The time T for a complete rotation corresponds to the advance of the wave by a full wavelength .

Figure 12.71 The correlation between the streamlines and the circular motions within travelling water waves. The horizontal row of points indicates volume elements at the water surface which are in their rest positions; the clockwise circular arcs connected to them show their paths. By connecting the small arrow points, we obtain the profile of the wave (which is travelling to the right) at the end of the next time interval. The circular motion is shown only for each second velocity vector arrow. D D 2 D .For gh u c % c (12.31) (12.30) D (12.32) p ; in a trough, 4s and : 2 25 . In a wave trough, But as a result, the 2 D ) of the water particles c w 6 w w/ T D r c %. 1km, 32 km/h). 1 2 : the travelling wave : D D thus c gr D g 2 stagnation pressures 2 D p ; orbital velocity D c c 2 7sand w c : ,wefind 5 acts to produce a deepening; at a crest, D moving with r ). 1 D T p T 12.72 , pulls downwards. A static pressure , and at the crest of a wave, a velocity of c and 50 m, that . w ,andatacrest, 2 p tends to produce a flattening. In a trough, the sur- D %w r C 2 2 D w/ 10.12 p T c The orbital motions ( 2 2 D C between the crests of the wave and its troughs. It follows p 2 and D c h p D 1 %. 1 u . These velocities produce w 1 2 pushes upwards. This static pressure results from the vertical p 1 10.11 w r p 142 km/h; or (For example ground swells, with D 2 g 1 The “observer in the moving frame” can thus apply the knowledge gained from p the stagnation pressure ity of c Thus, a volume element of the water in a wave trough has a veloc- individual liquid particles are zippinghigh past velocities the (Fig. observer to the left at and so the pressure face of the liquidmutual is equilibrium: subject The to difference twoi.e. of pressures the with two forces stagnation that pressures, are in as seen by an observer who is 6 Figs. Figure 12.72 To simplify the computation,and we air. assume Initially, a werelative to will surface those ignore between of the the water that density water. the Furthermore, and from observer motions now is of onthis moving we the “observer to assume in air the the right movingits at frame”, outline the the appears wave phase as to velocity a be whole frozen is in at its rest; motion. % For small orbit amplitudes distance from Travelling Waves and Radiation 12 338 Part II 12.21 The Origin of Waves on Liquid Surfaces 339

In deriving this equation for the calculation of the static pressure, the surface tension was neglected in comparison to the weight of the liquid. This is permissible down to wavelengths of the order of 5 cm. For still smaller waves, the term 2=% should be added into Eq. (12.32), and then we obtain

g 2 c2 D C : (12.33) 2 % Part II The phase velocity c for surface waves as derived from this equation is shown later as a graph in Fig. 12.74. – When the ﬁrst term is dominant, we speak of gravity waves. When the second dominates, we have capillary waves. For capillary waves alone, we ﬁnd 2 c2 D : (12.34) %

Derivation The static pressure p D 2r%g which results from the height difference h between the wave crests and the troughs of the waves can be neglected; instead, we consider the static pressure which is due to the curvature of the wave crests and troughs, p D =r C =r D 2=r (from Eq. (9.9;see Fig. 12.73). It provides the counter-force to the difference of the two stagnation pressures, i.e. the pressure p p D 2%wc.Weﬁnd 1 2 C12.15. Equation (12.36)is 2 important for the demonstra- 2%wc D : (12.35) r tion of how surface waves For the circular orbits of the water particles, we found Eq. (12.32). For break, Figs. 12.19, 12.22 and w D c, a circular orbit of diameter 2r D = provides a good ap- 12.23. It is derived here for proximation to the curvature of a sinusoidal wave. Inserting w D c and amplitudes comparable to the r D =.2/ into Eq. (12.35), we obtain Eq. (12.34). water depth, but holds also for smaller amplitudes. The Equation (12.32) remains applicable down to a water depth of h tsunami phenomenon can 0:5. In the opposite limit of a vanishingly small water depth h,the also be understood from en- propagation velocity of the shallow-water waves becomes indepen- ergy conservation: The wave dent of ; the same value holds for all wavelengthsC12.15: loses kinetic energy near the shore, due to the reduction of 2 c D gh : (12.36) c, and this is transferred to its potential energy, so that its Consequences: Long ground swells of small amplitude can rise up height (amplitude) increases. to disastrous heights when they run onto a beach with a gentle slope See also Video 12.2 (http:// tiny.cc/tfgvjy).

Figure 12.73 The addition of two static pressures produced by two curved surfaces. The liquid, bordered here by dashed lines, has an arbitrary shape. Its surface I presses upwards and is pressed upon (owing to the isotropy of pressure) by the surface II. iling (12.37) 8–10 km/h), are : 0 % may be different. Then : 0 % C gh % is presumed to be present. of the water at rest. Below D and 0 2 h % % c 2 C or of the water is reduced, we arrive at . The amplitude of the gravity waves g g 2 h % 0 0 h in a wave trough which is moving to the % % 2 c ), we have . For such an observer, a stagnation pressure D C c 2 2 D % % c % D w 2 12.33 D is in equilibrium with a static pressure. The latter ground waves c 2 2 ; for an observer who is “moving with the wave”, this c c c C % c 2 : Not far from the mouths of rivers, especially in Scandina- D : Fresh water with a lower density forms a layer above salt , a second fluid of density . There, with the decay of the wave at the left side and its 2 / % c 2 12.71 %. 1 2 Dead water of right at a velocity a wave trough, thethe water left is at flowing a along velocity a horizontal line. It flows to velocity is thus suddenly brought to a stop by an apparently invisible force, while sa vessels no longer obey theExplanation helm. water with a higherregion density. between the The two vessel layers.along reaches Its this down motion interface, into generates which high-amplitude the howeverwater remain waves interface surface. invisible to The the visible water eyeremains surface above (interface practically the between at water rest. and air) wave The motion; vessel this must causes providesimilar its all to the noticeable the energy slowing formation for down. of this ato This flow the resistance formation case for of is objects vortices behind in thus 3. it. a For fluid flow demonstration due experiments,slow we propagation sometimes need velocity. waveswater, with Then mark a we the very interface can with pour aluminum a dust, and layer insert of a petroleum flat excitation onto arises as with all gravitycrests of waves the through waves the and height their difference troughs; between thus the has become practically equal to the depth vian fjords, one canShips frequently observe which the are phenomenon moving of “dead slowly, water”. i.e. at 4 to 5 knots ( at the interface between thethrough layers, there the can periodic be condensation waves.clouds. of They can water be in seen the2. form of ‘ribbed’ cirrus Examples We offer three examples: 1. In twotemperature layers differences, the of mass densities the atmosphere (one above the other), as a result of Derivation In deep water,Fig. the gravity waves propagate to the right as shown in the limiting case of buildup on the rightout side circles. of each – wave When crest, the the velocity depth vectors trace Then, instead of Eq. ( Thus far, we have neglectedabove the the fact liquid that surface. a This second mediumneglected medium is its air, is effects. and located we We have nowof specifically relax density this constraint. Above the liquid (tsunami). The reducedeffect so velocity that near the the waterat from shore its the causes rear front; of a a thea “pile-up” short swell very time, overtakes and large the can water volume inundateup the of nearby to (normally depths water dry-land) of regions thus more than reaches 10 the m. shore in Travelling Waves and Radiation 12 340 Part II 12.22 Dispersion and the Group Velocity 341

bar into the interface region. It can produce waves of high amplitude there, while the surface of the petroleum, its interface with the air, remains practically at rest.

12.22 Dispersion and the Group Velocity

The content of Eq. (12.33) is graphically represented in Fig. 12.74: For surface waves, the phase velocity c depends on their wavelength Part II (top image), and the waves exhibit dispersion, deﬁned as the quotient dc=d, which is a function of their wavelength (bottom image). – If dispersion is present, then the phase velocity c D can always be determined experimentally when the waves are excited at a frequency and their wavelength can be measured7. Two examples can be seen in Fig. 12.75. When the excitation is limited to a single frequency, this instantaneous image of the travelling wave (Fig. 12.1) corresponds to the curve of a sinusoidal oscillation (Fig. 4.12). If waves are excited at

C12.16. An example of the dispersion of these waves is shown in Video 12.2,where a short rectangular pulse, produced by dipping an excitation bar into the liquid just once, is stretched out into a long pulse (see also the “aged capillary waves” Figure 12.74 The phase velocity (top) and dispersion (bottom) of ﬂat, nearly in Fig. 12.81). Short waves sinusoidal surface waves on water with various wavelengths as used for propagate faster than longer demonstration experimentsC12.16 waves.

Video 12.2: 7 This procedure is often considerably more precise than a direct measurement of “Experiments with water the phase velocity from the distance travelled and the time required. The method waves” using (KUNDT’s) dust ﬁgures has long been applied in acoustics, and today, an http://tiny.cc/tfgvjy. analogous method is preferred for electric waves of short wavelength. c ˙ )is )has A and C (12.38) of the travel- has travelled ) is assumed to .Thenasaresult 8 B d Within the resulting . In the instantaneous 9

. A on the waveform. It is / c there is a wave travelling has travelled a somewhat d ; the shorter wave ( d c : and c c . 12.76 ). After a propagation time of marker d d C group velocity D 0 tant in limiting cases. If the fre- c instantaneous images and c B is a “beat curve” (something like that in D A c (curves , while the maximum t d and the phase velocity c , a prominent point on the resulting waveform, ), imagine that in Fig. , but rather at the and D d s are transformed periodically into one another along = 12.38 c and the phase velocity Measurement of the phase velocity of sinusoidal capillary ). The longer of the two sinusoidal waves ( d , the two sinusoidal waves cannot be distinguished in any form. , it is marked with the double arrow 1. This maximum lies over / 11.11 A A . . 11.10 D , both maxima have moved to the right. The maximum t 0 12.74 along a distance convenient to use a maximum (crest) of the wave image the wave crests In order to ﬁndmoving the to velocity the with right, which we the need resulting wave to (curve deﬁne a To derive Eq. ( wave to the right, whichwaves. has resulted Its from the ‘snapshot’Fig. superposition image of two sinusoidal have the wavelength in Fig. , they produce wavelengths between r Corresponding to normal dispersion in optics (Vol. 2, Sect. 27.2). See also This term can lead to misunderstandings; note the last paragraph of this section. waves on a water surface. We can see the shadow of a 5 cm long ruler. 9 Fig. 8 Figure 12.75 two or more frequencies, then the ling waves correspond to theof graphs sinusoidal which result oscillations from of superposition differentdispersion is frequencies also and present, amplitudes. thening If during waves, the the shape propagation of ofexcitation the the frequencies instantaneous result- in images the changes. ratioS 1 With two : 2, for example, the two curves the path of propagation. These changes of shapequencies are involved lie impor e.g. within a narrow region between d of dispersion d e.g. the highest wavewhich belongs crest, to does not move at the phase velocity Travelling Waves and Radiation 12 342 Part II 12.22 Dispersion and the Group Velocity 343

Figure 12.76 Wave graphs (“snapshot images”) to illustrate the deﬁnition of the group velocity. We have chosen d D tdc here. Part II

shorter distance s0 D .c dc/t. The lead .s s0/ D ds D dct gradually approaches the value d. This case is sketched in the three lower curves: The point of equal phases is now at the wave crests ı and e.Thatis,the maximum, the marker of the wave group, has not moved along a distance ct, but only along the smaller distance s D .ct /. Therefore, the velocity of the marker, i.e. the group velocity, is

ct c D ; t and this leads to Eq. (12.38), since we chose ds D dct D d.

The consequences of Eq. (12.38) can be – quantitatively! – explained by a demonstration experiment. The necessary apparatus is described in Fig. 12.77: Two waves of differing wavelengths are represented by the shadow images of two gears, B and C. Black gear teeth mean wave crests, and the white gaps between them are wave troughs. These “waves” do not travel as in Fig. 12.76 along straight-line paths, but rather on circular paths. – The two gears are mounted on the same shaft, one behind the other, and can be rotated independently. Then in the shadow image A, the beat curve resulting from their superposition can be seen; it exhibits four wave groups. The gears are driven by a slow-running synchronous motor M. The velocities c and .c C dc/ of the two gears can be conveniently adjusted by using pulley wheels of different diameters driven by elastic belts. A pointer Ph permits us to measure the phase velocities of the individual waves with a stopwatch. c Ac- inte- ). b). On ,which 12.77 . 12.78 OURIER . This narrow , and the phase ) or the shorter c B C12.17 ( (Fig. 0. The groups remain and to demonstrate D c a. Every sinusoidal wave c ; 12.78 0 is an electric motor D M c very narrow band . d d ,asinFig. 12.15 c). We made use of this fact above, in order and ) for the group velocity 12.78 is smaller than the phase velocity 12.12 c 12.38 The quantitative demonstration of the group velocity (details spectral line ) has the greater phase velocity. In the first case, the , the group velocity becomes C overtakes the groups. In the second case, the groups over- . One can thus not consider the group velocity to be simply ( c d d Ph / ˙ d D d gral). For the purposes ofreplace computations, we this can as dense an series approximation narrow region with (Fig. two spectralto lines derive that Eq. fall ( within the both sides of the sineis wave frequency, it very includes narrow a region comparedregion d is to filled its with a mean dense wavelength series of spectral lines (a F fixed at one spot. A strictly sinusoidal wave trainginning in and the no mathematical end, sense either hastain in no a time be- single or in space.that occurs Its in spectrum nature will however has con- it a beginning appears and an in end. its Therefore, spectrum as a the groups have exactly thec same velocity as the phase. In the limit experimentally. As a result of thisgroup, approximation, we instead obtained of a a sequence single cording to of Eq. identical 12.38, groups a (Fig. group velocitywhose is spectrum defined only includes for a wave groups narrowand range of wavelengths between Figure 12.77 We can choose at will. whether the longer wave take the phase marker. In the limiting case marker are given in the text).lem here, The just “rectangular” as profileexample in of in other the Sects. waves geometric poses model no experiments prob- on wave theory, for the velocity of any arbitrary wave group. group velocity , 257 43 Phy- Travelling Waves and Radiation 12 sikalische Zeitschrift C12.17. E. Mollwo, (1942). 344 Part II 12.23 The Excitation of Waves by Aperiodic Processes 345

Figure 12.78 The spectra of unbounded waves (a and c), and of a short (d) and a long (b)wave group (short refers here to the spatial length of the group, not to its wavellength) Part II

12.23 The Excitation of Waves by Aperiodic Processes

The greatest contrast to mathematical sine waves is the propagation of aperiodic processes, e.g. the practically aperiodic wave group shownatthetopofFig.12.79. Such wave groups have a broad, continuous spectrum; strictly speaking, they contain an unlimited range of wavelengths on each side of the mean wavelength (Fig. 12.78d). If such an aperiodic wave group passes through a medium without dispersion, that is a medium in which the phase velocity is the same for all wavelengths, then the shape of the group along its path through

Figure 12.79 The time dependence of the production of gravity waves by an aperiodic disturbance of the water surface at the time zero (‘snapshots’, extended by computed values to the right of the arrows, because the wave tank was too short); upper image, 1.3 s after a single (i.e. aperiodic) pulse of the excitation bar (in a wave tank as described in Sect. 12.21) , 12.79 ,i.e.their https://en. normal . b. Therefore, spectral appa- for all the waves experimen- c 12.79 . 12.78 . (Their dispersion is an aperiodic process (in optics, for exam- tting a very thin layer of fatty-acid molecules ). In their spectra, they are represented by 12.79 d ) increase with increasing wavelength.) – At the C convert c 11.4 , then the shape of the group will continually change The shape of the waves can be observed through windows . Along an elastic cord, an aperiodic wave group propagates . and ( C12.18 12.80 Here, we havea used straight a bar for waveinto wave the tank surface excitation. of ca. the This water.can 3 m The bar be formation long is of suppressed perturbing thrust and capillaryonto by once, waves the 60 cm aperiodically, pu liquid surface; deep, usually,water it with is sufficient toon dip the one’s sides hand of intothe the the mirror images tank. of a If longThese no fluorescent were tube windows photographed reflected are in to the available, obtainWe water Fig. one can surface. could consider observe a similar experiment on a larger scale: In Fig. first, after several days,ods waves of of about lengths 20 arounddecreasing s wavelengths several and appear. 100 periods. m For The andm.wikipedia.org/wiki/Swell_(ocean)# more peri- waves details, see which e.g. follow later have gradually imagine that theHorn. zero point Then at when thecentimeters the Atlantic amplitude upper propagates is left as free is far of as a storms, the a storm south ground coast zone swell of near England. of Cape a At few These observations are veryalone instructive; suffices they to show that dispersion the medium remains unchanged.termine the common In phase thistally. velocity case, Transverse waves one along can ancf. elastic Fig. readily cord de- offer a good example; Figure 12.80 without any change infollows from shape Eq. (here, ( a 10 m long helical spring); its velocity ple “incandescent light”) into periodicmatic waves light”). (in It optics is “monochro- not(e.g. at glass) all a necessary particular toSect. geometric give 16.10). the form dispersive (e.g. medium Dispersion assoidal can a (“monochromatic”) waves. prism; however Even never cf. shortwave produce Vol. wave groups 2, strictly trains produced sinu- from by long dispersionbetween consist of waves from a range longer; at itsbehind front, the longer group. waves form, while shorter ones collect If, in contrast, anwith dispersion aperiodic wave groupalong passes its through path. aa medium This water can surface, be Fig. phase seen velocities for examplebeginning in of gravity an waves experiment on into (upper the left), water surface. an After excitationwave 1.3 bar group. s, we is can In thrust observe the a course nearly aperiodic of time, the group becomes longer and a narrow band, as shown in Fig. ). , 450 12.32 4 Optik Travelling Waves and Radiation 12 (1948/49)). (E. Mollwo, observed with gravity waves of wavelengths 5–20 m and used to verify Eq. ( C12.18. A very well-formed (circular) wave train has been 346 Part II 12.24 The Energy of a Sound Field. The Wave Resistance for Sound Waves 347

Figure 12.81 “Aged” capillary waves travelling to the right on a water surface; at the left side of the group, the wavelength is approximately 1.7 cm

ratus that are based on dispersion, for example prism spectrographs, have a limited resolving power, =d.

For each short wave train from a long wave group produced by dis- Part II persion, we can specify a group velocity c. For the case of gravity waves, c is always less than their phase velocity c at all wavelengths, but c increases along the direction of propagation of the group. For this reason, the wave group in Fig. 12.79 becomes longer and longer, the further it moves away from its source.

The dispersion of the capillary waves can be treated in a similar manner, but now, dc=d is negative, so that the dispersion is “anomalous”: the short waves travel faster than the long waves (Fig. 12.17). At the front C12.19. Such an “aged” of a wave group, new crests are formed. At ﬁrst, we can discern around 15 wave crests; but the shorter waves are damped much more strongly by wave group of capil- frictional effects in the surface layers as they propagate through a wave- lary waves can be seen in length than the longer waves (cf. Sect. 9.5, point 5). Therefore, the short Video 12.2 (at 2:30 min.); see waves at the front of the group decay more quickly. After a propagation also Fig. 12.17. time of around 2.5 s, an aged group exhibits the form shown as a photographic image in Fig. 12.81. At the left, waves of about 1.7 cm wavelength Video 12.2: remain. Explanation: At a wavelength of D 1:73 cm, the curve of the phase velocity c has its minimum (Fig. 12.74, top). In the region of this “Experiments with water minimum, dc=d 0; thus, the group velocity c and the phase velocity c waves.” are practically identical; the aged group in Fig. 12.81 can then continue on http://tiny.cc/tfgvjy its way without any further noticeable change in its shapeC12.19. All of the observations described in this section can be conveniently ex- “All of the observations de- perienced on the smooth water surface of a pond. There, one frequently scribed in this section can sees a very noticeable phenomenon: If a small object (a stick or a ﬁshing be conveniently experienced line) is moved relative to the water surface within its plane, we can see standing waves in front of it, i.e. waves moving at the same speed as the on the smooth water sur- object. They occur just when the relative velocity u > 23cm/s,i.e.whenit face of a pond.” exceeds the minimum value of the phase velocity in Fig. 12.74. Then, the slowest waves can no longer ‘run ahead’ of the moving object.

12.24 The Energy of a Sound Field. The Wave Resistance for Sound Waves

The energy density ı of a sound ﬁeld is deﬁned by the quotient

Energy of oscillation within the volume V ı D : (12.39) Vo lu m e V (12.40) (12.42) (12.44) (12.43) (12.41) pressure ,SPL).Then is defined by . The greatest de- 0 velocity amplitude ; kinetic energy. The (known technically u ), but are practically 0 10 p : and contains the kinetic en- c irradiance perpendicular to it. Then ı % : A 2 0 D 2 : eflection, all of its oscillation ; / : 2 0 Vu 0 % c u % p 2 field is the sum of the oscillation Act is tA ı sound pressure level c % Act 2 1 ı ı 1 2 . D or D D 2 1 D e . C12.20 D and density 2 E E E kin ı D tA V E ı Irradiated surface area D Incident radiation power e is “irradiated”. Its E D A e E , they convey to this surface an energy t of the sound waves, will be called . The area sound pressure amplitude Act The details of divergence, radiation power and related concepts can be found in the ratio i.e. the entireume energy which was previously contained in the vol- From this, we obtain for the irradiance the important relation: still plane waves, which strike a surface within a time same is true for sinusoidal sound waves. The maximum velocity of the air(technically: their particles, “rapidity”), i.e. will the be denoted by Its units are for example W/m a quantity of air of volume as the viation of the air pressure from its value in air at rest, i.e. the amplitude Starting from the potential energy,tain after an a expression for brief the computation acoustic we energy ob- density: thus The oscillation energy in the sound energies of all the volumethe elements direction of of the air propagation that ofsinusoidal are the oscillation vibrating sound can along waves. be The computedof energy either its of potential as each energy the or the maximuma maximum value of simple its pendulum: kinetic energy. At Thinkenergy its of is maximum in d the formits of rest potential position, energy; its when energy is itthe all total is kinetic. energy passing is At composed through of all both intermediate potential positions, Vol. 2, Chap. 19. 10 and its acoustic energy density The waves area beam presumed with a to small divergence (opening be angle weakly divergent, i.e. they form ergy ; S p is equal to ı ). 12.29 Travelling Waves and Radiation 12 cf. Eq. ( C12.20. Note: the sound pressure (acoustic radiation pressure) 348 Part II 12.24 The Energy of a Sound Field. The Wave Resistance for Sound Waves 349

Derivation 1 Beginning with Eq. (5.8), we ﬁnd Epot D 2 V p0, and the energy density 1 V ı D p : (12.45) 2 V 0 We use the modulus of compression of a gas,

p K D V 0 V

and the velocity of soundC12.21 C12.21. Derived in an anal-

s ogous way as for a solid rod Part II K (Sect. 12.17); however, the c D : (12.46) % modulus of elasticity E of the solid material must be Combining these equations yields Eq. (12.44). replaced by the modulus of compressibility of the gas. During each sinusoidal oscillation, the velocity amplitude u0 and the maximum deﬂection x0 are related to each other by the equation

u0 D !x0 (4.12) .! D 2 D circular frequency/:

We thus obtain a third expression for the acoustic energy density, this time containing the frequency of the sound waves:

1 ı D %!2x2 : (12.47) 2 0

The above equations are by no means applicable only to air; they apply to every medium that carries sound waves. The energy density increases as the square of the frequency. As a result, high-frequency sound waves give rise to striking phenomena, for example cavitation in liquids. In Fig. 12.82a, standing acoustic waves ( 1cm)are produced by reflection from a ground-glass plate serving as a mirror. We place a soap film in the region of the sound waves. It cuts the wave crests and troughs into strips perpendicular to the plane of the page. In the crests, the film becomes cloudy due to cavitation, i.e. the segregation of gas bubbles. This can be projected as a shadow image onto the glass plate (Fig. 12.82b). All three characteristic quantities of air vibrations, namely the amplitudes u0 of the velocity, p0 of the pressure changes and x0 of the deflection can be measured directly.

1. The measurement of the velocity amplitude u0 is carried out by making use of hydrodynamic forces.

Example The RAYLEIGH disk. In the sound field, a thin disk the size of a coin is hung suspended, free to rotate. It carries a mirror for a light pointer and is protected from air currents by a small gauze cage. Let the surface normal of the disk be inclined at an angle # of ca. 45° relative to the direction of ound b. (12.49) (12.50) (12.48) 10.16 : ) to obtain 2 . Ã 2 2 ) % % 2 2 . The reflected and 12.44 c c C 12.45 1 1 , one generally uses a ca- % )and( % , one places tiny dust par- %: 0 1 1 ). They exhibit nearly the 0 c p c K # reflectivity resultant wave Â p 10.3 12.43 sin 2 or D D 3 r % of ı c 4 3 (compare Vol. 2, Sect. 12.7.) D D C12.22 energy density of the sound waves). 0 M 0 p D u ı ) with Eqns. ( maximum deflection x pressure amplitude 12.46 . ı reflection coefficient Sound waves of high energy density cause cavitation in a soap 1 cm; the source is the same as in Fig. Incident radiation power Reflected radiation power C12.23 D Radius of the disk, is D D R r R acoustic wave resistance Z ( propagation of the soundthe waves. disk with The the airflow well-known ofThe streamline disk the pattern experiences as waves a passes shown torque in around Fig. It is called the the incident waves superpose to give a ticles into the sounda field microscope. and observes The theirfriction small (viscosity) pendulum of spheres orbits the with aresame gas pulled amplitudes (Sect. (maximum along deflections) by aselements the the of surrounding internal the volume air.energy However, densities this method is applicable onlyWe at combine high Eq. ( pacitive microphone. 3. To measure the film, which is cutwaves at ( an angle by a field of high-frequency standing s This wave resistancetween determines two the media. reflectioninterface at If with the a a medium interface plane havingratio wave be- a is different wave incident resistance, perpendicular then the to the This ratio of the pressure amplitudethe to the velocity amplitude is called 2. To measure the Figure 12.82 . 2 // 2 Z ). C corre- 1 for elec- Z % 12.50 c / 0 ’s formu- /=. 2 D Z =. Z 0 1

Z RESNEL p .. Travelling Waves and Radiation D D 12 las (Vol. 2, Sect. 25.8) is R This yields Eq. ( tromagnetic waves (Vol. 2, Sect. 12.7). The reflection coefficient derived from F resistance sponds to the wave resistance Z C12.23. The acoustic wave C12.22. This torque is maximal when the disk is inclined at an angle of 45°the relative undisturbed to airflow. 350 Part II 12.25 Sound Sources 351

In the technical and acoustics literature, for a sinusoidal sound wave of radiation power WP 1 or pressure amplitude p1, the absolute values are 2 often not quoted, i.e. for example WP 1 in watt or p1 in newton/meter , but instead only the relative values. These are compared to a reference power WP 2 or a reference amplitude p2, which must be precisely speciﬁed in every case. Then, either the quantity

WP p x 1 1 D 10 log D 20 log (12.51) WP 2 p2

is computed, or else the quantity

1 WP p Part II y 1 1 : D ln D ln (12.52) 2 WP 2 p2

Both of these quantities are pure numbers. They are combined with the number 1 as multiplier and we give the number 1 two new names, namely in the ﬁrst case decibel (dB), and in the second case Neper (Np). – If for 2 example we use a reference pressure of p2 D 1N/m (D 1 Pa), then the 3 2 speciﬁcation “60 dB” implies a pressure amplitude of p1 D 10 N/m . These special names for the number 1 have an advantage: They remind us which equation for the power or pressure comparison has been employed. Their great disadvantage, however, is that one can interpret decibel or Neper (and, later, the phon) incorrectly to be units, similar to the ampere, kilogram, candela etc. In technology, the (effective) reference pressure is often taken to be p2 D 2105 N/m2. Then the quantities x and y, i.e. the logarithms of the relative measured pressures, are quoted as absolute sound-pressure levels! Often, decibels are also used to denote orders of magnitude, in contrast to the usual form. Then, for example, 80 dB D 108,20dBD 102, 30 dB D 103 etc.

12.25 Sound Sources

In the ripple tank, we could get a good intuitive picture of the mechanism of wave emission. We could see how the excitation bar dips rhythmically into the water surface and displaces some water at the frequency of its vertical vibrations. This experiment can be applied in analogous fashion to the spatial emission of longitudinal elastic waves in air, water, etc. If a sphere changes its volume in the rhythm of sinusoidal oscillations, one is dealing with an ideal sound emitter,thebreathing sphere. All the points on its surface vibrate with the same phase, and the result is a completely symmetric emission of spherical waves. This ideal sound emitter has yet to be technically implemented. But some solutions to this problem approach the ideal rather closely. Thick-walled containers with a vibrating membrane are good examples in this connection. The membrane is best driven electromagnetically from within the container. Applying this principle to acoustic signals in water, using membranes of around 50 cm diameter, power levels of the emitted sound waves in water of up to 0.5 kW have been achieved. The “membrane” or diaphragm is in this example a steel plate of ca. 2 cm thickness! One which To this end, strings b). Now the vibrations are milar considerations apply to h is thus excited to forced vibra- are, in contrast, the open ends of short, thick oscillat- ng, the air on its lower side, and 12.84 a, a piece of twine is held at the right by a hand. Very poor emitters vibrational mode, the membrane of a sound source . In this case, the diameter of the membrane must be , the black disk represents the cross-section of a vibrat- The sound emission of a vibrating string . 12.84 . We can thus consider the emission of the waves into 12.14 12.83 simplest 12.12 is a ratherpoint skewed is way only of the expressing relatively good the emission situation. properties of the The box; essential the often hears that “The vibrations were amplified by resonance”. This Figure 12.83 tion, we will considersurface its area. amplitude to Thenthe be we constant-phase constant emission have over of physicallyFig. the waves quite from whole similar the conditions aperturea of to spatial cone a under the slit right in in circumstances, similar Fig. to what is shown In the oscillates with the samenodal phase lines all except at over its its rim. surface, and Furthermore, exhibits as no a rough approxima- case of resonance to obtainup large to amplitudes. have The weak emitterto is damping that then and of set its theexamples: resonance tuning frequency fork is or matched string.1. To illustrate In this, Fig. we offer two It then “displaces”, roughlythere, speaki a wave trainleaves, begins again with roughly speaking, a an crest. empty spacethere, on a At its wave the upper train side, begins same and with time,difference a of the trough. practically The 180° string in two every waves direction havealmost a and completely cancel phase by each other interference.a Therefore, very the poor vibrating emitter string oftuning is sound forks waves. Si For practical applications, theforks must vibrations therefore of first be strings transferredone to and arranges good emitters. a of suitable mechanical tuning tuning connection fork between and the a strings good or tions. emitter, whic Under some circumstances, one can make use of the special a multiple of the emitted wavelength. Passable sound sources are also the ing air columns. are often employed in musical instruments. In Fig. ing string which is perpendicular tois the just beginning plane a of vibration the in the page. direction The of string the arrow, downwards. Two fingers of the othervibrate like hand a rub violin string, its butattach it left emits its end. practically right no end sound. The to Thencannister twine we a begins with good to emitter, a e.g.emitted membrane a as short (Fig. sound metal waves or which can cardboard be clearly2. heard at A some distance. tuning forkat is one mounted end, on usually a called short a wooden resonance box chamber or which is sound open box. Travelling Waves and Radiation 12 fied by resonance’. This is a rather skewed wayexpressing of the situation.” “One often hears that ‘The vibrations were ampli- 352 Part II 12.25 Sound Sources 353

Figure 12.84 Coupling of a poorly-emitting string to a membrane which is a good emitter Part II

Figure 12.85 Improvement of the sound emission from a tuning fork by a wall on each side C12.24 (W. BURSTYN) C12.24. W. BURSTYN constructed among other things electric musical instruments at the beginning of the 20th century.

resonance is simply an aid to transferring the vibrations. We can demonstrate this with a surprising experiment. We bring one tine of a tuning fork as shown in Fig. 12.85 into the gap between two walls which are large in comparison to the wavelength of its tone. The tuning fork can now be heard at some distance away. This is because the interference of the waves from the inner and outer sides of the tine is now considerably reduced and the tuning fork has thus been made into a tolerably good emitter.

In the case of musical instruments, e.g. those of the violin family, the situation is rather complex. The strings and the sound box form a confusingly coupled system (Sect. 11.15). The box itself has a whole series of resonance frequencies. In producing its forced vibrations, certain frequencies of the vibrating strings are thus strongly favored. Figure 12.86 shows a vibration curve and the associated vibration spectrum of a tone from a violin. The sound box of a violin is stiffened from within by the sound post. The top plates, considered as membranes, are by no means small compared to all the wavelengths used in music. Therefore, the emission can exhibit strongly-preferred directions.

With the topic of violin characteristics, we come to a discrimina- tion which is of some importance technically, between primary and secondary sound sources. Primary sound sources (emitters) have to produce vibrations with a certain spectral composition. We condone the fact that every individual primary sound source, for example every musical instrument, has a right to its characteristic spectrum; or, stated physiologically, to its own particular timbre. The situation is quite different for secondary emitters. A typical modern represen- ) right shows ). A sin- ; a periodic 12.87 12.11 ‘bang’ or . Figure 11 ‘boom’ ’s angle (Sect. ) and the frequency spectrum ( ACH left C12.25 ) using the litudes can propagate at higher velocities than RANZ ACKHAUS , we encountered shockwaves and conical waves. They and Supersonic Velocities The muzzle report of a gun The vibration curve (at 12.10 In the acoustic investigationsound of waves bullets, of it large must amp be taken into account that Journalists will then report that “the sound barrier has been broken”. such a conical waveovertaken the from muzzle a report bullet, (‘bang’)of of just the the at sound gun. wave the cone The is moment opening called when angle M it has schlieren method.) and the conical wave fromblack a cloud bullet at (the left isexplosion). gas (This from and the the powder followingare picture photos taken by C. C gle conical wave is heardseries by of the these ear waves as sounds a like the tone from a trombone. 11 Figure 12.87 In Sect. Figure 12.86 tative of this latter groupfree is choice in the their loudspeaker. frequency spectra.the Loudspeakers They form have are no of required sound toto waves emit them, – – without in the giving preference electrical to oscillations any which partial tone are components. sent 12.26 Aperiodic Sound Sources of a violin tone (H. B are generated when anexceeds object that moves of aperiodicallyten the and occurs sound its in waves velocity a the (or whiplash, air. other a bullet, waves). or Asrapidly an moving examples, bodies aircraft This generate ﬂying we conical case at waves could supersonic of- speed. mention These the end of , 17 note 1 analysis, see ) of the vibration OURIER 11.3 Travelling Waves and Radiation 12 partial tones, of the g curve shows the amplitudes of the fundamental tone and the harmonics, referred to as considerably from instrument to instrument. (After Fig. 11 from H. Backhaus, Die Naturwissenschaften the fundamental is 392 Hz. The harmonics determine the timbre, which can vary of the d-string of anStradivarius Antonius violin dating from 1707. The frequency of C12.25. The frequency spectrum (F Sect. 811 (1929).) 354 Part II 12.27 Sound Receivers 355

Figure 12.88 Shock waves produced by two simultaneous sparks with different electric currents (larger at the top) Part II

the normal speed of sound. Near electric sparks, one can readily observe sound waves whose velocities are around 500 m/s (Fig. 12.88).

12.27 Sound Receivers

Sound receivers can be divided into two groups in the sense of limiting cases: Pressure receivers and velocity receivers. 1. Pressure receivers: The majority of pressure receivers consist of membranes fixed at their edges. They can be attached within capsules, walls, funnels etc. Examples: Microphones of all types, as well as the eardrum. All pressure receivers carry out forced vibrations in the sound field. Their amplitudes are independent of their orientation within the sound field; the air pressure is independent of direction (pressure isotropy; cf. Sect. 9.3). Every living-room barometer demonstrates this. A barometer is in the end just a pressure receiver for longitudinal waves in the air. But in the case of atmospheric pressure fluctuations, we are usually dealing with oscillation processes of extremely low frequency. Technically, microphones are much more important today than all other types of pressure receivers. Radio broadcasting and sound recording have raised the standards here enormously. Over a broad frequency range from around 100 up to 10 000 Hz, they are required to maintain the original amplitude ratios. As with all forced oscillations, this requirement can be bought only at the price of a greatly reduced sensitivity. 2. Velocity receivers. In the case of velocity receivers, the velocity amplitude of the alternating air flow of a sound wave is used to produce forced oscillations. This can be most readily made clear by giving an experimental example. In Fig. 12.89, we see a thin glass fiber about 8 mm long which is set up as a small leaf spring perpendicular to the direction of the its 1024). D 10 . Imagine two of the oscillat- . If it is placed 0 x and therefore larger % )). Therefore, pressure receivers directional receivers 12.47 – characteristic property: We find that (compare Eq. ( A fine glass fiber as a de- 0 x velocity receivers ing air or water particles. Air has a low density Figure 12.89 tector of motion (velocity receiver)reality, it (in is only 0.028 mm thick) range from about 20 Hz upencompasses to a around 20 spectral 000 Hz. range The human of ear at thus least 10 octaves (2 12.28 The Sense of Hearing Hearing and our auditory organsphysiological are and for psychological research. the Nevertheless, most forpurposes, part physical we the want to objects summarize of thethem. most important Likewise, facts in concerning optics oneat should least know in the general properties of terms. the eye, 1. Our ears react to mechanical vibrations in the very broad frequency Velocity receivers can be usedsuch as fibers oriented symmetricallya to moving body. the If long thereboth axis is receivers a on straight-line will each course answer to sideviations the with of from sound the the source, same straightforced amplitude. course amplitudes of will the Sideways give two de- rise fibers. to inequalitiesPressure and in velocity the receivers are, asery mentioned, momentum limiting cases. transfer Ev- fromnoticeably pressure deformed by requires the awall pressure. must wall remain small which The compared is to forced the not amplitudes deflections of the propagating sound waves (microprojection!). Periodic changes inair the pressure have no influence atalternating all air on flow this in fiber. the direction Onit of the along the contrary, by the oscillating internal air friction particles and pulls (here thereby using excites a it reed to pipe forced as vibrations is sound a source, typical close velocity to receiver. the Itand–for fiber). at the This same fiber time exhibitsamplitude an depends important on its orientation inwith the its sound field long axis parallelfiber to remains at the rest. direction of sound propagation, the deflections sound waves in water. Pressurelocity receivers receivers in in water. air can also become ve- can be constructed successfully for air, but only with difficulty for Travelling Waves and Radiation 12 356 Part II 12.28 The Sense of Hearing 357

The upper limit of this range decreases with increasing age of the hearer. 2. Sinusoidal vibrations produce a sensation of a pure note in the ear. Every note has its particular pitch. The pitch of a note is a sensory perception and as such is not accessible to physical measurements. The pitch of a note depends in the main on the frequency of the waves, but to some extent also on the strength of irradiation at the ear. Unfortunately, we generally speak of the frequency of a note. This is of course convenient, but it is a rather lax form of expression; what is meant is always the pitch of the note as experienced by the ear when it is excited by a sinusoidal wave of the given frequency at Part II a medium irradiation strength. 3. In the most favorable frequency range, our ears can distinguish two frequencies that differ by only 0.3 %. The ear thus has a “spectral resolving power” =d of around 300. This corresponds in optics to what one can achieve with a glass prism of a basal thickness of about 1cm. 4. The ear responds to non-sinusoidal vibrations by perceiving a tone. A tone is independent of the phase differences between the individual sinusoidal partial tones. This is a fundamental insight based on the observations of GEORG SIMON OHM. It corresponds to HELMHOLTZ’s interpretation that the inner ear functions as a spectral apparatus for sound waves (cf. Sect. 12.30). A musical chord corresponds to a frequency spectrum with a particular structure, characterized by the ratio of the frequencies and amplitudes of its spectral lines. The absolute value of the fundamental frequency is unimportant. Two sinusoidal oscillations of nearly the same energy density with a frequency ratio of 1:2 always yield the chord called an “octave”, etc. 5. The most important tones are the syllables of language. – In normal reading, the eye is stimulated by a temporal sequence of two- dimensional symbols. They are distinguished by a spatial sequence of individual elements, namely letters or syllables. Often, we can ‘hear’ the writer speaking when we read a text written by someone we know. In normal hearing, however, the ear is stimulated by a temporal series of air-pressure variations. These have spectra of differing forms. Voiceless consonants are characterized by broad, continuous spectra with a variety of shapes. Voiced consonants and vowels can be represented as frequency spectra. One finds the same spectral lines independently of the pitch of the voice (bass, tenor etc.) in regions which are characteristic of the particular vowel. These are called the formant ranges (Fig. 12.90). They are finally just the damped normal modes of the oral cavity etc., which are periodically excited through impulses of air pressure from the larynx along the lines of Fig. 11.16. In general, the frequency and intensity of this excitation is constantly changing. These changes determine the intonation of the speaker. If in contrast the excitation is continuous, with a constant frequency and intensity, we experience the solemn intonation of a priest at prayer. . )or 2 C 1 , but also a third 1 from which sound difference frequencies and 2 ). The impulse frequency of direction , point 2). ). This yields a script whose eech along with a Description of 11.12 12.91 s. 2 . It contains apparatus to produce tones 12 . Difference frequencies can be readily : Representation of this vibration curve in the RENDELENBURG 1 T ) (cf. Sect. 2 Right ). 2 EMPELEN 1 : Vibration curve of the vowel ‘a’ as sung by a male voice 12.19 .K ERDINAND Left ). At frequencies of several 1000 Hz, differences in the 12.92 OLFGANG V His book on “The Mechanism of Human Sp the frequency (2 note of frequency the larynx was 200 Hz. (recording by F and hissing sounds whichoperated could by be the activated hands by andmachines fingers. openings make and Newer use versions keys of of electronic such devices. speaking 6. At higher energy densities, the ear can hear It hears not only two notes of frequencies 8. With two ears,waves we are can arriving. recognize This works the bestonset for or tones with and repetitions noises with ofis a characteristic sharp the details. time The decisive differencethe factor right between ear the by excitation theFig. same of portion the of left the sound ear wave and curve (compare of demonstrated using organ pipes.tion Occasionally, frequencies” still can other be heard, “combina- e.g. the “sum frequency” ( a Speaking Machine” was published in Vienna in 1791 by J.V. Degen Publishers. 12 Figure 12.90 form of a frequency spectrum. The principal formant range is clearly visible. The time sequence of the languagecontinually elements, recorded i.e. on their tape spectra, can (Fig. be to be of the order of some 10 7. The rise and decay times of the ear are not well known; they appear irradiation power due to thea role shadowing (cf. effect Sect. of the head also play play with dolls‘mama’ which and ‘papa’. have – a AW speaking bellows machine was in described in their 1791 by torsos and can say spatial sequence of elements consistsread, much of like spectra. e.g. Morse code, This but script only after canSpoken considerable be practice. language can be reproduced by mechanical means. Girls Travelling Waves and Radiation 12 358 Part II 12.28 The Sense of Hearing 359

Figure 12.91 A time sequence of a number of spectra (around 200) corresponding to counting from 1 to 3 in English. The spectra, which follow each other from bottom to top in a dense series, each have a height of only a few tenths of a millimeter in the image. The amplitudes are not represented as in Fig. 12.90 (right side) in terms of ordinate values of different heights, but rather as the darkness (density) of a photographic plate. Part II

Figure 12.92 Directional hearing Hold the two ends of a piece of hose around 2 m long in your ears and let an assistant tap on the hose near its midpoint. The apparent direction of the sound deviates from the medial plane of the head when the position of the tapping is more than 0.5 cm from the C12.26. At the left: midpoint of the hose. Our sense of hearing thus reacts to difference in the E. MOLLWO, Dr. rer. nat. arrival times of sounds of t D 3105 s. At t D 60105 s (corresponding (Göttingen 1933); at to 20 cm path difference, roughly the diameter of the head!), we localize the the right: H. KELTING, apparent sound source at right angles to the medial plane.C12.26. Dr. rer. nat. (Göttingen 1944). in Hz, Just 3 min I bright- (12.54) (12.53) 10 = /: I (compare irradiance D loudness L illumination 2 sources, as being ;seebelow . This large range 12 reference 12.93 are termed sound , but rather as a relative unit: W/m 2 eye I sources, to be equally bright. min I I P W , i.e. on our ears and eyes. (i.e. the incident radiation power log light . (This corresponds roughly to the A 2 10 D W/m L 12 (or sound level). It corresponds to the 10 sensory organs phonometry applies only to observers with normal ), even from quite different Receiver area 2 A two irradiance strengths of light, even when they orig- D irradiance strengths loudness energy current, measured in watt), but rather according depends on the frequency as seen in Fig. Incident radiation power in absolute units, e.g. in W/m D min two metry, one does not try to quote the irradiance strength (in- I D I min . I I . perceive in the perception of light by the eyes. The loudness cannot be phono /receiver area P strength so that they can be reproduced at anIn arbitrary place and time. tensity) practice would yield numbers between 1 and 10 typically for a younger person, ca. 20 years old). The ratio is made more compact byis using deﬁned decimal as logarithms. The quantity in multiples of an agreed-upon, small detection threshold of the human ear at the frequency 12.29 Phonometry Phonometry in acoustics corresponds toevaluate photometry a in type optics. of physical Both ergy/time radiation not according to its power (en- to its effect on our as photometry holds only for observers with normal sight Vol. 2, Chap. 29), hearing All sound perceptions, i.e.their notes, quality tones, ofquality, ‘pitch’ noises their (high, etc., have, low, hollow, besides brilliant etc.) a second equally loud. As is well known, the eyeIt is can capable of a corresponding comparison. inate from quite different types of W The irradiance strengths as judged by the ness physically measured, i.e.some it physical cannot unit, be notional determined quality more as of than a our sensory brightness multipleperceive organs. or of However, with any our other ears, we percep- can levels (cf. Vol. 2, Chap. 29). This ability ofa our phonometry sensory (sound organs measurement)surement) for and forms technical, a practical the purposes. photometrydetermine basis Both (light two make spatially for it mea- or possible setting temporally to separated up irradiance strengths Travelling Waves and Radiation 12 360 Part II 12.29 Phonometry 361

The loudness is thus a pure number. One combines it with the number 1 as multiplier and gives this number the name phon (cf. the end of Sect. 12.24C12.27). One thus says for example that the loudness C12.27. The unit phon is not of everyday speech corresponds to 50 phon. This means that ev- included in the SI (cf. Com- eryday speech has an irradiance strength as perceived by the ear of ment C2.14.). It is a unit 5 7 2 5 I D 10 Imin D 2 10 W/m (10 log 10 D 50). – The range of the subjective quality of loudness stretches from 0 phon (the level of the barely percepti- “loudness”, used e.g. by the ble) up to 120 phon (the noise level in a boiler factory or next to an American National Standards airplane, sometimes termed the threshold of pain for the ears). Institute and the international standards organization ISO

From the defining equation (12.54), it follows that if the irradiance strength (see http://www.iso.org/iso/ Part II as perceived by our ears is multiplied by a factor of 10, then the loudness home.html). It is defined as increases by 10 phon, since 10 log 10 D 10 1 D 10. – Examples: “the dB sound pressure level One very softly ticking clock has a loudness of 10 phon, ten such clocks (see Comment C12.28.) of together have 10C10 D 20 phon. – One roaring motorcycle has a loudness of 90 phon, ten of them together have 90 C 10 D 100 phon. a 1 kHz tone which sounds just as loud”. The spectral sensitivity distribution of the ear is illustrated by Fig. 12.93. The ordinate values show both the sound pressure amplitude p0 and also the irradiance strength I. The values of the irradiance strength or intensity apply to a cross-sectional area of a free sound field, not disturbed by a human head, for a wave which is incident perpendicular to the face of the hearer. With increasing loudness, the spectral sensitivity distribution changes; the curves become flatter. With a further increase, instead of hearing, the observer feels pain. In the range of their greatest sensitivity, our ears react to

2 Loudness W/m N/m2 2 phon 10 2 =^ 2 10 120 2W/m 0 p 1 10 Δ 10–2 80=^2∙10–4W/m2 1 –4 10 10–1

–6 –8 2 –2 C12.28. Today, loudness Intensity 10 40=^2∙10 W/m 10 is usually quoted in dB –8 10 10–3 (Eq. (12.51)). For this mea- ^ –12 2 sure, instead of the individual 10–10 0=2∙10 W/m –4 Sound pressure amplitude, 10 curves as in Fig. 12.93,which –12 10 10–5 depend on the age of the hearer, ﬁxed conventional 20 50 100 200 500 Hz 1000 5000 2000 frequency dependencies

Frequency 10000 20000 (so-called ‘A-weighting’, ‘B-weighting’, etc.) are used, Figure 12.93 The curves of spectral sensitivity of the ear for different val- and they are indicated by ues of the loudness. Along each curve, the loudness is perceived as the same as for a pure comparison tone ( D 1000 Hz) which is produced with an adding corresponding letters, irradiance strength (or sound pressure level, i.e. a physical stimulus!) corre- e.g. dB(A). For more details sponding to the given loudness. The bottom curve is the detection threshold on A-weighting, see for ex- of the ear of a young person (20 years old). With increasing age, the curves ample https://en.wikipedia. rise more sharply above 2000 HzC12.28. org/wiki/A-weighting ’s organ, 3000 Hz, the between the ORTI m; it is thus only accomplished R 10 ,i.e.avaluewhich 10 . Accordingtothis over a range of 1 2 I 13 N/m 5 10 takes place in C D ear 0 sable; one can still hear without an p ndolymph with its given energy density octave of the electromagnetic spec- takes place in the mosaic-like retina. It one eye each the e shows vividly just how astonishingly adaptable our ears are: 12.93 times smaller than the normal air pressure! . Both of these sensory organs behave, put succinctly, as though they 10 12 , without hindrance and without reflection losses. This is were measuring instruments with logarithmicadapted scales. to Both their are respective wonderfully purposes:as The well ear in for example a can regionas hear when of nearly the weak, waves diffracted, are freely reflected propagating. or scattered sound waves 10 by the eardrum and the lever system of the ossicles ı concept, the eardrum and thethat ossicles live would exclusively be in unnecessary water inanimals (dolphins has mammals and an whales). outer ear. Indeed, Thehave none degenerated auditory of canal, to these the mere eardrum remnants. and the ossicles The eardrum and the ossiclesner fulfill ear is merely filled the with following a purpose:800 watery times liquid, The greater the than in- endolymph, that of and air. its As density a is result, about the reflectivity Figure In the frequency range of– its can greatest deal sensitivity, with the variations in ear the – irradiance just strength like the eye air and the innerthe ear sound would wave be should nearly r 1.0 (cf. Eq. (12.50). Now, however, At the hearing threshold and the most favorable frequency, within the cochlea.To The first basilar order, membrane ittween is can two a be rigid part tubes described (the of as scala this a tympani organ. and delicate the separating scala wall vestibuli). be- In is 10 a few atomic diameters. In thea inner factor ear, the of amplitudes 30. are still smaller by at least trum. Nevertheless, weun-colorful can hues. distinguish Furthermore, a the multitudeis acuity of of not colorful the compatible and images with thatfects we the cannot see quality be understood ofif physically. the central lens They processes can of withinrole. be the the produced brain eye. only play an Both important ef- participatory The nerve excitation of the inner reacts practically to only amplitude of the motions of the eardrum is only about 6 13 Nerve excitation in the 12.30 The Human Ear The most essential part of ouris auditory a canal bony is labyrinth the containing “innerthe ear”, a petrous which bone. spiral organ The mechanical (the (sound)two cochlea) waves are paths: lodged directed in 1. to it through via middle the ear eardrum (ossicles); (tympanum) and and 2. thethe bones through head. of the the soft The tissue first and path the is bones dispen of changes in the air pressure of eardrum and the ossicles. Travelling Waves and Radiation 12 362 Part II 12.30 The Human Ear 363 humans, it is 34 mm long, and its width increases from the beginning of the tubes to their ends from 0.04 to 0.5 mm. HELMHOLTZ interpreted this membrane as a tiny vibrating-reed frequency meter, a spectral apparatus which is characterized by its particular simplicity (Sect. 11.12). He thought of the membrane not as a homogeneous band, but rather as a dense series of stretched strings, grown together at their ends (in place of the vibrating reeds or leaf springs). Their resonance frequencies were presumed to decrease along the length of the membrane, i.e. with increasing width. In reality, the basilar membrane lacks such a structure, but nevertheless it retains the prop-

erties of a spectral apparatus. Its function can be demonstrated using Part II a model which was designed for that purpose. It fulfills the necessary condition for demonstration models: The processes to be observed take place sufficiently slowly. The model is sketched in Fig. 12.94. Two metal channels with a rectangular cross-section are fitted at their sides with glass walls, and between them in the middle is a highly elastic partition, the artificial basilar membrane. Its width is delimited by a wedge-shaped metal frame. The width of this frame increases from left to right. For reasons of “mechanical similarity”, the membrane must have elastic properties which cannot be realized with a solid body; instead, we have to make use of an interface between two liquids of differing density and surface tension, e.g. benzene above and water below (with an added salt, MgCl2, to increase its viscosity). – The “stirrup” (one of the ossicles) at the lower left end can move the “oval window” back and forth in a sinusoidal motion by means of an eccentric, at frequencies between 1 Hz and 8 Hz. The vibrating oval window becomes the source of a wave group which travels to the right along the “basilar membrane”.

C12.29. Similar investigations were reported by H.G. DIESTEL (Acustica 4, 489 (1954)). There, he mentions an unpublished thesis by H. DORENDORF,inwhich similar model experiments were described (1st Physical Figure 12.94 A linear model of the cochlea in the inner ear. The brass plate at the right end is removable and has a cork gasket. Instead of the “round Institute, Göttingen (1950)). window”, a glass tube, bent upwards, is used. The artiﬁcial basilar membrane It may be that the measure- is 31 cm long and its width increases from 1 to 18 mm. The “stirrup’ is caused ments quoted here are at least to vibrate by an eccentric attached to an electric motor. In the human ear, the partially based on this thesis stirrup is the last of the three ossicles, the small bones which act as levers to (which has in the meantime reduce the displacement between the eardrum and the oval windowC12.29. been lost). , c - c ,we . a - a a b 4Hz. To 15cm , the next 12.96 b D - b tlines were visible. c b : The wedge-shaped mem- lling along the “basilar mem- ; not identically to a vibrating- (numbered 1 to 8) show photos 12.96 her wave group. – Thus, in spite 10 lling-in can be seen in Fig. 12.95 a 4 Hz), it reaches its maximum amplitude D spectral apparatus Distance marks (from the “oval window”) on, at the left of the dashed straight line Wave groups which are trave 0 c 5 b

2 2´´ 2´ 4´´ 4´ 4 3´´ 3 3´ 6´ 7 7´ 6 5´´ 8 8´ 5´ 5 1´´ 1 1´ Time in units of one-eighth of a period a of one-eighth of units in Time brane” of the ear model after being sinusoidally excited at provide a clearer overview, thea waves have black been color; emphasized in byWave trains ﬁlling the without with original this photos, subsequent ﬁ only their sharp ou (‘instantaneous images’) taken at equal timecillation intervals period. during one At os- the right, above the dashed straight line Figure 12.95 The eight upper lines in Fig. we see the beginningof of the still sinusoidal anot excitationbrane does of not carry the a ovalgroup. sine wave, In window, but this the rather example ( a basilar peculiarly-shaped wave mem- at the 10 cm distancethe mark. process, its Further group to velocity the decreases.it right, has At it the become decays 13 practically cm rapidly. zero. distancebrane” In Further mark, remains to the completely right, at the rest. “basilar mem- Now we come to a second,or decisive experimental distance result: mark The where position thetude wave depends group uniquely attains on itsbe the maximum seen frequency in ampli- of the original the photos excitation. in Fig. This can wave group appears and follows a similar path. Below the line brane thus acts as a reed frequency meter, but in a similar manner. can see the remainderFrom of line the 1 wave group from the previous period. Travelling Waves and Radiation 12 364 Part II Exercises 365

Distance marks (from the “oval window”) 5 10 15 cm 20

6.3 4.0 3.5

Frequency (Hz) 2.7

Figure 12.96 ‘Instantaneous images’ of wave trains on the model basilar membrane at the moments when they reach their maximum amplitudes, for four different frequencies of the sinusoidal excitation. The wave groups re- Part II main similar to each other. They are simply stretched longer with decreasing frequency. The group in the second line corresponds to the group in the fourth line in Fig. 12.95; its wave crest has however not been ﬁlled in with black color in this image.

What it accomplishes can be described physically only as a preliminary decomposition14. The high resolving power which is in fact observed in human hearing cannot be understood physically. As in the case of the eye, we ﬁnd also for the ear that central processes, which take place in the brain, must play a major role. These remain today outside the realm of our scientiﬁc understandingC12.30. The so- C12.30. For further reading lutions to the great problems in biology and neurophysiology will on the subject of hearing, probably be found only in some still-distant future. see e.g. E.A. Lopez-Poveda, A.R. Palmer and R. Meddis, The neurophysiological bases of auditory perception.New York: Springer (2010); ISBN Exercises 978-1-4419-5685-9.

12.1 A locomotive is approaching an observer, who is not moving, with a velocity of u D 72 km/h, and emits a whistle tone of frequency D 500 Hz. What frequency 0 is heard by the observer? (Sect. 12.2)

12.2 A plane wave on water strikes a wall at perpendicular incidence. In the wall are two slits with a spacing a, whose width is small in comparison to the wavelength of the water wave. Describe the interference maxima at a distance y a for small deﬂection angles. (Sect. 12.3)

12.3 A planar sound wave strikes a screen that contains four small apertures. They are arranged at the corners of a square whose sides have a length a. Where are the interference maxima with the smallest distance from the symmetry axis? (Sect. 12.3).

14 A preliminary decomposition selects from a mixture of waves of various lengths those that belong together in broad regions. One often uses “filters” for this purpose, e.g. in optics, a glass filter that passes only red light. . . D of D t d with E 12.20 12.66 c from the ) on which a in a glass have indices l 56 Hz (Radio c 12.25 12.57 . Determine the d D is reflected from B .Howlargemust B C b and 03 cm is incident on a cu- : of an auto that is travelling 1 Hz and u 3cm,asshowninFig. 9 D ) 10 D d and modulus of elasticity km/s). In deriving the answer, it is 1 of the prism in Fig. 1440 m/s. Determine the depth The online version of this chapter (doi: 5 12.17 3 ) D ) 10 D max c and ) ˛ . Compute the group velocity 3 n 12.15 d 6g/cm 12.20 : D 8.3 2 c C be, i.e. the spacing between two neighboring 12.18 ) n D D ) 12.17 % . (Sects. and 2 effect and is superposed with the incident wave, so n 12.22 ) . The frequency is N/mm u The frequency range of the human singing voice stretches In order to measure the velocity Determine the angle Two water waves of wavelengths Determine the longitudinal sound velocity A planar sound wave is incident at right angles on a diffrac- The sound pulse of an echo-sounder travels in a time A sound wave of wavelength 4 12.20 OPPLER 1, 2, 3 and 4. (Sect. 10 D 15 s from the surface of the sea to the sea bottom and back. The 5 : : which the pulse formedgates. (Sect. by superposition of the two waves propa- from about 80 Hz (bass)wavelengths. The up velocity to of 800 sound Hz is (soprano). 340 m/s. Find (Sect. the range of 12.11 rod of density useful to begin with theplane mathematical waves, superposition moving of in two opposite travelling helpful directions; Comment for C11.4 this. can (Sect. be waves and electromagnetic wavesVol. 2, in Chap. general 12. are Here, we treatedat need the only in know velocity detail that of in light, in air, they propagate velocity 0 the water. (Sect. 12.7 6 12.8 12.10 (Sect. that the observer measures beats with a frequency the incident sounddex wave of undergoes refraction total which reflection, was using found the experimentally in in- Sect. the lattice constant slits in the grating? (Sect. 12.5 grating, the interference maxima have a spacing Electronic supplementary material 10.1007/978-3-319-40046-4_12) contains supplementary material, whichable is to avail- authorized users. tion grating with narrow slits. On a screen at a distance 12.4 12.6 the auto. Thethe frequency D of the reflected wave is increased due to velocity of sound in water is towards an observer, a radio wave of frequency Find the angle under whichm diffraction maxima occur for the orders bic lattice with a lattice constant of 12.9 of refraction Travelling Waves and Radiation 12 366 Part II Thermodynamics III 367

Part III vrdylf oti neteeylrenme fidvda atoms. the individual example, of For number in large encountered extremely amounts an The contain life atoms. everyday of composed are substances All aldthe called min topei rsue h he quantities three The pressure. atmospheric ambient atri rsn nyi h oi om ri nylqi rgaseous, or liquid only is or form, solid the thenwesaythatitisina in only present is matter rsn ohi iudfr n sags hni a w hss All phases. mass two their has it to then addition gas, in a as have, and substances form liquid in both present perature nGra,i smr opcl xrse as expressed compactly more (SI). Units is of System it International German, the of In part is term awkward rather Fundamentals tt hc hrceie h nebeo l h molecules. the the only all measure of can ensemble We the molecule. characterizes single which a state for defined can- velocity, be temperature or and even Pressure position not exact direction. its its them nor of magnitude one its any neither for spec- time cannot given We any individuals. at single ify of fate the and behavior the about particles individual of numbers enormous molecules. concept liq- the of use amounts will definite we for gases, and bodies and solid uids for term collective a tedious. As and time-consuming eval- is quantitative results a their and of uation complicated, of- setups This in experimental insulators. thermodynamics, makes perfect ten no In are there required. electromagnetism, to usually contrast are measurements se- of tedious and ries long instead, all of experiments; clear-cut for areas straightforward, other valid structure the qualitative are to its laws contrast physics, important in Unfortunately, most of phenomena. Its natural branches all technology. for and importance science fundamental of is Thermodynamics Definition Remarks. Preliminary 13.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © fara omtmeaueadpesr is pressure and temperature room at air of hs nec ui ilmtro i,teeaetpcly2 typically are there air, of millimeter cubic each in Thus, N T V ipesaevariables state simple n hyaesbett oepressure some to subject are they and , D fSubstance’ of ‘Amount Concept the of hroyaisi h n el ihtebhvo of behavior the with deals end the in Thermodynamics Number ubrdensity number N olsItouto oPhysics to Introduction Pohl’s fmlclsi h volume the in molecules of igephase single ol e m lu Vo or antb eonzddrcl from directly recognized be cannot uniiso state of quantities V ftemte,eg ae,is water, e.g. matter, the If . muto substance of amount tsy ohn whatever nothing says It . M N ,avolume V Stoffmenge O 10.1007/978-3-319-40046-4_13 DOI , p D o xml the example for , . V V 2 , : 5 p V and 10 )I the If .) : ,atem- 5 .(This 25 (13.1) T = 10 m are 16 3 . 13 369

Part III ;andfor . Guided ) and gases time ). 131 °C). cold is thus a mea- 13.2 appears reddish n > 2.2 2 of the objects. The means that it is to be : : A 1 N C13.1 n mol (unit: kg/mol) and the molar D 23 . In thermodynamics, a fourth n temperature . = N just like other physical quantities, 10 M mass or quantity as the base quantity, namely D 022 : constant: /mol). . m 6 3 A one N M N D and comment C2.2 in Sect. temperature C. The amount of substance A D 12 of particles: was defined as a physical quantity in 1971, as N 3.3 n , the other with the perception of N (unit: m VOGADRO n = (Video 13.1) V warm ); bimetallic strips become curved (Fig. D of Temperature C13.2 is the A m 13.1 V A . In kinematics, a second quantity is needed, the N a unit (see Sect. at lower temperatures and yellow at higher ones ( wires become longer and stretched(Fig. rubber bands become shorter expand Quantities which refer tolar”, the e.g. amount the of molar substance mass are termed “mo- This definition of the amount of substance Here, volume used in physical-quantity equations in particular for quantitative statementsand always as a numerical value sure of the number by these sensory organs,to we their can ability order to objectsgin call of in these forth a perceptions a series is called according feeling the of ‘warm’ or ‘cold’. The ori- temperature, qualitatively defined in“cause” this of numerous way, other isdependent phenomena, useful of our including also perceptions. many Changes as that inchanges the the in are temperature in- also cause In the skin onbranes, the besides surface pressure of and oursensory pain bodies organs. receptors, and we in One have type someperception another of mucous sort these of mem- of reacts to external stimuli with the quantity is required, the 13.2 The Definition and Measurement In geometry, we measure In order to permit a simpleto description first of relationships order which on depend theamount number of of substance particles involved, thean additional concept base of quantity the with1 the mol base of unit a “mole” substance (symbol:atoms contains mol). in exactly 12 as grams many of particles as there are length dynamics, a third quantity, the 1. the dimensions of objects. With increasing temperature, metal 2. the absorption of light. Thus, for example, HgI ount . .For 1 T V

K physicists thermal 1 5 V 1 3 did not include 10 is “the number of n is negative. l OHL T 23 ˛

: 1 l 1 Fundamentals D D 13 rubber, Video 13.1: “Model experiment on thermal expansion and evaporation” http://tiny.cc/nggvjy The thermal expansion of a solid body as wellmelting as and its evaporation are demonstrated in a model experiment. C13.2. This thermal (length or volume) expansion is described by the expansion coefficient ˛ C13.1. P of substance in his textbooks, although he was in other cases very open to accepting new definitions of quantities and units. Here, however, he “saw no advantage inas it”, he wrote in thethe preface 12th to edition, and hetinued con- to use the mole“individual as unit an of mass”. – We have nevertheless decided to introduce this convention into the text, as it hasbeen now in effect for more40 than years. We note however that this quantity is still not familiar to many this definition of the am and is not very carefullyfined de- in most textbooks. One finds again and again for example the incorrect statement that It can itself depend upontemperature. the For iron, for example, at room temperature, ˛ moles”! 370 Part III 13.2 The Definition and Measurement of Temperature 371

Figure 13.1 A stretched rubber band (length 50 cm) draws together by about 3 cm when it is heated to ca. C90°C. We must increase its load from 2 up to 2.2 kg in order to restore its original length.

Figure 13.2 Bending of an electrically- heated bimetallic strip (at the left seen from the front, with the strip lying in the plane of the page; at the right seen in proﬁle). It consists of two layers of sheet metal, made of a nickel-iron alloy, which Part III are welded together; one of the strips contains 6 percent by weight of man- ganese. The strip has been slit so that it can be heated by an electric current.

3. the electrical resistance of metals, which increases linearly with temperature at not-too-low temperatures. 4. the electric voltage between two different metals which are in contact (Fig. 13.3: the thermocouple).

This list could be continued almost indefinitely: The majority of all physical and chemical phenomena show some dependence on the temperature. Each of these could be used to define a measurement procedure for the temperature and to construct a measuring instrument, called a thermometer, for the temperature. In daily life, we often use the volume change of a liquid,asinamer- cury thermometer, with a temperature scale suggested by the Swedish mathematician and surveyor A. CELSIUS in 1742. It contains one hundred divisions between the temperature of melting ice and that of boiling water, and is familiar today to every school child. Hg thermometers can be used between C800 and 39 °C. For lower temperatures down to 200 °C, thermometers filled with pentane are used.

Hg thermometers for use up to 300 °C are evacuated; at higher temperatures, vaporization of the Hg is prevented by filling the thermometer tube above the liquid with nitrogen gas at pressures up to 100 times atmospheric pressure (107 Pa). – Addition of a small fraction of thallium allows mercury to remain liquid down to 59 °C, so that it is usable for temperature measurements down to this temperature. )is,for 13.5 e values known as “fixed points”. , or also resistance thermometers. . 13.3 15.6 . It shows the scale of a Hg thermometer of the 13.4 An electrical ther- ). The gas thermometer belongs among the most important is presumed to be the “International Temperature Scale of Compare also Fig. . T 19.6 C13.3 mal contact with the objectmeasured, to while be point 2 isa held reference at temperature of 0for °C, example using an ice-water bath mometer for demonstration experiments. It consists of two wires, one made of silverother and of the Constantan, which aredered sol- together at the pointsand 1 connected and to 2 a voltmeter.contact The point 1 is brought into ther- Figure 13.3 defined by the volume expansion ofarbitrary liquids choice are of seen to the depend substances onused (e.g. the for Hg), the and tube. on the type ofThe glass temperature defined withsufficiently gas low thermometers gas (Fig. densities,of practically the independent gas of used.temperature the should nature however A be conceptually notmentally fully-satisfactory only independent of practically, definition the but of substance also usedgoal the funda- in the was thermometer. This attained bySect. the “thermodynamic temperature scale” (see All thermometers are calibratedaccurately today reproducible using temperatur legally determinedThese and fixed points (meltingdeveloped points, with vapor a pressures, considerable amount etc.) ofing have tedious aspects effort. been are important The for follow- this work: A quantitative definitionliquid-expansion of thermometer is the – temperatureplicability in – spite using not of completely a its satisfactory.referring great This Hg to practical can Fig. ap- or readily beusual other seen technical by design, andmometer to which has its been left, calibratedonly the with the scale its divisions aid. of on In anwhile the the alcohol scale those range ther- of of shown, the the Hg alcohol thermometer thermometer are are uniform, not. The temperatures Besides liquid thermometersranges, or electrical outside thermometers their aremocouples limiting as often shown temperature used. inFor Fig. These even can be higherusing ther- temperatures, “radiation thermometers” optical playChap. an temperature 28). important measurements role (see Vol. 2, atures 1990” (ITS-90), which contains fixed points ranging from 0.65 Kelvin measurement instruments which cantemperature be scale used based on to the establish thermodynamic a scale. practical The currently best representation of the true thermodynamic temper- ). 15.6 ition, so that their dif- Fundamentals 13 are employed; they are more convenient to use. Today, for most demonstration experiments, electrical resistance thermometers with digital temperature indication couples are connected in oppos ference voltage is measured (as in Fig. C13.3. The two thermo- 372 Part III 13.2 The Definition and Measurement of Temperature 373

Figure 13.4 Right: The scale of a mercury thermometer with uniform divisions between 0 °C and 100 °C; at left: the divisions of an alcohol thermometer which has been calibrated using the mercury thermometer Part III

Figure 13.5 Schematic of a constant- volume gas thermometer. By adding mercury, the volume of the gas can be held constant and the pressure can be read off the height h of the mercury column. From the pressure, the temperature of the gas can be calculated according to Sect. 14.6 (cf. Sect. 14.9, Point 3)

up to the highest temperatures that can be measured by applying PLANCK’s radiation lawC13.4. The unit 1 kelvin (K) has been defined C13.4. A more detailed as the 273.16-th part of the temperature of the triple point of water. treatment can be found in: In daily life, we use the CELSIUS temperature scale. Its relation to W. Blanke, Physik in unserer the KELVIN scale is Zeit 22, 13 (1991). English: t T see bipm.org. In the year D 273:15 2000, a further international ı C K temperature scale (PLTS- (273.15 K is the melting temperature of ice under atmospheric pres- 2000) was adopted, which sure). extends the scale to the range of very low temperatures Science and technology have spoiled us by providing readily usable (from 1 K down to 0.9 mK). and accurate thermometers, no less than by e.g. the construction of precise clocks and electrical multimeters. Nevertheless, the fundamental question of the measurement procedure and the calibration of the instruments should not be neglected; otherwise, one could easily overlook the great efforts that have been expended in the past to develop reliable measurement technology. 1 T . .The (13.4) (13.2) (13.3) C13.5 2 which is M 2 /; . This can to describe 2 2 T and c 1 T M T / . and 2 2 c , 1711-1753) 1 2 T c M / : ), we write 2 M 2 2 , which lies between c T M T C D 13.3 1 C 2 c / , top). – Let the temperatures 1 1 1 : M T ICHMANN T 2 2 M M c . . Q of different composition and ini- 13.6 D changes in temperature D D T D cM . 1 2 2 , and their masses 1 1 T 2 T T D (G.W. R Q 2 2 M T 1 c Q 1 M 2 c which is given up by the warmer body 1 M M and C 1 1 The same process c 1 1 D C Q T T Performing mechan- , and instead of Eg. ( 1 / 1 T T M 1 c heat Center: 1 The flow of heat from , then after the transfer, 1 2 M T T Bottom: . 1 . Top: The heat of Heat and Heat Capacity D M M 1 hold. Rather, we require two factors 1 c T . , will be observed. It is however not simply a mean value; the 1 2 not T T > 2 we use the name For the product two bodies are broughtical into transformations close nor thermali.e. phase contact. solids changes are are Neither presumed chem- After assumed a to to certain remain time, take a solid, ‘mixing place, liquids temperature’ remain liquid, etc. the process, and we must write does taken on by the cooler body 2 or, rearranged, In words: relation when thermal contact is established is equal to the heat Q T Figure 13.6 the temperature between two bodies tially at different temperatures (Fig. of the two bodies be and 13.3 The Definitions of the Concepts The concept of ‘temperature’all the by processes itself associated isalready with be not seen sufficient from to a describe simple example, namely the equalization of a hot to a cooltact gas is container established. when directwhen con- the two containers area connected metal by rod ical work transfers energy fromhigh a pressure gas to under a gasinitially at lower pressure; if pro- ’s rule”). In ICHMANN Fundamentals ICHMANN 13 1753, he was fatally injured while carrying out an experiment with lightning. posed this rule in 1747/48 (“R C13.5. R 374 Part III 13.3 The Definitions of the Concepts of Heat and Heat Capacity 375

The word heat is unfortunately used in different ways. Only in special cases does it refer to a particular form of energy, just like the words potential, kinetic, electrical and magnetic energy. Most often, the word ‘heat’ characterizes that special type of energy which can be transferred from a particular quantity of matter to another quantity of matter: This is a transport process which is driven by a temperature difference alone, without any other contributing factors. We refer to such a transport process for short as thermal. – In the special case mentioned above, one refers to heat as the kinetic part of the internal energy (cf. Sect. 16.1). Heat transport can take place in various ways: either through conduction, when the two quantities of matter or their containers are in direct contact or via a thermally conducting material (Fig. 13.6,top and center); through radiation, if they are separated by a vacuum (or in practice often by air) (see Vol. 2, Chap. 28); or through convection (see Sect. 17.6). The italicized words “without any other contributing factors” will be illustrated by comparing two examples: In Fig. 13.6 (top), a container Part III of warm gas I is brought into direct contact with a container of cool gas II. Instead of the direct contact through the walls of the containers, a metal rod M (a “good conductor of heat”) could also be used to connect the two containers; cf. Fig. 13.6 (center). In both cases, the temperature T1 is initially higher and the temperature T2 lower. Heat is conducted from the container I to container II. In Fig. 13.6 (bottom), each of the gas containers has a sliding wall in the form of a piston. The two pistons are connected rigidly to each other by a glass connecting rod (a “thermal insulator”). The pressure of the gas in container I is higher than in container II.Aftersome sort of catch (not shown) is released, the two pistons will move to the right. In this process, T1 will decrease and T2 will increase. Energy is removed from the left-hand container, because the gas it contains is performing work; the energy is added to the right-hand container, since its piston, in moving to the right, performs work on the gas in this container. Here, a “process of work” is taking place, i.e. a “transfer of work”, which also leads to changes in the temperatures. This example illustrates that heat and work (or energy) are essentially similar. – Likewise, when a quantity of matter is heated through friction, mechanical work is being performed1. In practice, heat transfer processes play an important role both in the kitchen and in the laboratory. We could think for example of an electric immersion heater in water. The container is presumed to be thermally insulating,i.e.itshould allow no thermal transport due to temperature differences between its contents and the surroundings. Thermally well-insulated vessels have double walls made of a poorly- conducting material such as glass. The space between the walls is

1 Or else electrical work, when an electric current is passed through a resistive conductor and thereby heats it through ‘electrical friction’ (cf. Fig. 13.2, and Vol. 2, Sect. 1.12). c .The (13.6) (13.5) .The specific T Ã M Ws specific heat 6 12 84 02 0 2 9 kmol : : : : : : , the factors in a substance 5 9 6 10 f 13 30 L 10 Â Q IUt , and the walls are D with the mass Ã E n was recognized as early as . Then, as the Ws ; : kg 5 . In retrospect, we may never convection T 047 247 17 26 34 036 T 10 : : : : : : . f 2 0 5 1 3 Specific and molar latent heatmelting of l 4 Â Q energy Q They are valid in every case only AYER n . C13.6 ttles”), which can be found today in M ) M and D D 5.2 Ã heat capacity C13.7 is thus a unit of energy, e.g. joule (J) or C c K as far as possible. We will often make use heat some other quantity. Suppose for example Q OBERT 2 5 5 7 : : : : kW s 24 24 26 49 75 kmol C 132 Â radiation is called the referred to T increases the temperature of the substance by lists some values molar heat capacity = Ã (often simply called the “specific heat”), we define Q Q K . 13.1 897 385 128 85 69 16 kW s Q : : : : : : kg Specific and molar heat capacity at 20 °C c 0 0 0 0 1 4 Â capacity Table be able to fullyToday, the appreciate similarity the of achievementssince heat been of to considered scientific to the be pioneers. course. other “self-evident”; forms it The has of unit become of energy a heat matter has of long within a limited temperature range. and as the 1842 by the physician R watt-second (W s) (see Sect. evacuated to prevent heat transfer by ‘silvered’ (with a thin filmheat of transfer silver, by copper or aluminum) to reduce of such containers (“Thermos bo almost every household. The essential similarity of Now that we have introduced the quantity of heat also acquire their physicalheat, significance. i.e. heat They refer to the quotient that an electric heater produces the energy which contains the amount of substance energy transfer is thermal,heat i.e. the energy is delivered in the form of ). Ã .For ,this 98 54 45 11 02 2 : : : : : : 13.1 kg m 78 18 26 63 58 kmol quanti- 13.1 Â 207 Molar mass M amount of , vice versa M molar (cf. Sect. Specific and molar heat capacities and heats of melting of some solids and liquids n quantities, i.e. quan- Mass Fundamentals D 4.2 J. m 13 Lead NaCl Benzene Water Aluminum Copper Substance ties referred to the amountsubstance of and the values in Table can readily be verified. –numerical The value of the molar mass was previously termed the “molecular weight” (see also comment C16.1). Table 13.1 C13.7. “Molar” quantities are referred to the substance n C13.6. The heat unit “calorie”, historical but still used in daily life, is given by 1 cal Making use of the “molar mass”: M tities referred to the mass of a substance, can readily be converted to specific 376 Part III 13.4 Latent Heat 377 13.4 Latent Heat

C13.8 (JOSEF BLACK, 1762) . In our experiments thus far, the sub- C13.8. J. BLACK (1728– stances have undergone no kind of transformation. Solid bodies re- 1799) carried out investiga- mained solid, liquids remained liquid and gases remained gaseous. tions on thermal equilibrium The composition of materials also remained unchanged, both their and discovered, indepen- chemical composition as well as their crystal structures. We now re- dently of J.C. WILCKE,the lax this restriction, and allow phase transitions to take place. Then concepts of latent heat and a substance can accept or release energy in the form of heat without specific heat. changing its temperature. In this case, the heat transferred is called latent. We offer two important examples: 1. Specific heat of vaporization and condensation. In Fig. 13.7,we see a container which is partly filled with water and then evacuated. A manometer is attached to the container so that we can observe the pressure inside it, and also an adjustable spring valve. In addition, the container is equipped with an electric heater, insulated from the outside world. After switching on the heater current, we observe that the water be- Part III comes warmer, and with its increasing temperature, its vapor pressure also increasesC13.9. The vapor is continuously in contact with the C13.9. Vapor refers here to liquid water and is in equilibrium with it. The resulting partial pres- the gaseous phase of water. sure of the water vapor is termed the saturation pressure or the vapor pressure. Numerical values can be found in Fig. 14.3. At a certain pressure p, the spring valve opens and the vapor escapes continuously. From this moment, the temperatures of both the water and of the vapor remain constant, and the two temperatures remain equal. – Deduction: The vapor which is escaping must be continually replaced; water must be constantly passing from the liquid to the vapor phase. The heat Q which is being passed into the water is consumed by the process of evaporation without any increase in the temperature, i.e. it is taken up as latent heat. The amount of water which has evaporated, measured in terms of its mass M, is found to be pro-

Figure 13.7 The measurement of the heat of vaporization (H D electric heater inside the insulation). The manometer M indicates normal air pressure when its connecting tube is open to the room air. It thus measures the entire pressure of the vapor, not just its excess pressure over the normal air pressure. . (13.7) (13.8) – In the next . to it. Then we Q ; : C13.10 The specific heat of heat of vaporization of the substance which is often used. It contains M to be the quantity 13 °C, vapor pressure at 18 °C 13.8 Q Q D Added heat Added heat of the evaporated liquid of the melted substance which has been added. Thus, we form the M M is equal to the specific Q Cl) shows measurements of the vaporization of water 5 Mass Mass H 2 Pa). The liquid is sprayed out of the small nozzle under its specific latent heat of vaporization D D 14.3 5 f v A cooling bottle with liquid l l 10 specific latent heat of melting 24 : 1 own vapor pressure.heat The of surface vaporization, struck sotain by that temperatures it the below is spray 0 cooled. has °Cis In in to used this the to provide way, laboratory. provide the onea a In piece can local of medicine, readily anesthesia black by ob- this paper, cooling. method breathe on (Demonstration: it Spray and observe how it becomes frosty.) Every liquid whichNumerous is types evaporating of takes coolingoratory, up the apparatus cooling heat are bottle sketched from based inliquid ethyl on its Fig. chloride (boiling this surroundings. temperature fact.D In the lab- Specific heat of melting and crystallization. chapter, Fig. at temperatures between 0 °C and 374 °C. a thermos bottle filled with coldthe water. water It in condenses there, the and process.the warms From temperature the increase, mass of wecondensation. the can condensed water calculate and the value of2. the heat of melting is determined in essentially theof same vaporization. way as We the determine specific the heat mass define the For demonstration experiments, water vapor (steam) is passed into has been melted by applying the quantity of heat and call it the Figure 13.8 The heat required forthe vaporization vapor can back be toheat regained a of by liquid. condensation condensing Apart from its sign, the specific latent portional to the heat quotient ethyl chloride (C f . for ’or va- l origi- 14.4 amount ’, which l latent heat ’ denotes and OHL v refers to the and is then used the sym- , e.g. the heat v l ), P molar 14.3 ,i.e.referredtothe OHL 13.8 ’ refers to the ’ here; we have used Fundamentals L r entire body ’ here, with a subscript 13 L termed the The subscript ‘ “vaporization”. The symbol ‘ of substance n the more relevant ‘ stands for “latent heat”. The symbol bol ‘ nally used the symbol mentioned. In Eq. ( also refer to the latentan heat of of melting of an ice cube,this and should be specifically (similarly to heat capacities; see previous section). It may porization mass of the body; compare also Sects. (specific) latent heat of C13.10 P the latent heat of melting (or fusion). We again use ‘ ‘ for “fusion”. 378 Part III 13.4 Latent Heat 379

Table 13.1 contains examples, along with the correspondingly- defined molar heat of melting Lf D Q=n, again at normal atmospheric pressure. The latent energy contained in the liquid can be completely regained when it solidifies. The heat of crystallization is, apart from its sign, identical to the heat of melting. For a demonstration experiment, sodium thiosulfate (Na2S2O3 5H2O), used as a fixing agent in pho- tography, is well suited as the substance to be melted or solidified.

The melting point of this salt lies at C48.2 °C. The melt can be strongly supercooled. It can be maintained for days at room temperature. If the melt is “seeded” with a small crystal, the process of crystallization begins, accompanied by a considerable rise in the temperature. This process can be used to evaporate ether and make it visible at some distance by igniting the ether vapor. – Technically, this transition can be used to manufacture heating pads. The salt is ﬁlled into a rubber pouch and melted by immers- ing in hot water. It will then maintain its temperature for some time at C48 °C (the “hold point”).

3. Heat of transition. In Fig. 13.9, a piece of carbon-containing sheet Part III iron (0.9 percent by weight of C) is heated by means of an electric current until it is glowing yellow. After the current is switched off, it cools rapidly and darkens. As it passes through a temperature of t 720 °C, it again ﬂashes brightly: at this temperature, it undergoes a phase transition – delayed by supercooling – from the form of iron called the phase into a less energetic mixture of carbon-free iron and Fe3C (cementite). At this transition, a considerable amount of heat is released.

Without the supercooling, the temperature decrease would simply be brought to a standstill for a short time: the “hold point” is characteristic of a phase transition.

Summary: Adding heat to a system can not only increase the temperature of some substance, but also can induce a transition between phases at constant temperature in the interior of the material. In both cases, energy is stored within the substance. All forms of energy stored within a material are termed “internal energy” U.Itis thus qualitatively distinguished from the potential and kinetic energies which the substance may possess as a whole. More details will be given in Sect. 14.3.

Figure 13.9 Demonstrating the heat of transition ). 13.2 The online version of this chapter (doi: (Sect. 1 K 5 10 is 1 ˛ Calculate the influence of thermal expansion on the movement of a pendulumtemperature clock increases whose by pendulum 20 K. is Fortreated made simplicity, as the of a pendulum mathematical steel, gravity may pendulum. if be expansion The the linear coefficient of Electronic supplementary material 10.1007/978-3-319-40046-4_13) contains supplementary material, whichable is to avail- authorized users. Exercise 13.1 Fundamentals 13 380 Part III h okcerb rvdn kth tcnb eni Fig. in seen be can it sketch; a providing for expression by this make clear to work want we the examples, previous with as Just ancntn uigtemto ftepso,a hw ytecurve the by shown as piston, re- 1 the not of motion does the pressure of during The non-uniformities constant occur. local main can no temperature that or slowly density so pressure, motion place The take work. to vol- mechanical pressing supposed its performs increasing is thus is right, and cylinder the process, to the a piston in the ume in displaces It contained piston. is a against which substance working A fIelGases Ideal of State of Equation the and Law First The ntemcaiso oi ois edfie h work the defined we bodies, solid of mechanics the In the discussions. and preparatory This provide thermodynamics. will of energy section Law of following law First the the of is summary quantitative that a conservation, obtain to is goal next Our Technical and Expansion of Work 14.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © usac.Frti ae h ocp of concept the case, this For substance. hycnpromwr nyb aigueo a of use making by only work perform can They ntcnlgclapiain,almcie okin work machines all applications, technological In oueb uhn h itnott h oiin1 nteprocess, the In initial displacement 1. its of position occupies work the to It the out performs piston cylinder. it the the pushing substance into by working pressure volume the cycle, constant the of at piston. (stroke) a flows interval and time valve first outlet the an and In inlet an with machine a of cylinder endfie.I sepandi Fig. in explained is It defined. been A rdc frei h ieto ftemto ie itnemoved”, distance times motion the of i.e. direction the in “force product h hddarea shaded the h rdc “pressure product the rags hnfrtewr,w obtain we work, the for Then gas. a or ::: d s W D ntefiue h oko expansion of work The figure. the in 2 D oueelement volume R Work F d s below (Sect. p h xaso curve. expansion the 5.2 . ie area times olsItouto oPhysics to Introduction Pohl’s d W V .Teforce The ). / C14.1 D , Z A 14.2 p hni seetdb liquid a by exerted is it when ” d V h pe mgsso the show images upper The . F p : 1 technical W a tefb xrse as expressed be itself can V 1 R D ntepso,ie the i.e. piston, the on 1 2 p O 10.1007/978-3-319-40046-4_14 DOI , d R V flowing work eidccycles periodic pA orsod to corresponds d s W W r since or, working techn sthe as (14.1) 14.1 has . . Fig. in substance working the e.g. input is which Energy vention: con- following the on based is quantities energetic of the Determining C14.1. hc sperformed is which W sion the while positive), are integral the in factors (both negative is d integral, the of limits reversed the to owing since positive, F W performs expansion of work the substance, working the to output is that energy the while tive, V D D o a h poiesign. opposite the has now 14.1 noapriua system, particular a into sngtv.Referred negative. is Z Z 2 1 14 stknt eposi- be to taken is , 1 2 oko compres- of work p p d d V V hc it which on tis it signs 381

Part III :the :Itis p V d d V p 2 1 2 1 R R D techn 2 2 W V V 2 p 1 p decreasing = const 2 1 p V = 1 0 p const V the expansion curve. The working substance alongside diagram for defining technical work diagram for defining the work of expansion, V V - - p p ) ressure. ressure. ressure. The working substance flows in at constant p p The working substance is pushed qut at constant p The working substance expands with decreasing , e.g. a condenser, or into the free atmosphere. 2 p 18.2 , e.g. a steam boiler, and is received by a vessel with a low, constant 1 p the shaded area enclosed belowin the this expansion example curve. performs The not(cf. Fig. only working lifting substance work, but also work of acceleration pressure flows through the machine fromsure a pressure vessel with a high, constant pres- shaded area enclosed Figure 14.2 Figure 14.1 in , OHL ECHT Continued The First Law and the Equation of State of Ideal Gases 14 edition. This modification corresponds to a suggestion made by K. H technical thermodynamics conventions of the time, have all been removed from this some cases in earlier editions, which corresponded to the 1985 that sign changes were needed. (Dr. rer. nat. 1930) of the author, who pointed out in a detailed letter as early as a friend and former student books. The opposite signs whichwereusedbyP sistently defined here, to conform to this convention and to agree with other text- C14.1 The signs have been con- 382 Part III 14.1 Work of Expansion and Technical Work 383 working substance gives up energy. In the second interval, the inlet valve is closed and the working substance expands further, pushing the piston to position 2; its pressure decreases from p1 to p2.In this process,R the working substance performs the work of expansion 2 W D 1 p dV on the piston. In the third interval, the outlet valve is open and the working substance is pushed out of the cylinder at the constant pressure p2 by the piston. It regains the work of displacement p2V2 from the piston. The working substance thus performsR two portions of work on the 2 piston, namely p1V1 and 1 p dV. While it is flowing out of the cylinder, it receives the work p2V2. Thus, in sum, the working substance performs the following technically useful (or for short technical) work on the piston: R R 2 2 : Wtechn Dp1V1 1 p dV C p2V2 D 1 V dp vertical area shaded area horizontal corresponds in rectangular V 12V p 12p 1 2 rectangular area 1 2 Fig. 14.2 to: area under the alongside the Op22V2 Op11V1 expansion curve expansion curve Part III (14.2) So much for this specific example based on a cylinder and piston. In an analogous manner, we quite generally distinguish two different cases: 1. A confined working substance expands and performs work on the external systemC14.2: C14.2. Don’t let yourself be confused by the different signs of W and Wtechn. Z2 Both of these expressions are Work of expansion W D p dV : (14.3) negative; they thus describe work which is performed by 1 the working substance on the external system. 2. A working substance flows through some arbitrary machine, increases its volume in the process from V1 to V2, and reduces its pressure from p1 to p2. It performs the

Technical work Wtechn D V dp (14.4) 1 on the external system. The relation between these two types of work can be seen from Eq. (14.2); it is

Wtechn D W p1V1 C p2V2 : (14.5)

In ideal gases (Sect. 14.6), at constant temperature, p1V1 D p2V2.In this case, there is no difference between work of expansion W and technical work Wtechn. , , S . )or , its 14.1 V .Itis 9.5 C14.3 (14.6) , the entropy H . Among these, we U , as we have already men- thermal equation of state D , the enthalpy ,wehave: state functions . Suppose that a quantity of heat U , thus increasing it by the amount . Think of a steam engine, or of . These readily measurable quan- , we can specify its volume n W T W C14.4 14.1 C Q . A small number of state variables is always . This can be seen in an example: In Fig. F simple state variables . V Internal Energy U d internal energy U the confined gas, e.g. by an electrical heater which p 13.1 R and the First Law , and its temperature p D W applied to . In the form of an equation is U will encounter the internal energy sufficient to specify all of therelations observable in and a measurable (macroscopic) quantitative thermodynamical system. and the free energy is not shown into the the figure. external system Some as part work of this heat can be removed increasing the area of a liquid surface (cf. surface work, Sect. of the output of electrical energy,the e.g. by energy a which thermocouple. was Theof input rest the of as system heat as can be stored in the inner parts particularly simple in the casethree of an state ideal variables gas. are –mines When known, the any the third, two independently of thermal of the equation anytaken changes of place of state in state deter- that the maythe meantime. have changes of The state onlyproperties, may prerequisite e.g. change is the the microcrystalline that chemical structure none of composition of the or substance other the work performeddiagram. is represented This by areaample however the on depends the shaded on shape area the– of in “path”, the There curve i.e. is the which in a p-V one leads this unique phase from ex- relation is state present; between 1 to it the state is state 2. called variables the when only We return to considering Fig. General In addition to the simplea thermal number state of variables other state mentioned, variables there or are 14.3 The 14.2 Thermal State Variables For every amount of substance Q pressure tities are called the tionedinSect. The defining characteristic of aof state the variable process is or that the itindependence “path” is taken is independent by not previous the changeswork of case state. for This other important quantities, e.g. the )) must 14.6

(4), pp. 423– )! The remarks of the atoms or American Journal 69 N depend on whether ", or the number of The elusive chemical 14.7 n not The First Law and the Equation of State of Ideal Gases 14 of Physics 434). potential be taken into account (see e.g. Baierlein, Ralph (April 2001), " C14.4. The formulation of the First Law (Eq. ( C14.3. The amount of substance particles molecules involved is also assumed here to remain constant. When it is not,additional an state variable, the chemical potential given below the equation are only exemplary. does the energies considered are input into a system orput out- by it. This applies also to the special case shown in Eq. ( 384 Part III 14.3 The Internal Energy U and the First Law 385

For the special case of work of expansion R Q ) C ( . p dV/ ) D ( U Heat input Energy Increase of .not C output as external work D the internal energy a state variable/ .not a state variable/ .state variable/: (14.7) Equation (14.6) is called the First Law of thermodynamics. As long as processes proceed without any temperature changes of the substances involved, the sum of potential and kinetic energies in mechanics remains constant. The same is true in electrodynamics for the electrical and the magnetic energies and for combinations of these four forms of energy. For these energies, the “law of energy conservation” has been thoroughly verified. The First Law of thermodynamics includes an additional energy form in these sums, which can be thermally taken up by a substance from its surroundings or given up to the surroundings, and in the process can be consumed completely or partially to change the internal energy U of the substance by an amount U. Part III The internal energy has thus far been only qualitatively introduced in Sect. 13.4. The First Law permits us to define its changes quantitatively (corresponding to changes in the potential energy in mechanics). The First Law also implies the assertion that the internal energy is a state function. It declares that: Given a system1 in a particular state 1, characterized by state variables p1, T1 :::, through inputs and outputs of heat Q and external work W (of arbitrary kinds), the system will pass in sequence through the states 2, 3, :::Finally, it will arrive back at its initial state 1. Then, one will find experimentally – without exception – that the sum of all the energies thermally input to and output from the system is equal to the sum of all the work input to and output by the system. During all the changes of state of the system, no energy is lost or gained. The internal energy in state 1 at the end is exactly the same as it was at the beginning. It is determined only by the state variables (p, T,...). We first apply Eq. (14.7) to a system which consists only of a single chemically uniform substance. We know the value of its heat of vaporization Q D nLv. A liquid is presumed to evaporate at a constant pressure, called its saturation pressure (Fig. 13.7). The measured heat of vaporization can be decomposed into two terms according to Eq. (14.7) in the following way: 8 9 ˚ « < Increase U = Work of expansion of the internal energy Heat input Q C D W Dp.Vvapor Vliquid/ : in the transition liquid ; ! vapor:

The increase of the internal energy is U D Uvapor Uliquid.Itarises in particular from an increase in the potential energy of the molecules,

1 As the “system” in physical terms, one means all the substances and objects which are included in the considerations at hand. D Pa are 5 U 10 v H , Q/M=l before entering a second ves- 1 p into . 2 p Temperature : A working substance specific heat of specific heat vaporization constant pressures and U/M W/M Δ Water – Enthalpy M 1 10 100 220 14.4 Pa. Above 374.2 °C, 5 Saturation pressure and the pressure 10 1 V 0 100 200 300 374°C and the pressure 5 6 6 6 6 2 kg V Ws 2.5∙10 2.0∙10 1.5∙10 1.0∙10 0.5∙10 through a machine I specific , it has a volume . All three is explained in the text (fine print). A working substance flow- M shows this decomposition for the vaporization of water in The specific The performance of work by a flowing working substance. The W M of a vessel working substance, as mentioned above. All such cases can 14.3 ,areshown. out . The two loaded pistons indicate that M of the internal en- II of water at different ; on exiting U flowing v expansion ergy, and the work of M meaning of ing through the machine has a volume Figure 14.3 0; vapor and liquidcalled become the “critical indistinguishable. temperature” of This water. temperature is quantities are referred to the mass of the vaporduced. pro- Thus, the quantities, referred to the mass temperatures, and its decomposition into two components, the increase sel maintained in the two vessels. Figure 14.4 the temperature range from 0 to 374.2vapor °C. pressure In increases this up range, the to saturation 228 14.4 The State Function In many applications ofa thermodynamics, we needbe to reduced make to the use scheme of sketchedflows in Fig. which mutually attractbe each performed other. against the Theto saturation make work pressure room of of forFigure more the expansion vapor vapor, has that in to is order continually being produced. – latent heat of vaporization l The First Law and the Equation of State of Ideal Gases 14 386 Part III 14.4 The State Function Enthalpy, H 387

M could for example be a steam engine of arbitrary construction, or a pneumatic hammer. Both supply technical work Wtechn to the external world. M could also be a compressor which inputs the technical work Wtechn into a quantity of working substance. M could be a mixing machine which increases the temperature; the energy required is input as work. M could also be a heating or cooling apparatus, which adds or removes heat. Fi- nally, various possibilities can be combined; one could for example equip a compressor with a cooling system.

For the treatment of ﬂowing working substances, we introduced the concept of technical work. From its deﬁning equation (14.4), it follows that W D Wtechn C p1V1 p2V2 : (14.5)

We insert this value into Eq. (14.6), i.e. QCW D U2 U1, and obtain

Q C Wtechn D U2 U1 p1V1 C p2V2 (14.8) or Q C Wtechn D .U2 C p2V2/ .U1 C p1V1/: (14.9) Part III U, p and V are state variables. As a result, the sums in parentheses are also state variables. These sums have been given the name enthalpy H; thus

H D U C pV : (14.10) Enthalpy D Internal energy C Work of expansion

The enthalpy is a much-used energetic state variable or state function. It is applicable to ﬂowing working substances in cases where one would use the internal energy U for conﬁned workingR substances. With the enthalpy H and the technical work Wtechn D V dp, Eq. (14.9) takes on the form R Q C V dp D H Technical work performed Enthalpy (14.11) Heat inputg C D on the external system(!) increase:

Example of an application: If a vapor is produced at its saturation Rpressure, then p is constant (Fig. 13.7). At constant p,wehave V dp D 0. Therefore, Eq. (14.11)givesQ D H; in words: The heat input for the vaporization is equal to the increase in the enthalpy of the substance resulting from the vaporization. Thus, this energy which is required for vaporization (the latent heat Lv) is also referred to as the enthalpy of vaporization; and the same applies to the heat of fusion Lf or enthalpy of fusion. ) is const const V 13.5 D pV D ( c V p (14.12) (14.14) (14.13) Ã Ã C (14.15) T T U and , we can now : D : H p T H c const . Then, instead of D : p pV Ã : : C T V const @ @ D U p const const into a physically impeccable D Ã D Â D T V V p H @ @ added , the enthalpy Ã Ã const Â and the enthalpy U T T H Temperature increase U Q Temperature increase D p @ @ of the enthalpy @ @ T U C Ã Â Â H of the internal energy T U V U Heat 1 1 @ @ M M Temperature increase @ @ at constant pressure, i.e. U Ä of the substance is increased. However, at p Â D D c 1 M U M p C V c c D p p Ä Increase c specific heat capacity Mass 1 M of the substance of the substance Increase D D M M c at constant volume, i.e. V ), we can write: c V c Mass Mass 14.13 p Â Â c D D Eq. ( Derivation Start with the definition of the enthalpy, p V c constant pressure, the substance canrise. expand during In the additionwork temperature to of the expansion. increase Inan other of words: increase its At of internal constant the energy, pressure, internal there instead of energy is also The heat put intotributed the quite substance, differently when e.g. thethan from system when an is it held electric at is heater,by constant at is volume sufficiently constant dis- rigid pressure. containeronly walls, At the the internal constant energy temperature volume, rises enforced and thus Secondly, a specific heat cast the concept of form. – Up toby now, we means of have the defined equation the specific heat of a substance The difference of the two specific heats is found to be c increased. As a result, we need tocific define heat two different specific heats: first, a spe- or or 14.5 The Two Specific Heats, If we know the internal energy The First Law and the Equation of State of Ideal Gases 14 388 Part III 14.5 The Two Specific Heats, cp and cV 389

In general, the internal energy U of a body or an amount of substance depends both on T and also on V. Therefore, we obtain Â Ã Â Ã @U @U T V dU D @ d C @ d (14.16) T VDconst V TDconst and from this, Â Ã Â Ã Â Ã Â Ã @U @ @ @V U U : @ D @ C @ @ (14.17) T pDconst T VDconst V TDconst T pDconst

Combining Eqns. (14.12), (14.15)and(14.17) yields Eq. (14.14).

So much for the now flawless definitions. For a reasonable comparison of the specific heats of various substances, one requires their molar heat capacities (Eq. (13.6) from Sect. 13.3). Then, for different materials, we are dealing with the same amounts of substance n or with the same numbers N of particles.

Figure 14.5 shows typical examples of the molar heat capacities Cp for some solid materials. Only at high temperatures do they approach a constant value. With decreasing temperature, they are strongly re- Part III duced.

The two specific heat capacities of gases, cp and cV, play an important role. Unfortunately, only one of them, namely cp, the specific heat at constant pressure, can be measured reliably. – The fundamentals of the measurement procedure are explained in Fig. 14.6C14.5. A steady C14.5 The temperature of flow of gas passes through a coiled tube S within a calorimeter K;this the calorimeter increases is an apparatus for determining heat capacities, here a water calorime- only slightly above room ter, in which the heat input Q is obtained from the temperature rise temperature during the experiment. Therefore, the cooling rate of the hot gas –200 0 +200 +400 +600 +800 °C in the calorimeter remains Cp R practically unchanged, and Pb 3 (T1-T2) quickly becomes con- Ws kmol∙K Cu stant. Thereafter, both Q 2∙104 and M increase linearly with p C Diamond 2 time.

104 1 Molar heat capacity 0 200 400 600 800 1000 1200 K Temperature

Figure 14.5 The temperature dependence of the molar heat capacity Cp for three solids. If the measured points are plotted against a rescaled temperature axis, T= , then with a suitable choice of , one can make the curves nearly identical. This ‘universal curve’ is called the DEBYE functionC14.6. The tem- C14.6. P. Debye, Ann. Phys. perature parameter is the DEBYE temperature of each of the solids (Pb: 39, 789 (1912). D 88 K; Cu: D 315 K; Diamond: D 1860 K) (see also Sect. 16.4, final part. Here, R is the universal gas constant; cf. Eq. (14.22)) . 2 / 2 ttle T 1 , known T V . c = M ,areanawk- p p c c D D (or, equivalently, using knowledge V V H volume c c = p of the flowing gas. The c D M Q D were obtained in this manner. , Point 4). It is however expedi- ). 14.9 14.1 enhances heat exchange with the water in 14.10 S (Sect. ). Not shown is the thermometer for measuring the , the specific heat at constant V 17.9 contains some specific heats that were measured in c of this kind are suitable as practical laboratory exer- , which is in a double-walled vessel like a Thermos bo and K 14.1 . One measures the ratio Scheme of the measurement of the specific heats of gases at con- V of Ideal Gases. Absolute Temperatures c 13.3 ) and computes from it the specific heat adiabatic exponent V . The values listed in Table C p c = p as the ent to discuss them later (Sect. Measurements of C stant pressure. The flowmeter operatesa according glass to the tube “Rotax” whosepropeller-like principle: wings. diameter In The increases float rises conically, highervolume/time). with there The increasing coiled is gas tubing flow a ratethe float (gas calorimeter with short, (see Sects. temperature rise of the water There are several useful procedures for measuring ward matter. The heatmore taken than up that by absorbedare the by walls in the of general gas the largerbe confined gas than avoided container in only the is it. by quantities The limiting to corrections the be measuring measured. time (to This less can than 10 cises; but as lectureeffect. demonstration Table experiments, they havethis a way. boring Measurements of Thermodynamics received a considerable clarification through thevestigation inof gases, beginning with the thermal equation of state for 14.6 The Thermal Equation of State The temperature ofthrough the the calorimeter; gas likewise the is mass measured before and after it passes seconds). As atermining result, usually an indirect route is followed for de- Figure 14.6 of the water and the known heat capacity of the calorimeter vessel. heat deposited in the calorimeter is then The First Law and the Equation of State of Ideal Gases 14 “Measurements of this kind as lecture demonstration experiments, they have a boring effect.” are suitable as practical laboratory exercises; but 390 Part III 14.6 The Thermal Equation of State of Ideal Gases. Absolute Temperatures 391

Table 14.1 Heat capacities of various gases

Gas Mass density % Molar Speciﬁc and molar heat capaci- at 0 °C and mass ties at 20 °C 5 cp 10 Pa cp cV Cp CV D cV Á Á Á Á kg kg kW s kW s m3 kmol kg K kmol K He Mon- 0.179 4.003 5.23 3.21 20.94 12.85 1.63 Ar 9 atomic 1.784 39.94 0.523 0.317 20.94 12.69 1.65 H = 2 Di- 0.0899 2.016 14.3 10.1 28.83 20.45 1.41 O 2 ; atomic 1.429 32.0 0.918 0.655 29.37 20.98 1.40 air 9 1.293 29 1.005 0.717 29.14 20.78 1.40 CH = 4 Poly- 0.7168 16.04 2.22 1.697 35.59 27.21 1.31 NH 3 ; atomic 0.771 17.03 2.16 1.655 36.78 28.18 1.31 CO2 1.977 44.01 0.837 0.647 36.83 28.48 1.29 ideal gases. – Up to now, we have encountered the “ideal gas law” only for the special case of constant temperature. It states: For an Part III ideal gas at constant temperature, the quotient ‘(pressure)/(mass density)’ or the product ‘(pressure) times (speciﬁc volume)’ is constant. Or, in the form of an equation, p pV D pV D D const: (9.11 to 9.13) % s M

(p D pressure, M D mass of the gas conﬁned in the volume V, % D M=V D mass density of the gas, and Vs D 1=% D V=M D speciﬁc volume of the gas).

For air at 0 °C, one finds experimentally the quotient of pressure/mass density to be Â Ã pV Ws D 7:84 : (14.18) M 0 ıC kg This quotient of pressure/mass density has been measured over a wide range of temperatures, not only for air, but for many other gases. Some of the results are summarized in Fig. 14.7 (upper part). For all the gases, one observes straight lines when the values are plotted against the temperature. The slopes of these lines differ from one gas to another, but the extrapolation of all the lines to lower temperatures cuts the axis of temperature at the same point, namely at 273.2 °C. In the lower part of the figure, instead of the quotient pV=M, the ordinate represents the quotient pV=n (n is the amount of substance). This produces an essential simplification: Now, the slope of the lines is the same for all gases. A single straight line can be used to fit the measured points for all the different gases. Its intersection with the abscissa at 273.2 °C remains the same. Thus, this temperature, 273.2 °C, has been found to be a distinguished value with a universal significance; it is called the ‘absolute zero’ of temperature. in has m M (14.19) ELVIN (28) (44) ; a numerical V (16) 4 2 2 N scale are exactly the CH . There are many pos- CO (lower part) where the allow us to define a tem- : 15 14.7 : without negative values and ELVIN 14.7 273 nRT Pa.) C 5 D ). C t scale. As we have already mentioned 10 Temperature ı pV Absolute temperature (4) D He 4 T K 2 2 ELSIUS –200 –100 0 100 200 300 400 °C Ne N He H CH 0 100 200 300 400 500 600 K Exercise 14.1 –273.2 3 1 3 2 40 20 –300 was measured at the temperature of melting ice is kg

kmol Pa m

scale (‘absolute temperatures’) is likewise shown in Pa m M 5 n 5 n

= The equation of state of ideal gases and the deﬁnition of the 10 10

p·V p·V ), there is a simple expression relating the temperatures as pV . Temperatures measured on this scale give a simple form to : The point on the line in Fig. amount of substance of the gas conﬁned in the volume ELVIN 13.2 D 14.7 n example is given in ( ELVIN g/mol (previously, they were calleddimensionless ‘molecular numbers) weights’, (1 atm and were quoted as a great advantage: The units of the K quotient same as those of the C The K measured on these two scales: (Sect. Fig. the equation of state of ideal gases: absolute temperature. The small numbersupper in graph parentheses are at the the numerical margins values of of the the molar masses of the gases, perature scale in “absolute”practically independent terms, of materials i.e. properties sibilities for this deﬁnition.K The one in use today goes back to Lord The experimental results shown in Fig. Figure 14.7 associated with a temperaturehave of been equally 273.15 K. permissible. But Any the other value value chosen by would K The First Law and the Equation of State of Ideal Gases 14 392 Part III 14.6 The Thermal Equation of State of Ideal Gases. Absolute Temperatures 393

Other formulations are in common use and are expedient:

pVm D RT (14.20) and

%RT p D (14.21) Mm

(Vm D V=n D molar volume, Mm D M=n D molar mass with M D the mass of the gas, % = mass density)C14.7. C14.7. For the deﬁnition of “molar” quantities, see The proportionality factor R is called the universal gas constant. Comment C13.7. It can be determined experimentally from the slope of the lines in Fig. 14.7. The result is Part III

W s R D 8:31 : (14.22) mol K

Equation (14.19), the equation of state of ideal gases, also contains the amount of substance n of the molecules conﬁned within the vol- C14.8 ume V . If we insert n D N=NA (Sect. 13.1) into Eq. (14.19)and C14.8. The volume of an at the same time use the abbreviation R=NA D k, then we obtain an ideal gas with the amount additional formulation of the equation of state of ideal gases, namely of substance n D 1mol can be readily computed from Eq. (14.19); under so- pV D NkT (14.23) called “standard conditions” (or “normal conditions”, i.e. T D 273 K and (N D the number of molecules conﬁned in the volume V). p D 1.013 105 Pa), it has the value V D 22.4 liter. The new constant k which appears here is thus This is the standard molar volume V D 22:4 liter/mol. R 8:31 W s mol m k D D – thus, as long as one can : 23 NA mol K 6 022 10 apply Eq. (14.19), the amount or of substance n or the molar mass Mn D M=n can be obtained very simply, by W s k D 1:38 1023 (14.24) measuring the mass of the K gas in a certain volume under standard conditions.

23 1 (NA D AVOGADRO’s constant D 6:022 10 mol ).

This universal constant k is called BOLTZMANN’s constant (the mo- tivation for this name is given in Sect. 18.4). . . : 2 2 2 O p p C than the physical 1 and 2 p 1 at a given p D nnot calculate Water vapor 62.7 hPa V 1013 hPa) in : p is nearly 0.4. At 2 D and depends only 14.8 p body produces just as ’s law of the addition of Carbon dioxide 53.3 to that of O 2 in the volume ’s law, leads to two surpris- p ltitudes merely according to ALTON ng is no longer possible. At still ALTON type of molecules Oxygen 140 add as “partial pressures” to give the 2 p ’s law: At the temperature of the human ), the pressure ungs at various a and 1 p 14.23 757 Nitrogen ALTON .ThisleadsustoD N . D ). At body temperature, the vapor pressure of water p 37 °C, the air pressure at sea level ( Schematic of the C 9.32 of a person is composed of the following partial pressures at high altitudes as it does at sea level. One therefore ca 2 lungs The air in the lungs thus has a considerably higher concentration of CO Partial pressure Gas A piston pushes theinto gas the from left chamber, the while rightResult: the chamber temperature In is through the being a left held chamber,The valve constant. we two – now pressures have a pressure of partial pressures. We explain it by referring to Fig. Two different gases (which doare not conﬁned react in chemically two with equal-sized each chambers other) at the pressures ing demonstrations: addition of partial pressures total pressure An example of D body, i.e. the the composition of air in the l atmosphere. This, together with D Figure 14.8 2 air outside. The ratio ofhigh the altitudes, a partial pressure person breathes ofincreases faster CO still and further more deeply. withmuch Nevertheless, increasing CO this altitude, ratio since the At an altitude of(see 22 Fig. km, thealone atmospheric is pressure just is asgases only high. in 62.7 the hPa As lungs drop aonly to result, with zero. the water The partial lungs vaporlower pressures of and external pressures, of a breathi the person the human body are other begins thus topressure boil, ﬁlled of i.e. the the water vapor it containspressure. becomes higher than the external air Boiling means that bubbles ofwhen vapor the form vapor pressure within of apressure the which liquid. acts liquid on It becomes it, equal for occurs example to the pressure the of external the surrounding 14.7 Addition of Partial Pressures AccordingtoEq.( criteria. temperature is independent of the on their number The First Law and the Equation of State of Ideal Gases 14 394 Part III 14.8 The Caloric Equations of State of Ideal Gases.GAY-LUSSAC’s Throttle Experiment 395

1. At normal atmospheric pressure, water boils at 100 °C, and carbon tetrachloride (CCl4) at 76.7 °C. – We arrange these nearly mutually- insoluble liquids in two layers, with the heavier CCl4 below, and heat them in a water bath; then boiling begins at the interface already at a temperature of 65.5 °C! – Reason: At this temperature, water has a vapor pressure of 256 hPa and CCl4 has a vapor pressure of 757 hPa. The sum of these two partial pressures, according to DALTON’s law, gives a total vapor pressure of 1013 hPa, equal to the ambient atmospheric pressure, so that the formation of bubbles and boiling can begin. 2. We immerse a test tube ﬁlled with air, with its open end downwards, in a ﬂat dish full of diethyl ether ((CH3CH2)2O). Immediately, air bubbles out of the opening! It is displaced by the partial pressure of the ether vapor ( 580 hPa at 20 °C).

14.8 The Caloric Equations of State

of Ideal Gases. GAY-LUSSAC’s Part III Throttle Experiment

Besides the simple state variables p, V and T, we have deﬁned other state variables (or state functions, also called thermodynamic poten- tials; for example the internal energy U and the enthalpy H). They in turn are functions of p, V and T. This dependence can be represented by caloric equations of state. In these, one of the three simple state variables can always be substituted by the other two. In general, caloric equations of state thus contain two simple state variables. For example, in order to describe the dependence of the internal energy on the temperature, we make use of Eq. (14.17). It contains the dependence of the internal energy of an amount of substance n on its volume at a ﬁxed temperature, which is the same before a process begins and when it is completed (independently of whatever changes take place during the process). We thus need the quantity Â Ã Â Ã @U U : @ i.e. the limiting value of V TDconst V TDconst It has to be determined experimentally. For the measurement of U, we can use the relation

U D Q C W : (14.6)

For measurements on gases, we employ the apparatus which is sketched in Fig. 14.9: Two steel cylinders, I and II, are in a water-bath calorimeter (the thermometers, thermal insulation and stirrer are not shown in the figure). I contains air at high pressure (e.g. 152 105 Pa; under these conditions, air is still nearly an ideal gas), while II is evacuated. When the throttle valve between the two cylinders is is Q This . (14.25) is equal ,i.e. I .–When C14.9 exchanged U 0 , its tempera- Q . In practice, I . Thus, also in must be added II D . As a formula: D T II Q U Q , we can draw two con- . In words: A confined of the gas decompressed : 7 K. The result is that the Q throttled release 0 U D D USSAC U -L const is decompressed and expands. It AY ssed through a throttle. The gas is D 0, and therefore T I Ã D thereby increases by 52 kg and V U : @ @ 4 Q II Â , in this example D . The kinetic energy of the air jet is converted I II T T M ) is simplified to is taken up from the walls of cylinder by turbulence and internal friction into internal en- (1807): At constant tem- D Q II I The internal energy U of an ideal gas is independent of 14.6 T The throttle experiment of USSAC -L 0, Eq. ( AY 2 liter, a mass of D D through the throttle. Experimentally, one finds equal to the change in the internal energy means that the internal energy ofcompression. the air did not changevolume, during pressure, its and de- density at constant temperature clusions: this demonstration experiment, the net quantity of heat From the throttle experiment of G with the air is overall zero. process in detail.each equipped The with cylinders an electric themselvesis thermometer. serve opened, When the as the throttle calorimeters, valve air in cylinder ture decreases by ergy. The temperature of To understand the demonstration experiment, it is best to follow the to or removed from the calorimeter. Then we have one finds perature, the internal energy ofis an independent ideal of gas its pressure(Above: and schematic; density. below: aexperiment. demonstration Each cylinder has aV volume of a heat capacity of 2093 W s/K) L.J. G Figure 14.9 quantity of heat which isto taken the up quantity by of the heat air which from is cylinder released in cylinder produces a jetquantity and of heat performs work of acceleration. Thewithin cylinder equivalent opened, the pressurework and being density of performedwork). the on Such air the a decrease decompression outside without is called world any a (for short: external before. In order for this to hold, a quantity of heat gas in a calorimetersupposed is to decompre be at the same temperature after the decompression as W 0. one upper D T . Here, the 14.9 The First Law and the Equation of State of Ideal Gases 14 result due to its symmetric construction. the experiment, In the actual demonstration experiment (lower part of the figure), we still see the same whole setup is inside calorimeter, which indicates no temperature change during C14.9. This experiment is not shown as a demonstration. It is illustrated in the part of Fig. 396 Part III 14.8 The Caloric Equations of State of Ideal Gases.GAY-LUSSAC’s Throttle Experiment 397

1. In ideal gases, the internal energy U contains no potential energy which depends upon the spacing of the molecules. Therefore, in describing ideal gases, we can neglect the forces between the molecules and treat them as vanishingly small. 2. In ideal gases, the internal energy U depends only on the temperature, and thus not on two, but only on one of the simple state variables. As a result, in Eq. (14.12), we can leave off the condition V D const; similarly, we can omit p D const in Eq. (14.13), since pV also depends only on T. Then we obtain

@U Mc D or U D Mc T C U (14.26) V @T V 0 and @H Mc D or H D Mc T C H : (14.27) p @T p 0

Every energy is deﬁned only with respect to an arbitrarily-chosen zero point; think for example of the potential energy of a stone which Part III has been lifted to a certain height. Thus, we can agree to take U0 and H0, the zero points of the internal energy of an ideal gas and its enthalpy, to be their values at the temperature of absolute zero3.Then for ideal gases, we obtain the simple caloric equations of state

Internal energy U D McVT ; (14.28)

Enthalpy H D McpT ; (14.29) or, for the corresponding molar quantities,

Internal energy U D nCVT ; (14.30)

Enthalpy H D nCpT : (14.31)

It is important not to overlook the essential assumption underlying these expressions: In integrating the starting equations (14.26)and(14.27), cp and cV were taken to be constant.

The enthalpy H and the internal energy U differ by the quantity pV (Eq. 14.10). For an ideal gas containing an amount of substance n, we have pV D nRT. Then we ﬁnd

n.Cp CV/T D nRT ; or

Cp CV D R : (14.32)

3 The magnitude of the constants U0 (or H0), i.e. the energy of a substance at absolute zero, is well known today. It is equal to the mass of the substance multiplied by the square of the velocity of light. U0 is therefore very large; for example, for 1 mol of hydrogen, it is equal to 1:8 1014 Ws. ) of by uni- 9.11 ( (14.34) (14.33) isother- diagram, changes V - replaced p ,isdrawnin temperature is ) are hyperbo- 9.11 In this process, the . Therefore, the work Ws . mol K isotherms 79 : : : 20 : R p 1 p V D const Ws D mol K Ws p 1 D mol K d . They occur when the p V 14 external work W : d 35 d versal gas constant. : M pV V 29 8 V d D D V C p C , produces the ’s Law: . We have already encountered their equation under the For every ideal gas, the difference of its two molar heat . A transition, e.g. from state 1 to state 2, that is an OYLE within the volume of gas under consideration. There should be 14.10 Numerical example for nitrogen which is performed on the external system must be Isothermal changes of state held constant name B state. Such changes can in general be represented in a and the equations for thetwo of changes the of simple state state give variables. the These are relation presumed between to change formly no local differences in temperature, pressure oroccur density, such when as the might changes ofequations state are are very sufﬁciently rapid. simplepletely Unfortunately, these ideal only gases. in For these, thestate one changes: limiting usually distinguishes case ﬁve of types of com- 1. From it, by differentiating, we obtain The origin of pressure from thetreated random thermal in motion previous has also sections. been The graphs of Eq. ( las. One of these curves, which are called Fig. and the isothermal compressibility 14.9 Changes of State ofIn Ideal addition Gases togases, the we thermal must consider and as a caloric third equations case the of equations state for for ideal In words: capacities is equal to the uni W internal energy U of the gas remains unchanged mal expansion The First Law and the Equation of State of Ideal Gases 14 398 Part III 14.9 Changes of State of Ideal Gases 399

Figure 14.10 An isotherm at 22 °C

input of a quantity of heat Q. Quantitatively, both for the work of expansion and for the technical work, we ﬁndC14.10 C14.10. In general, the external work W and the technical work Wtechn are different quantities. In the special case V2 p1 : W D Wtechn DQ DnRT ln DnRT ln (14.35) of isothermal changes of state V1 p2 of ideal gases treated here,

however, they are equal. Both Part III the internal energy U and the Derivation Z2 Z2 enthalpy H remain constant nRT dV V W D p dV; p D ; W DnRT DnRT ln 2 : during such processes. V V V1 1 1

In such a process then, the entire quantity of heat Q which is input to the amount of substance n of the ideal gas is converted into work. Conversely, in an isothermal compression, the work performed on the gas is completely converted into heat and given off by the gas. 2. Isobaric changes of state. They take place at constant pressure. Their equation is T D const:; (14.36) V i.e. the volume V increases proportionally to the temperature T (Fig. 14.11). The transition from the state 1 to the state 2 or vice versa is represented in a p-V diagram by a straight line parallel to the abscissa. In an isobaric expansion, for example, the amount of substance n of the gas performs the work

W D p.V2 V1/ D nR.T2 T1/: (14.37)

During the expansion, the enthalpy of the gas increases by H D nCp.T2 T1/, and the corresponding quantity of heat must be input to the gas. The ratio of work performed and heat input is

W nR.T T / R C C D 2 1 D D p V ; (14.38) Q nCp.T2 T1/ Cp Cp . V - p volume (14.40) (14.41) (14.39) of the without ex- constant volume ordinate These occur /: . 1 : T : 1 (i.e. thermally isolated), that is 2 T . const V D , the corresponding quantity of heat D nC (Video 14.1) . These take place at p W T Q D U decrease ). Heat must be input to the gas; it is converted , V C . = p 14.12 An isochore C An isobar be- performed, since the volume remains constant. 0. They play an important role in physics and technology. D removed not D Q Isochoric changes of state Adiabatic changes of state 4. On cooling (the transition goingnal from energy state is 1 to decreased the andbe state the removed. 2), corresponding the quantity inter- of heat must Work is changing heat with the surroundings with On expansion of the gas, its pressure drops not only due to its Pressure and temperature in antional isochoric to change of each state other. areis The propor- represented transition, by e.g. from adiagram a (Fig. straight state line 2 parallel toentirely the to to state internal the 1 energy, giving an increase of In an isobaric volume must be 3. Their equation is Figure 14.11 between two (lightly drawn) isotherms Figure 14.12 or, with tween two (lightly drawn) isotherms The First Law and the Equation of State of Ideal Gases 14 Video 14.1: tively. has a small heat capacity,temperature the changes during compression and expansion are demonstrated qualita- state” http://tiny.cc/yggvjy Using an air pump which “Adiabatic changes of 400 Part III 14.9 Changes of State of Ideal Gases 401

Figure 14.13 Adiabatic curve for a monatomic gas with D 1:66 (Table 14.1)

increase, but also due to the associated cooling of the gas. The corresponding curves are called adiabatic curves (Fig. 14.13) and fall off more steeply than a hyperbola. Their equation is given by4

pV D const (14.42) Part III

(POISSON’s Law).

Derivation We refer to Fig. 14.14. We can replace the adiabatic expansion by an expansion from 1–3 at constant pressure (isobar) and a pressure reduction from 3–2 at constant volume (isochore). Along the path 1–3, that is the isobaric volume increase, the quantity of heat dQ1-3 D nCpdTpDconst must be input to the gas to keep its pressure constant. Along the path 3–2, that is the isochoric pressure decrease, the quantity of heat dQ3-2 D nCVdTVDconst must be removed from the gas to keep its volume constant. The sum of these two quantities of heat must be zero, since all together, we want to reproduce an adiabatic change of state, in which no heat is exchanged with the surroundings. We thus arrive at

Cp.dT/pDconst DCV.dT/VDconst : (14.43)

The two temperature changes result from the thermal equation of state for ideal gases, i.e. from pV D nRT.Weﬁnd p dV V dp .dT/ D and .dT/ D (14.44) pDconst nR VDconst nR or .dT/ V dp VDconst D : (14.45) .dT/pDconst p dV Continuing, with Eq. (14.43) we obtain dp C p p D p D : (14.46) dV CVV V In words: Along the adiabatic curves, the differential pressure change is by a factor greater than along an isotherm (Eq. (14.33)).

4 Instead of the volume V, one can also use the molar volume Vm D V=n,ifthe amount of substance n of the confined gas is separated out of the constant. . ) ) ) 14.15 14.19 set the (14.47) ( 14.42 ( not ), we have: the thermal (Fig. 14.46 ; ) polytropic 1 2 value must be used, called : nRT nRT const : Œ ln const D D D const smaller V D D 1 2 . They take place when ln ˛ V V 1 2 pV p p pV C p D ln ; instead, a und ), we obtain by integrating: The derivation of the adiabatic A polytropic 14.46 equal to ˛ greater than along an isotherm. Polytropic changes of state ˛ or From Eq. ( 4) : exponent Thus, when the thermal insulation is not perfect, onethe can polytropic exponent. Then, insteadAlong a of polytropic e.g. curve, Eq. the differential ( pressuretor change is by a fac- Making use of the equations falls off less steeply than an adiabatic curve. Its equation is Figure 14.14 exponent Figure 14.15 During the expansion, the pressure fallsand due to the the associated increasing cooling. volume lation, Because of this the cooling insufficient is thermalpansion. however insu- As less a result, than the in curve referred a to perfectly as a adiabatic ex- Additional equations which arestate important can be for found adiabatic in the changes following of Point5. 5. insulation is not sufficient to guarantee an adiabatic change of state curve for a diatomic gas ( 1 The First Law and the Equation of State of Ideal Gases 14 402 Part III 14.10 Applications of Polytropic and AdiabaticChanges of State. Measurements of 403 along with Eq. (14.47), we obtain the following relations which are useful for applications:

Â Ã˛ Â Ã ˛1 T V 1 p ˛ 2 D 1 D 2 (14.48) T1 V2 p1 and for the external work which is performed during the expansion: " # Â Ã ˛1 p V p ˛ W D 1 1 1 2 ˛ 1 p1 R p V p V Dn .T T / D 1 1 2 2 : (14.49) ˛ 1 1 2 ˛ 1

The technical work Wtechn is in this case larger than W by a factor ˛: " # Â Ã ˛1 ˛ ˛ p2 Wtechn D p1V1 1 : (14.50) ˛ 1 p1 Part III

For adiabatic changes of state, in all of these equations we must replace ˛ by D cp=cV. Then, for the external work performed during an adiabatic expansion, we derive from Eq. (14.49), second term:

W DnCV.T1 T2/: (14.51)

Derivation of Eqns. (14.49)and(14.50):

Z2 Z2 V1˛ V1˛ W D pdV D const V˛dV Dconst 2 1 : (14.52) 1 ˛ 1 1

˛ ˛ Furthermore, from Eq. (14.47), we set the const D p1V1 D p2V2 ,and from Eq. (14.19)wesetpV D nRT, and rearrange to obtain Eq. (14.49). We then arrive at Eq. (14.50) by starting with Eq. (14.49) and making use of Eq. (14.5).

14.10 Applications of Polytropic and Adiabatic Changes of State. Measurements of Ä D cp=cV

The changes of state described in Sect. 14.9 are important for numerous applications. Here, we can give only a few examples: 1. The measurement of a polytropic exponent ˛.InFig.14.16, air is conﬁned in a glass vessel (V D several liters) at a small overpressure water in mm : column 230 Pa 3 p D ), the ratio 100 3 that we are ^ = p 1 1 1 ˛ p p p 77 M 14.47 ^ = 23 can therefore be ) ^ = 3 3 D D p (along the isochore p – 1 / Mass p )and( V ( 14.17 (from Point 1 to Point 2 3 isoth 2 polytr / / 1 p 14.19 p d d Volume Volume increases ˛: expansion. We would simply isoth . mm water column). Both the ) . polytr p pressure 4 D ) p (d polytropic Atmospheric

results; in the example, Pressure p in Eqns. ( lines. From this ﬁgure, we read off isoth 3 polytr p p p V d d d isothermal = p 100 mm water column). The valve is opened straight D h (Pressure Š The measurement of a polytropic expo- ), since the thermal insulation offered by a glass vessel Measurement . As a result, the pressure Pa ( 23 mm water column). We could have reached Point 3 im- 3 14.16 14.17 10 D the polytropic pressure drop the isothermal pressure drop ˛ h D 10 Pa) 1 Š As found by computing d mediately through a slow of the polytropic exponent as in Fig. units: 1 mm water column polytropic curves and theapproximated isotherms as short in Fig. nent have to allow exactly thein same the amount case of of air the to rapid escape polytropic expansion. theThe vessel pressure as changes aresure small (atmospheric compared pressure, to the i.e. ambient 10 air pres- ( Figure 14.16 p Figure 14.17 and then immediately closedpletely when released. the The expansion overpressure is in has Fig. been com- of the two pressureseeking, drops and thus is the polytropic exponent is not perfect. The airby inside an is adiabatic not expansion, i.e. asertheless, with strongly perfect cooled less thermal as air insulation. it would leaves Nev- isothermal be the vessel than if the expansion had been from Point 2 to 3)ture. as the Again, air an gradually overpressure warms back to room tempera- The First Law and the Equation of State of Ideal Gases 14 404 Part III 14.10 Applications of Polytropic and AdiabaticChanges of State. Measurements of 405

˛ 100 : In the example, D 10023 D 1 3. The air thus expands in Fig. 14.17 with the polytropic exponent ˛ D 1:3.

2. Measurement of the adiabatic exponent D cp=cV from the velocity of sound. With ‘perfect’ thermal insulation, the expansion in Fig. 14.17 is adiabatic. The measured exponent ˛ must then become equal to the adiabatic exponent air D 1:40. Indeed, it has often been attempted to measure in this manner. However, it is not simple to eliminate all possible perturbing heat inputs. – This is more readily accomplished with very rapid expansion processes. They can be found for example in sound waves, both propagating and standing waves. One can obtain with good precision from measurements of the velocity of sound. For the sound velocity in gases, we found s K c D : (12.46) sound %

Here, % is the density of the gas and K its modulus of compression: Part III 1 dV 1 D : (14.53) K V dp

For an adiabatic expansion (˛ D ), we have (Eq. (14.46))

dV 1 V D ; dp p and thus K D p : (14.54) Inserting Eq. (14.54) into Eq. (12.46) yields r p c D : (14.55) sound %

Numerical example At 18 °C and p D 105 Pa (1000 hPa), air has a density of % D 1:215 kg/m3. The velocity of sound is measured to be csound D 340 m/s; from this we ﬁnd D 1:40. The velocity of sound can best be determined using standing waves of a known frequency. (“KUNDT’s dust ﬁgures”, Sect. 11.7).

The velocity of sound csound decreases with decreasing temperature. This can be seen by rearranging Eq. (14.55) using the equation of state of ideal gases, (14.21)C14.11: C14.11. The velocity of s sound csound in an ideal gas R is thus independent of its csound D T : (14.56) density %! Mm

3. The production of high temperatures through polytropic compression. Before European matches were introduced, the fire pump was in ): A piston is 14.18 , we could attach a piece of cotton S The non-physicist often thinks that a cylinder of Compressors A compressed- A Malaysian fire pump, replicated here as a glass (Video 14.2) energy of a quantity of airits at pressure a and constant density. temperature isby Thus, independent expanding when isothermally, of it compressed gives up air none of performsbut its work instead own internal takes energy, the energy fromair some motor, other as source. we Ainto shall compressed- work. see, is simply a machine which converts heat compressed air is anthe energy wound-up storage spring of system, afor similar pocket the for watch. following reason: example This Air to interpretation is is a false, nearly ideal gas, and therefore, the Figure 14.18 wool dipped in carbon disulfidepiston. or Then diesel the oil heat to ofmixture the compression to bottom burst would of into cause the flames. the air-vapor Figure 14.19 air motor. For isothermal operation, the cylinder is surrounded by an electric heating mantle. 14.11 Pneumatic Motors and Gas These are not onlymatic jackhammer), technically but important they machines areof (e.g. also view of the very physics. instructive pneu- from the point widespread use along the Malaysianneo; Archipelago, it especially is in often Bor- called a ‘pneumatic lighter’ (Fig. model. Instead of the tinder mixture which has been injected into the cylinder. thrust into a woodenniting cylinder. a The piece air of indiesel tinder the attached engines cylinder to is make the heated, use bottom ig- of of the the piston. same Today, process to ignite the fuel-air The First Law and the Equation of State of Ideal Gases 14 “The non-physicist often “Compressed-air motor” http://tiny.cc/0ggvjy Video 14.2: This interpretation is false”. compressed air is an energy storage system, similar for example to the wound-up spring of a pocket watch. thinks that a cylinder of 406 Part III Exercises 407

Isothermal operation is exempliﬁed by the cylinder of a small machine as shown in Fig. 14.19, when it is surrounded by an electric heating mantle whose heater current is controlled so that the air ﬂow- ing into and out of the cylinder is kept at the same temperature, i.e. T2 D T1. Then the expansion of the air is isothermal,andwe can apply Eq. (14.35) for the technical work performed:

Wtechn DQ :

(See the derivation in Sect. 14.9.)

In words: In isothermal operation, the heat Q which is input is completely converted into work, and the machine represents the ideal limiting case of 100 % mechanical performanceC14.12. If the heat- C14.12. The “mechanical ing mantle is left off, the machine operates in a polytropic mode; in performance” quoted here the limiting case of good thermal insulation it is adiabatic, and thus does not contradict the Sec- without heat exchange with the surroundings. In this latter limiting ond Law. It should not be case, we ﬁnd for the technical work confused with the efﬁciency of heat engines. POHL dis- Wtechn D H

cusses the compressed-air Part III (follows without computation from Eq. (14.11)forQ D 0). motor in this connection in more detail in Sect. 19.7. In words: When a compressed-air motor is operating adiabatically, the work performed is equal to the decrease in enthalpy of the compressed air. The air leaves the machine cooler than when it enters; that is, T2 < T1, but this loss of enthalpy will be replaced afterwards by heat exchange from the surrounding atmosphere, so that T1 is restored. In the adiabatic mode, the compressed air must merely lend some energy in advance.

The enthalpy reduction in a thermally insulated motor can be used to cool gases, e.g. for the liquefaction of helium. One then refers to ‘cooling by performing work’ or an ‘expansion machine’ (G. CLAUDE (1870–1960); cf. Sect. 15.6).

The operation of a compressor is just the reverse of that of the motor. When its cooling is not adequate, the machine driving the compressor must increase the enthalpy of the compressed air while compressing it, and the additional work performed for this is lost afterwards when the compressed air cools uselessly back to ambient temperature in its pressure vessel.

Exercises

14.1 The molar mass of oxygen is Mm D 32 g/mol. n D 2kmolof O2 gas (DO M D 64 kg) is at a temperature of 27 °C, i.e. T D 300 K, 3 in a volume V D 300 liter D 0:3m . What is the pressure of the O2? (Sect. 14.6) 36 : 1 D ˛ ) 40 l contains 14.7 D 2 V 10 l contains oxygen at a pres- D 1 The online version of this chapter (doi: V Pa. What is the total pressure when 5 10 Pa. Another cylinder of volume 5 10 ). To what fraction of its original volume must air be com- A gas cylinder of volume 14.9 nitrogen at a pressure of 8 sure of 15 the two cylinders are connected to each other? (Sect. pressed in orderthe to process is reach polytropic, a with a temperature polytropic of exponent of 500 °C? Assume that 14.3 Electronic supplementary material 10.1007/978-3-319-40046-4_14) contains supplementary material, whichable is to avail- authorized users. 14.2 (Sect. The First Law and the Equation of State of Ideal Gases 14 408 Part III 51PaeCagso elGases Real of Changes Phase 15.1 Gases Real .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 15.1 Figure gases ideal of state of equation the know now We h eu,adFig. and setup, the eosrto xeietwt oetefr.Figure effort. modest a with experiment demonstration a ofilteaprts( bar (1 apparatus the fill to ftevlm ftecnndgs h tt aibe tti point this at variables state The gas. confined termed the are of volume the of led togdfraino h uv sse.At this seen. of is neighborhood the curve the a of exhibits isotherm deformation strong a already hycnsilb ecie yteequation the by described be still can They ttmeaue above temperatures At chosen. appropriately been have axes coordinate whose nalcss o abndoie(CO dioxide behavior qualitative carbon same For the exhibit cases. isotherms all The the in im- of most determination gases. The experimental real the describe observations. of is on quantitatively these rely must of to we portant therefore, state, of state; changes their of without For equation applicable etc.) results. adiabatic experimental (isothermal, new transformations various the state of equations of the derive types to us permit to suffices knowledge critical h netgto fpaecags(semi-schematic). changes phase of investigation The .ForCO 15.2 on finflection of point D T C olsItouto oPhysics to Introduction Pohl’s rtclpitK point critical cr umrzsterslsi a in results the summarizes 10 0°,teiohrsaestill are isotherms the °C, 80 D 2 5 h rtcltmeaueis temperature critical the , Pa). 0 K 304 real . 31 ae,teei ogenerally- no is there gases, ihahrzna agn:In tangent: horizontal a with 2 h rsuei independent is pressure the , ı in exhibited be can they ), C pV /; m D O 10.1007/978-3-319-40046-4_15 DOI , os.At const. C15.1 p - C V hyperbolic 15.1 m , 1°,the °C, 31 isotherms n that and C diagram S 0°C, 40 sused is shows . od ihtegsisl.As itself. gas the with do to nothing have gas” “real and gas” “ideal terms The C15.1. emd“real”. termed is it then point, boiling its of neighborhood the in e.g. equation, that of predictions the from deviates its behavior If “ideal”. termed is it gases, ideal of state equa- of the tion obeys gas a as long 15 409

Part III is D n ,all l ; = ˛ m .The V V D . m V 15.1 exhibit a similar K . At the point ). : . When the volume is 1 ˇ ˇ ; 20 °C, beginning at a large Pa kmol C liquefied 5 = . They show the limits of the 3 ) and the gas has a molar volume 1 K 10 ˛ is considerably less compressible 6 2 : 096 m : 73 0 D envelope of the coexistence region D Pa at the point- 5 cr cr ; p m 10 V diagram of carbon dioxide with appropriate axes , chemist in Belfast, 1813–1885). ( 1 : (steep rise of the isotherm!). m V 2 /kmol (abscissa value of point - 3 p or surface, i.e. it becomes A NDREWS 46 m : A 0 /kmol (abscissa value of point of the curve). Along this segment, the form of the carbon 3 D g ˛ ; m interface V HOMAS 048 m : region in which the liquidAt and the the gaseous left phases of can exist the together. dashed curve, there is only liquid; at the right of the molar volume.) At 0 °C, the liquid has a molar volume of two curves meet at the critical point of decreased further, the pressure remains constantˇ (along the segment- dioxide changes: An increasing fractionan is separated from the restof by the substance ispresent. in the Continuing liquid to phase decreasean and enormous the an pressure: volume interface Liquid CO requires is that no we longer apply than gaseous CO 0 and the critical molar volume is (T Below the critical temperature,different. the Let us phenomena follow become thevolume, isotherm completely at that isup at to the the lower value right: 58 initially, the pressure increases Examples for other substances can be found in Table All the other isothermsbehavior. below the The critical end point delimited points at of the their leftdashed straight, by curve, horizontal a defining sections dashed the are curve, and at the right by a dot- the critical pressure is Figure 15.2 Real Gases 15 410 Part III 15.2 Distinguishing the Gas from the Liquid 411 the dot-dashed curve, there is only gas. Above the critical point K, distinguishing between gas and liquid is no longer meaningful. For the straight-line segments (the “average lines”) of the isotherms, the abscissa values at their end points, e.g. ˛ and ˇ0,givethemo- lar volumes Vm of the liquid and the gaseous fractions (a numerical example is given in the caption of Fig. 15.2). For each filling and temperature of a container (in the example 3 sketched for Vm D 0:2m /kmol and 0 °C), the ratio of the lengths j/i gives the ratio of the amount of substance as liquid/amount C15.2 of substance as gas . –Atthecritical filling,whereVm;cr D C15.2. Derivation: The total 0:096 m3/kmol and T D 0 ıC(cf.Fig.15.4), amount of substance n is the sum of the fractional amounts 89 % of the substance Db 45 % of the volume is liquid of substance of the gas, ng, 11 % of the substance Db 55 % of the volume is gaseous: and the liquid, nl.Atamolar volume of Vm, ˛1 < Vm < ˇ The left-hand (dashed) limiting curve ends at a pressure of 5 105 Pa on 1 (Fig. 15.2), a part of the substance is gaseous, n D xn a point marked by a small circle in the figure. Below this pressure, CO2 is g a solid. At the same pressure, a second circle is drawn in to the left, and (0 < x < 1) and the rest is Part III a third is far to the right outside this region on the isotherm corresponding liquid, nl D .1 x/n. x is : ı 3 to 56 2 C(atVm D 3m /kmol). We will return to the significance of thus the molar fraction of the these circles when we deal with the triple point. total amount of substance that is gaseous. Then we have for For water, today still the most important working substance, some the combined molar volume: special symbols and terms are in use. Water vapor in a state outside Vm D xˇ1 C .1 x/˛1 or the limiting curves is called superheated steam, and on the limiting x D .Vm ˛1/=.ˇ1 ˛1/; curve it is called dry saturated steam; within the limiting curve, it is since ˛1 and ˇ1 are the molar termed wet steam. volumes of pure liquid and Wet steam is a mixture of water vapor and fine water droplets. It pure gas, respectively. For the appears to the eye as a white fog or as a white cloud. Superheated example shown in Fig. 15.2, and saturated water vapor are just as invisible as for example the air we find in a room. They lack the fine suspended water droplets which scatter x D i=.i C j/ and 1 x D light (see Vol. 2, Chap. 26). – The non-physicist usually thinks of j=.i C j/; = = water vapor as this visible wet steam (fog). and thus nl ng D j i.

Technically, the term speciﬁc steam content of wet steam is used. This refers to the ratio Â Ã Mass of the dry saturated steam i x D D (15.1) Mass of the steam and the water droplets i C j suspended in it

in Fig. 15.2. Along the left-hand limiting curve, x D 0; on the right-hand curve, x D 1.

15.2 Distinguishing the Gas from the Liquid

The isotherms of CO2 (Fig. 15.2) lead us to some important conclusions. Figure 15.3 shows only two of the isotherms, namely the 2 $ .The below 1 has now interface 2 in the form discontinu- 2 ). The phase ,weseeanew , all of the CO ˇ ˇ , the dashed curve continuously 15.11 ). In this process, the ı Pa. 15.11 and and allow the volume to . An interface forms sim- 5 ˛ 20 °C, ﬁnally arriving back Pa. From now on, we keep . Then we see no interface and 5 ˛ C d even when the pressure (here 15.10 , 10 ˇ , 40 °C. In addition, the two limit- 15.10 /kmol) and raise the temperature up 3 C , ı , ˛ . 227 m : /kmol) isothermally ( K 0 3 . Result: We have not observed any D ˛ gas), in contrast, takes place m V 057 m $ : (the limiting 0 2 between two ); thereafter, we compress the gas back to its original D Distinguishing be- K m , but nevertheless, beginning at the point , nor any fog, i.e. no precipitation of liquid CO 20 °C and the one at V present. We begin with the state C at the critical point 40 °C ( changes of the physical properties, e.g. the transition (solid both C In outer space, liquids with a large mass could be held together by their mutual formation liquid surface is not a shell which secures it again been liqueﬁed. It exhibits auid: characteristic property It of cannot every be liq- noticeably compresse at 20 °C) is increased to several hundred 10 of small suspended droplets. Nevertheless, all of the CO the volume constant andat cool our back starting to point, to volume ( liquid): The two phasesistence are by separated over the their limiting entire range curve of (Figs. ex- the critical temperature: In Figs. transition (liquid ends A liquid cannot exist alone, i.e. in an otherwise empty space The whole closed pathdirection, (cycle) i.e. in can the also order be followed in the reverse pressure increases up to about 150 Figure 15.3 tween gases and liquids:process A cyclic which includes the critical point isotherms of CO curve at the left isright dashed, dot-dashed) at the attraction (gravitation). But thenatmosphere. they would always be surrounded by a gaseous 1 has vaporized and no surfacevolume or constant ( interface remains. Now we keep the one at ing curves are drawn inWithin this and region the (the region boundedare “coexistence by region”), them the is gas shaded. and the liquid interface form. Result: Inous general, phase transitions take place with disappear increase. The pressure remains constant,of while an the increasing substance fraction “vaporizes”. At the state point Real Gases 15 412 Part III 15.2 Distinguishing the Gas from the Liquid 413

Figure 15.4 The liquid-gas phase transition in CO2 with increasing temperature. In I, all of the substance is finally liquid (danger of explosion!), in III it is finally all gaseous. However, in II, the continuous transition at the critical temperature can be tracked; tube II illustrates the critical filling at 0 °C; cf. Sect. 15.1C15.3. C15.3. It is worth the effort to reconstruct these three phase transitions using the diagram in Fig. 15.2;this will lead to a more complete understanding of the diagram. The case of tube I at first ply as the boundary between two phases of the same substance. On seems to contradict all of our its outer side, the same substance must be present in the form of sat- everyday experience! urated vapor, pure or mixed with another gas, for example ambient air. Only then is there an equilibrium, in which the same number of molecules move per unit time from the one phase to the other and vice versa. For water at room temperature, around 1022 molecules per second and cm2 do this!! (cf. Sect. 16.1, Point 1). This statistical equilibrium prevents one phase from growing at the expense of the Part III other. At the end of Sect. 9.9, we encountered the diffusion boundary between two chemically different gases as a kind of interface or surface. With the same justification, we can now consider the surface of a liq- uidtobeadiffusion boundary. It separates two chemically identical substances in physically different phases. These results can be very clearly demonstrated with three equal-sized glass tubes filled with different amounts of CO2 (Fig. 15.4). When the temperature is increased, the surface in tube I rises, in tube III it falls, and in tube II, at the critical temperature the surface approaches the center of the tube and vanishes there. That is, at the critical temperature, the molar volumes or specific volumes of the gaseous and liquid phases have become equal; the two phases are no longer dis- tinguishable. When the tubes are again cooled, the surface appears once again in the middle of tube II2. Its reappearance is announced by a glimmering layer of fog: In the statistical manifestation of the thermal motions, the phase transition appears first here, then there3. Sub-microscopic “nucleation centers” form through a local accumulation or accretion of molecules, and from them, tiny and at first volatile droplets are formed. Only at larger particle densities NV of the droplets do they merge together to form a surface, i.e. a boundary interface between the two phases.

2 Only there does the molar volume have precisely its critical value in the gravitational field of the earth; above the middle, it is larger; below, it is too small. 3 At the critical point, dp=dV D 0ordV=dp D1. That is, even minimal local variations in the pressure are sufficient to cause noticeable changes in the specific volume or its reciprocal, the density. (15.2) VA N D E R are set out in ). b 15.5 ) that it describes. . As “higher-order /: and than to the gaseous a , suffices to determine of R RT Equation solid D diagram runs parallel to the , with the exception of the substance / m b V . At lower temperatures, how- - gas, by contrast, we require at p 15.2 AALS gas m V real . W Ã molar volume of the gas 2 m a constant, namely numerical values D V n = are two constants which are characteristic of C V A liquid is a solid in a state of turbulence with b p single D Â m and VAN DER V . a gas, a A two-dimensional of State for Real Gases constants. Some . ideal equation of state. It is given by 15.1 three abscissa, and furthermore, it has a point of inflection there. Therefore, the At the critical point, the isotherm in the AALS model of the structure ofstatistical density a fluctuations liquid with Figure 15.5 In this equation, For every “crystalline” regions that continuallybut change always maintain their hexagonal sizes close packing and (Fig. shapes, 15.3 The All of the isotherms shown in Fig. W On approaching the critical temperature,ous there transition from is the certainly liquid a toever, continu- the the liquid phase isphase. much closer A to liquid the cantalline almost powder: be thought its of micro-crystalsfragments as have of a very decaying very short micro-crystals fine,ternation lifetimes, reunite microcrys- and to in the an form uninterrupted newputs al- micro-crystals. this another way: – Thevery small following but formulation still crystallineindividuals”, turbulence these elements are subject to mutual, progressivetations. motions and In ro- two dimensions,steel-ball this can molecules be in convincingly a simulated flat with dish. On shaking and stirring, we find straight-line segments in the coexistenceto region, a can be good represented approximation by a third-order equation, the the particular type of molecules (i.e. the its equation of state.least For every Table Real Gases 15 414 Part III 15.3 The VAN DER WAALS Equation of State for Real Gases 415

Table 15.1 Critical state variables and VA N D E R WAALS constants for some real gases

Critical state variables VA N D E R WAALS Sub- Molar mass Temperature Pressure Molar vol- Constant Constant stance ume

Mm Tcr pcr Vm;cr a b Á 5 Á Á Á kg (K) (10 Pa) m3 105 Pa m6 m3 kmol kmol kmol2 kmol

H2 2.02 33 12.9 0.065 0.19 0.022 He 4 5.2 2.3 0.058 0.034 0.024

H2O 18 647.4 221 0.055 5.54 0.031 N2 28 126 34.1 0.090 1.36 0.039 O2 32 154 50.4 0.075 1.37 0.032 CO2 44 304 73.6 0.096 3.65 0.043 SO2 64 430 78.5 0.096 6.84 0.056 Hg 200 1720 1080 0.040 0.82 0.017

following relations hold there: Part III Â Ã @p D 0 (15.3) @ Vm cr

and Â Ã @2p D 0 : (15.4) @ 2 Vm cr With these two conditions, from Eq. (15.2) we obtain

. 2 / a D 3 Vmp cr (15.5)

and 1 . / : b D 3 Vm cr (15.6) The values listed in Table 15.1 for the constants a and b have been cho- sensothattheyfittheVA N D ER WAALS equation of state (15.2)tothe measured isotherms over the widest range possible. With the values of a and b fixed in this way, the condition (15.5) is well fulfilled, but the condition (15.6) only to a relatively poor approximation. For CO2,forex- 3 ample, the measured value is .Vm/cr D 0:096 m /kmol; but in Table 15.1 we find 3b D 0:129 m3/kmol. The VA N D ER WAALS equation is just an approximation. Strictly considered, each different gas, due to the individual properties of its molecules, would require its own particular equation of state. We cannot demand too much of an equation that neglects these many individual properties! (Exercise 15.1)

The VA N D E R WAALS equation of state differs from the simple ideal- = 2 gas equation by the additional terms a Vm and b. Their physical significance is readily seen. Let us start with the internal pressure, = 2 the term a Vm. Under otherwise similar conditions, a lower pressure would be measured externally if the molecules were to experience a mutual attraction for each other. Therefore, in the equation of state for real gases, we must add a correction term to the measured pressure p when the molar volume Vm becomes small and the molecules are on average close together. The force that corresponds to the , : ) - b D ,as held 14.11 D 15.6 W : OULE ELVIN held con- , we write molec ; ). m equation of m of the whole V V V 14.8 AALS .IntheJ internal energy . In this case, for W (later Lord K enthalpy H 0) and furthermore no U 14.4 const was derived for becomes small, we can Throttle D : molecules themselves is a reducing valve which D 0 . gas Q tional to the molar volume m M D VA N D E R C15.4 HOMSON pV ) W , was already discussed in Sect. . That is, instead of W C ’s experiment (Sect. 9.24 Q 4). ; but that has two disadvantages: First, HOMSON D D Q and W. T -T , under the same conditions there will be U 14.9 USSAC in Fig. . In principle, this can be demonstrated with process. acts in the same sense as the force applied gas. -L T B 2 m or, using the abbreviation const OULE : AY V . The equation OULE = b ideal a molec ; m V isenthalpic experiment, the “machine” its molecules themselves Experiment by a quantity which is propor ), we find gases, in contrast of (the factor ‘const’ is : gas const b 14.6 real molec . In this type of expansion, the temperature of the gas remains ; 4 The other possibility, namely HOMSON m m m no longer neglect the proper volume of the state, we return to G external work is performed (expansion through a throttle valve, the setup sketched in Fig. it is relatively insensitive; andduce second, a it gas is jetfriction superfluous with to back kinetic first into energy pro- disadvantages, internal and J.P. then energy. J convert that In energy order by to avoid both of these V T replaced the expansion of a confined gas with its constant by the expansion of a gas flow with its corresponds to the general scheme of Fig. stant, i.e. an Their experimental arrangement, shown schematically in Fig. a model gas. Therandom volume thermal in motions which was the taken molecules to can be carry the out volume their container. When the molar volume of the V we did for ideal gases.to We must the decrease the molar volume available externally by a piston ( From this fundamental experiment,a it gas follows without that changing we can its expand internal energy This can be thewith case the surroundings when (adiabatic process, there is no exchange of thermal energy Eq. ( 0) temperature changes constant if it is an For V (isothermally-operated compressed-air motor). 4 15.4 The J Now that we have knowledge of the internal pressure Now to the term prevents the formation ofchannels of a a jet, porous disk. e.g. It a is narrow thermally well opening, insulated. or As a the result, fine Real Gases 15 C15.4. Both forces are thus directed from the surface (container walls) into the interior of the gas. 416 Part III 15.4 The JOULE-THOMSON Throttle Experiment 417

Figure 15.6 The expansion experiment of JOULE and THOMSON (1853). The throttle consists here of a porous sintered glass disk. It is fused into the glass walls of the apparatus. The two thermocouples are wired in opposition, so that the voltmeter Th indicates the difference T of their two temperatures directly. the gas cannot take up any heat from its surroundings; in Eq. (14.11), i.e. Q C Wtechn D H, Q D 0. During the expansion, Wtechn D 0(no moving parts, no acceleration!), so that H D 0and

H D U C pV D const : (15.7) Part III

As a result of the constancy of the enthalpy, the internal energy U,and with it the temperature, can increase or decrease simply by changing the quantity pV during the expansion. Some data from a measurement are shown graphically in Fig. 15.7. Usually, the expansion (p < 0, i.e. negative) causes a cooling effect (T D T2 T1 < 0). However, in certain ranges of temperature, heating is observed, e.g. with air at a pressure of 220 105 Pa, above an “inversion temperature” of about 230 °C (Point b in Fig. 15.7)C15.5. The JOULE- C15.5. The inversion tem- THOMSON effect T=p observed in particular cases must thus be peratures of H2 and He the sum of two contributions, a cooling effect and a heating effect. are 200 K and 50 K, Usually, the cooling predominates; but the heating effect can also be respectively. In order to the larger of the two. obtain cooling with the JOULE-THOMSON effect, In order to elucidate these two contributions, we start with the ﬂow of an these gases must therefore ideal gas. A gas containing the amount of substance n of mass M passes be precooled, for example by through the throttle. At the pressure p1, we assume its volume to be V1. adiabatic expansion (see also Sect. 15.6).

Figure 15.7 Measurements of the JOULE-THOMSON effect for air at various starting temperatures and pressures p1. Outside the range a-b and at p1 D 220 bar, heating occurs (1 bar D 105 Pa). 2 m > V coun- V ,is 2 1 = 1 p U m V :The 1 2 ), illustrates M p T than that C15.6 2 expanded u are based on 1 3 D at the right, compressed 15.7 15.8 1 . The expanded than that of an D T higher . The gas leaves s 1 V IEMENS T = 2 lower > . Therefore, the work , at the same number u gas is / by the gas 2 1 3 throttle is b effect, in particular for T and thus D 2 real real gas by the compressor, m U and p V performed on the 2 1 D equation which describes them. =. U real gas of its molecules. The greater the V m 1 1 V < M p U HOMSON N 2 1 equation is, as we have already em- u AALS ), U -T 1 3 compressed at the left by the compressor, Due to the term b W ), observed on expansion. As a result of the Pa flows through a tightly wound, mul- D . 15.7 AALS b 5 p 15.7 OULE W gas. As a result, according to Eq. ( which results from the mutual attraction of the 10 e itself being warmed (S gases permits no change in the temperature on 2 m V cooling VA N D ER , the pressure of a = on the gas a 2 m ideal V = VA N D ER expanded a in the . The gas leaves the throttle at a lower temperature, it is processes for the liquefaction of gases 1 T ); for a real gas, , the more the pressure is reduced relative to that of an ideal work, i.e. an attractive force between its molecules, is lacking of the molecules, the pressure of a < . Then from Eq. ( 2 ). The 9.8 V of Low Temperatures and Liquefaction of Gases 2 V T N 2 . Therefore, from Eq. ( p 15.7 2 , is greater than the work of displacement performed by the gases on expansion must therefore be connected with the additional V internal 1 and 2 V p 2 1 internal pressure ideal gas at the same number density of an ideal gas. For the ideal gas, we have The internal pressure phasized, only an approximation.application. We should not expect too much from its During its expansion, no working is performed; the prerequisite for perform- in an ideal gas, andno there moving parts is and no the externalplacement molecules work performed are since not the accelerated. apparatus The contains work of dis- expansion without performing externalreal work. The cooling orcorrection terms warming of real gas by the compressor is less than the work of displacement U compression gas. Thus, the work of displacement the throttle at a higherpredominate temperature, over it the has cooling been described heated.The in quantitative This the mathematical heating previous may paragraph. formulationnot of yield these considerations satisfactory does the results; effects on in the(Fig. particular, pressure, it in stark gives contrast no to dependence the of experimental results equation of state of molecules explains the cooled. This cooling can, however, be overcompensatedis by a due heating to effect which the correction term density of displacement performed on the just the same as the displacement work performed (cf. Sect. performed by the gas, p and cooled gas can exit theflows between Dewar vessel the at coils its of top. thefollows copper it, On tubing the at and the way cools same out, the tim it gas which tilayered copper coil withinvessel”). At a its transparent lower end Thermos is bottle a (“Dewar fine nozzle, the a demonstration experiment showing the liquefaction ofa air. pressure Dry of air around at 150 the liquefaction of air, hydrogen, and helium. Figure 15.5 The Production Some important coolingbymeansoftheJ , ) ILLIAM IEMENS ILHELM S W (Sir W ARL Real Gases ERNER VON 15 IEMENS studied chemistry, physics and mathematics in Göttin- gen. (1823–1883), the brother of W C15.6. K S 418 Part III 15.5 The Production of Low Temperatures and Liquefaction of Gases 419

Figure 15.8 A demonstration experiment showing liquefaction of air by the LINDE process. The cooled but not yet liquid fraction of the air flows upwards between the coils of the copper-tubing heat exchanger and escapes into the room. This precools the air which follows on its way into the vessel (“countercurrent heat exchanger”; see Sect. 17.6.). The copper tubing has an outer diameter of 2 mm and an inner diameter of 1 mm. The nozzle (throttle) consists of its flattened end. (Video 15.1) Video 15.1: “Liquefaction of oxygen” http://tiny.cc/8ggvjy The experiment becomes more straightforward when the flattened end of the copper tube in the video is tercurrent process, Sect. 17.6). After a few minutes, the expanded replaced by an adjustable gas has been cooled to its boiling temperature. Then its liquefaction needle valve which can be controlled from above and begins at a constant temperature. One can observe a liquid emerg- Part III ing from the tube, at first as a fog, then as a continuous liquid jet; which serves as throttle. it quickly fills the lower part of the Dewar vessel. – In this process, only a small fraction x of the gas flowing into the apparatus is liquefied (x 0:1). The greater portion (1 x 0:9) flows out of the vessel as gas, taking with it the enthalpy of condensation of the liquid produced.

We can consider the entire LINDE apparatus, that is the throttle and the countercurrent heat exchanger, to be an isothermal expansion machine: The gas flowing into the apparatus and the gas flowing out have the same temperature T. The gas flowing in, at a pressure p1, brings its enthalpy

HT;p1 into the vessel, while the gas ﬂowing out at the pressure p2 takes . / its enthalpy 1 x HT;p2 with it. The enthalpy xHliquid remains with the fraction x which is liqueﬁed. Then we can write the enthalpy balance as

. / HT;p1 D 1 x HT;p2 C xHliquid

and from it, we ﬁnd the fraction x whichisliqueﬁedtobe

H ; H ; x D T p1 T p2 : Hliquid HT;p2

NumericalÂ Ã example for the liquefactionÂ Ã of air H H : 5 Ws ; : 5 Ws 20 ıC D 4 634 10 20 ıC D 5 028 10 M 200 bar kg M 1bar kg Â Ã H liquid : 5 Ws ; : : : 193 ıC D 0 922 10 and thus x D 0 096 0 1 M 1bar kg C15.7. See for example Besides air liquefiers, there are today technically highly-developed G.K. White, P.J. Meeson: liquefiers for hydrogen and helium. Those in use in modern laborato- Experimental Techniques in ries yield many liters of liquid per hourC15.7. By reducing the pressure Low-Temperature Physics, above liquid helium and thus lowering its boiling point, one can ob- 4th ed., Clarendon Press, tain T 1 K. Temperatures between 1 and 0.4 K can be reached by Oxford (2002). . C15.8 rectifi- ndustrial transitory ; : 2 2 :Atagiven He isotopes 15.9 4 and 35 %and N 63 % N He and 3 to transfer the temperature 2 2 of gases, in particular for sep- bout 1.33 liter/kWh (the maximum effect, i.e. according to the scheme tilizers!) and oxygen for example in H or tritium, was carried out on an i 3 separation He. It is circulated in a closed-cycle cryo- is rather small in the ideal case, namely 3 He is currently again rather expensive. 5 3 HOMSON air -T . The yield of liquid air from all the different pro- . It is required only in order to compress the two gases OULE 15.8 3 and the Separation of Gases countercurrent heat exchanger ). , but rather by using mixtures of in the gaseous phase: 37 % O in the liquid phase: 65 % O 17.6 . It is based on the fact illustrated in Fig. alone This rare isotope of helium became more widely available, and affordable, since He 0.014 kWh/m arating air into oxygenin and the nitrogen. synthesis of Nitrogen ammonia iswelding (fer and used for for medical example for purposes. the – separation The amount of of work required showninFig. possible yield from an ideal process wouldLiquefied be gases 5.3 are liter/kWh). indispensableand cooling research resources today. Liquefaction for on technology aamong technological other scale things is for required the cesses is roughly the same. It is a temperature, air (like manyhas a other different mixtures composition ofcentages in of different the the liquid substances) amount and ofboiling the point), substance gas for (mole phases. example, percent), consists In air of: per- at 83 K (its from their partial pressures up toEvery atmospheric pressure. gas separationmolecules. is hampered For by thisand reason, the separates one thermal the first motionscooling cools gas of can the mixture in air the at principle untilone a be it uses a carried low liquefies out temperature. without expending A work, if The actual separation process is known under the name of cation (Sect. stat, alternating continuously between the liquid(similarly and to the the vapor cooling phase substance– Still used lower in temperatures household (down to refrigerators). 0.005 K) can be obtained not with the manufacture of its parent isotope, 5 15.6 Technical Liquefaction Processes The liquefaction of gases haswell considerable as economic scientific importance, interest. as Theway various in processes which differ the mainly gas is inby precooled. the adiabatic Often, expansion precooling in is a accomplished bine: piston Then and the cylinder work machine performedThe or by last the in cooling gas stage a prior can tur- to beby liquefaction put using is to frequently the good implemented J use. – using the helium isotope 3 scale for military purposes. He 4 He- 3 Matter in the last OHL , Springer, 2nd Real Gases 15 “dilution refrigeration”, mentioned by P ed. (1996). editions, is very widespread today. More details can be found for example in the book by F. Pobell, and Methods at Low Tem- peratures C15.8. The production of low temperatures by 420 Part III 15.7 Vapor Pressure and Boiling Temperature. The Triple Point 421

Figure 15.9 The separation of air into its component gases by ‘rectification’. The gentle slope of the tube in the lower image indicates that the liquid current flow is maintained by gravity. Technical rectification columns are vertical, and the pure oxygen is drawn off as liquid at the bottom. The distribution of concentrations observed in such a plant is shown in the upper image; the difference in the vapor/liquid distributions of O2 and N2 near their boiling points is an example of their differing ‘relative volatilities’, often observed for distinct liquids, which can then be separated by distillation. Part III

The essential point of a rectification process is shown in Fig. 15.9: currents of liquid and gas phases flow in intimate contact with each other in opposite directions through a tube along which a temperature gradient is maintained. The liquid flows in the direction of increasing temperature. Nitrogen, which boils at 77 K, evaporates preferentially from the liquid current, while oxygen, boiling point 90 K, condenses preferentially out of the gas phase. When the flow rate is sufficiently slow, an equilibrium is established at every point along the tube, corresponding to the local temperature at that point. For example, in a section of the tube which is at 83 K, the liquid and the gas phases have the different compositions marked by dashed arrows in the figure, as mentioned above. In the technical implementation of rectification plants, care is taken in particular to provide a close contact and mutual mixing of the two oppositely-directed currents. The yield of pure oxygen from well-designed plants is about 2 m3/kWh (much less than the ideally possible yield of 14 m3/kWh).

15.7 Vapor Pressure and Boiling Temperature. The Triple Point

The p-Vm diagram of a substance (for example CO2,asinFig.15.2) does not permit us to discern an important relation, namely the dependence of the vapor pressure on the temperature. This relation is best represented in a p-T diagram. An example for CO2 can be seen in Fig. 15.10, and for water in Fig. 15.11. In both cases, the ordinate of the diagram is logarithmic, i.e. it shows increasing orders of magnitude. coexist of the liquid; of the solid phase. of the solid phase; so that the pressure ı boiling point melting point . sublimation temperature : the saturation pressure of the solid phase. Finally, . With 2 frost The phase The phase (Video 15.2) . slopes. Compare at these paired values can two phases of the substance 15.12 the full curve gives the axis becomes the abscissa. Thendashed for curve every shows value of the the corresponding pressure, the Fig. diagram of water. At the triple point, all three curves intersect each other with different diagram of CO and the dot-dashed curve shows the a linear scale on thedinate, or- instead of the logarithmic scale used here, the curves would rise very steeply. Figure 15.10 These diagrams each containdenotes three a curves. matchedOnly pair Every point of on the a variables curve pressure and temperature. These two figures can also be rotated by 90 Figure 15.11 in a stable manner, that is in equilibriumThe with dashed each curve other. correspondsliquefy to the the gas, pressure known whichcorresponds as is to the required the to saturation pressureformation pressure. of of solidification The of solid the curve gas, that is the the third, dot-dashed curveliquid phase corresponds to the solidification of the Real Gases 15 frozen. Note that the density of the solid phase isthan greater that of the liquid phase. “Liquid and solid nitrogen” http://tiny.cc/ihgvjy By “pumping off”, liquid nitrogen can be cooled and Video 15.2: 422 Part III 15.7 Vapor Pressure and Boiling Temperature. The Triple Point 423

Below a pressure of 500 105 Pa, the melting temperature depends only weakly on the pressure. For CO2, the melting temperature rises somewhat with increasing pressure; for water, it is lowered. The three curves have one common point of intersection, the so- called triple point. The data for the triple point are:

ı 5 for CO2 W T D 217 K .56:2 C/; p D 5 10 Pa ; ı for H2O W T D 273:16 K .0 C/; p D 611 Pa :

At the triple point – but only there – all three phases, solid, liquid, and gaseous, can exist together. They are in equilibrium; no one of the three phases can grow at the cost of the other two. In Fig. 15.2,weleft the three points marked by small open circles without explanation. Their meaning is now clear: they correspond to the triple point. At this point, at a temperature of T D 217 K (56.2 °C) and a pressure 5 of p D 5 10 Pa, the molar volume Vm

3 of solid CO2 is 0:034 m =kmol; 3 : = Part III of liquid CO2 is 0 041 m kmol, and : 3= : of gaseous CO2 is 3 22 m kmol

Away from the triple point, at most two phases can exist together; along the solid curve, only a solid phase and its saturated vapor. At pressures below 611 Pa, ice can no longer melt, it can only sub- lime (evaporate). Likewise, at atmospheric pressure (105 Pa), it is not possible to produce liquid carbon dioxide, only CO2 snow (the well-known ‘dry ice’) at T D 194 K (79.2 °C).

The production of dry ice is very simple (Video 15.3): The CO2 cylinders Video 15.3: which are commercially available have a pressure of about 50 105 Pa at “Solid carbon dioxide (dry room temperature; according to Fig. 15.10, they contain a mixture of liq- ice)” uid and gaseous CO . A thick cloth bag is attached to the cylinder valve 2 http://tiny.cc/lhgvjy and the valve is cautiously opened, so that gas from the cylinder ﬂows out In this video, showing through the bag. As it ﬂows out, the CO2 gas forms a jet and performs work of acceleration (external work to accelerate the molecules in the jet). In ad- part of a lecture given by dition, due to the JOULE-THOMSON effect, it also performs internal work, Prof. BEUERMANN,the i.e. work against the mutual attractive forces between its molecules. For preparation of dry ice is both reasons, the CO2 cools until it reaches the temperature corresponding explained and demonstrated. on the curve to the ambient atmospheric pressure of 105 Pa, i.e. 79°C (see Figs. 15.6 and 15.10, and Sect. 15.4).

The contents of Figs. 15.10 and 15.11 form the basis of GIBBS’ phase rule for a system containing a single substance: The number of freely disposable state variables is equal to 3 minus the number of phases which are in equilibrium. When all three phases are in equilibrium, no state variable is free; all three have their fixed values at the triple point. – When two phases of one substance are in equilibrium, only one of the two state variables p and T is freely disposable; the other is fixed on one of the curves in the p-T diagram. – In order to obtain only one phase of a substance, one can choose both state variables p and T arbitrarily; every pair p and T on the diagram is allowed; the values no longer have to remain on one of the curves. . structure 33 °C. The 72 °C without cubic . nucleation centers be exceeded without not , an enlarged section from supercooled 15.12 which act as 10 °C. Smaller amounts of water, 70 °C. – The vapor-pressure curve of Solid. Supercooled without passing through the phase transition ! ). dashed The undershot Liquid Liquids ). solid: Liquids may be strongly 20 °C and shaken or stirred, avoiding splashing, the water submicroscopic impurities ! 15.11 For comparison, the vapor- pressure curve of ice is drawn in as a thin solid line. vapor-pressure curve of supercooled water ( Figure 15.12 a characteristic andlar precisely-determined substance. quantity The for meltingmelting that temperature of particu- can the solid;reverse direction, that the is, situation its isconsiderably different: outer The layers melting become point liquid. can be In the If a boiling flaskbath containing at dust-free water iscan dipped be into readily a cooled cooling several to tenths around of aactual gram, solidification can temperatures be are supercooledvarious determined down by to the presence of 15.8 Hindrance of the Phase Transition The melting temperature of a crystalline solid at a given pressure is Tiny water dropletsfreezing. can The even water must be benucleation purified cooled of points impurities down for which could crystallization to ing. act by as repeated At freezingwith this and a temperature, melt- solidification point thea of supercooled ice liquid is crystallizes a continuation inuid of without the a bends curve for or the kinksFig. normal (cf. liq- Fig. liquid The process of crystallizationto initiated the at formation of these hexagonal centers ice. always leads Real Gases 15 424 Part III 15.9 Hindrance of the Phase Transition Liquid $ Gas:The Tensile Strength of Liquids 425 15.9 Hindrance of the Phase Transition Liquid $ Gas: The Tensile Strength of Liquids

The phase transition gaseous ! liquid can also be hindered by removing nucleation centers. Saturated vapors can be strongly supercooled, most simply by an adiabatic expansion. Here, too, the phase transition can be induced retroactively by introducing nucleation centers. Suitable for this purpose, among many other species, are ions of any type. At such nucleation centers, surfaces are formed from supersaturated vapors, leading to the precipitation of droplets as fog (This is applied for example in a cloud chamber used for the detection of ionizing radiation). Furthermore, the phase transition from liquid ! gaseous can also be strongly hindered by removing nucleation centers. For a demonstration, we fill a test tube which has been well cleaned using chromic acid with double-distilled water and heat it slowly in an oil bath. Part III The water can reach a temperature of around 140 °C without boiling. It persists in a quiet state, with evaporation from its surface. Then, suddenly, within the water a turbulent transition to vapor sets in; the contents of the tube are ejected explosively. Boiling water can produce a genuinely dangerous situation. That is why, in practice, a delay in boiling, accompanied by ‘bumping’, should be avoided if possible. A simple protection lies in using ‘dirty’ vessels. A better method is to introduce small, angular objects (“boiling stones”), for example small shards of porcelain. The pores in their surfaces permit the stabilization of small gas bubbles (i.e. by preventing their reabsorption), which then serve as nucleation points. To form such free surfaces within liquids, it is by no means always necessary to increase their temperature. One can also cause the liquid to rupture. This requires tensile forces of the order of 100 105 Pa (Sect. 9.5). Is there a prospect of increasing this value still further by removing perturbing nucleation centers? The VA N D E R WAALS equation replies positively: Fig. 15.13 shows the isotherm for water at 300 °C in a p-V diagram. It was determined experimentally, like the two limiting curves ˛00K and Kˇ. The isotherm ˛0ˇ0 shows the normal process, where along the linear segment of the curve ˛0ˇ0, a surface is formed. In this region, two phases are present, and therefore, we cannot apply the VA N D E R WAALS equation here. It is probable that the nucleation centers can be effectively removed so that the linear segment of the isotherms can be suppressed. If this is indeed possible, then only one phase will be present, and then we can apply the VA N D E R WAALS equation also between the limiting curves. This is indicated in the curve segment ˛"ıˇ.Thisrepre- sents the complete isotherm as calculated for 18 °C. It of course is an approximation, as is every application of the VA N D E R WAALS equation. For example, the point ˛ should lie on the left-hand limiting , D Pa 7 % AALS 10 ’s equa- W 2 D AALS ile or rupture p W , it follows that axis at a point 1000 bar). There- VA N D E R pressures, that is to m 15 K). b) Show that : 9.23 V 21 or 2 D VA N D E R 293 K and and above the curve segment T negative D "ıˇ . T hould exhibit a tens upper limiting temperature. This 100 N/mm D equation. – As a supplement to the text, we Pa ( (Video 9.4) 8 of nitrogen at AALS C15.9 m W , the isotherm intersects the V cr calculated in a) also fulfills the cannot correspond to stable states. Along this segment, an T of over 10 m 27 32 ı V Computing the tensile strength of “nucleus-free” water by ap- (this value was measured at . – Nevertheless, one result is valid: The isotherm leads 3 D 00 VA N D E R ˛ T From the measurements shown in Fig. are not equal, as they would be with a linear scale on the abscissa. – The of the 18° isotherm would fall practically along the axis of the abscissa, 026 g/cm : where the tensile strength thus vanishes. tion: at the value of using the equation ofmolar state volume of for crystalline ideal (solid) gases nitrogen, and with compare a it density with of the 1 strength of this order fore, nucleus-free water at 18 °C s plying the quote the following data:˛ˇ In “nucleus-containing” water,since the the linear saturation segment pressureder of to water show vapor at allThus, 18 the °C the data is areas only in below˛" 2.13 this hPa. the figure, curve In the segment or- curve abscissa segment has a logarithmic scale. equation of state to a good approximation. Exercise 15.1 curve at The tensile strengthperature. of all It liquids becomesexperimental fact zero decreases is also at with in agreement an increasing with tem- Figure 15.13 for certain values of the molar volume to increase in the molar volume leads to a decrease in the tensile strength. air at temperatures abovelike 0 °C an ideal and gas. pressures This upthe is molar to to volume 100 be bar quantitatively behaves confirmed: a) Calculate tensile stresses ; see also ). 8.2 9.5 ) 9.5 Real Gases 15 (see Sect. “The tensile strength of water” http://tiny.cc/3vqujy Video 9.4: C15.9. Comparable to the tensile strengths of solids (see Table Sect. 426 Part III Exercise 427

Electronic supplementary material The online version of this chapter (doi: 10.1007/978-3-319-40046-4_15) contains supplementary material, which is available to authorized users. Part III eta admMotion Random as Heat u ( Eqns. Combining gases, form ideal the encountered of already state have we of equation experimentally-determined the For pressure: the by for equation molecules following the the derived (Sect. replace we exper- ments, balls these model steel versions, of elastic simplest aid small, their the In with “thermal understood short iments. readily for motion, be random can of concept motions”, the that seen have We Molecular the on Temperature 16.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © Furthermore, h iei energy kinetic the xrse as expressed hseuto ttsthat states equation This eprtr fagsi eemndb h iei nryE energy kinetic the by determined is gas a of temperature oeueo nielgsi rprinlt h eprtr n is and temperature m the mass its to or proportional nature chemical is its of gas independent ideal an of molecule 2 stema au ftesurdvlct;telf iei hstwice thus is side left the velocity; squared the of value mean the is ( ( tions.) oeue and molecule, N % D stenme fmlclsi h volume the in molecules of number the is est ftegas, the of density Scale k steB the is E 9.14 kin of ,( ), OLTZMANN olsItouto oPhysics to Introduction Pohl’s u one 14.23 D h vrg iei nryE energy kinetic average the E m pV eoiyo h oeua rnltoa mo- translational molecular the of velocity p kin % u oeue hnteeeg a lobe also can energy the Then molecule. 2 D D )and( D 9.7 D D osat(q ( (Eq. constant 3 1 NkT Nm .Bsdo uhmdlexperi- model such on Based ). 3 % 2 3 V kT kT u 2 9.15 : : : yields ) V , m 14.24 h aso single a of mass the rconversely: Or . O 10.1007/978-3-319-40046-4_16 DOI , ))). kin kin feach of ( 14.23 ( (16.1) ( (16.2) 9.15 9.14 fits of The ) ) ) 16 429

Part III . ) ) D ,we pass M 16.3 14.56 (16.3) (16.4) (16.6) (16.5) ( u volume V within the /: gives for the 1 A ))). p RT 1 m mol : . Then, in a time molecules is M 14.22 .–Agasvolume is the total mass of ,ortheir 23 m amount of substance m N s 10 RT 9.8 : M A Nm 3 2 mass-current density D s p pN 022 : : m 4 6 m M : D m ) at a pressure RT M sound 0 of these . A more precise calculation RT M RT D c M ; r kT D m RT s p 3 Nm where = 4 14.21 1 : 4 , s m = molecules with the velocity : N 0 At r 3 0 M D constant p C16.1 D D ), it follows that the velocity of the gas D p Aut 2 p ,wehave ; 1 1 u sound V sound At number of molecules N . We refer to Sect. c V N M c At 16.1 29 q : 1 6 sound is the density of the gas. The equation of state D RT . Inserting these expressions into Eq. ( c 0 1 m D u m , we could also refer to the 1 is the universal gas constant (Eq. ( VOGADRO 1 D V RT M N R = rms . The total mass N m the molar mass u s D M A At molecules, each having a mass p D is the A % 1 4 mass M pM : n A N 0 = N . D M .–FromEq.( D total mass of the molecules which pass through the area % ). D t ,where n D At m M M The thermal motions of molecules during evaporation and ex- contains the molecules, and time ( ( Aut of the molecules, or the 1 % For the velocity of sound , approximately 1 6 obtain changes only the numerical factor; for the and of an ideal gas in the form of Eq. ( through an area yields the ratio t molecules has an average value of 1 Instead of the We list here a few applications of these equations: 1. pansion through a nozzle molecules V Then we obtain the corresponding current densities n Therefore, the velocity of the gas molecules is density ’ here. m in the chemical M ) is usually denoted 13 Heat as Random Motion (see Comment C13.7 in m 16 C16.1. The “molar mass” M Chap. Remarkably, in most physics textbooks in use today, the molar mass is not mentioned at all! simply by literature. In order to avoid confusion with the mass in general, we will continue to use the index ‘ 430 Part III 16.1 Temperature on the Molecular Scale 431

Example Water at 293 K (20 °C) and at its saturation pressure of p D 2:32 103 Pa. R D 8:31 W s/(mol K), Mm D 18 g/mol. Inserting these values yields the particle current density N 1026 : At m2 s At room temperature, every second around 1026 molecules escape from each m2 of water surface, and the same number returns from the saturated vapor above the surface. – In 1 m2 of surface area, only about 1019 water molecules are located within the surface layer (cf. Fig. 15.5). Therefore, the dwell time of an individual molecule at the surface is on the average only about 107 sC16.2. C16.2. This explains why water vapor in bubbles is Equation (16.6) leads to the ratio of the times t1 and t2 in which, quickly liqueﬁed when the at a given temperature and pressure, equal volumes of two different bubbles shrink, as we can gases escape through an opening: conclude from the observations in Video 9.4 (http:// s r tiny.cc/3vqujy,seetheex- t1 Mm;1 m1 D D : (16.7) planation beside Fig. 9.16). t2 Mm;2 m2

We could also think of the Part III bubbles which occur in cavi-

From this relation, the molar masses Mm (or the molecular masses m) tation and cause strong local of different gases can be compared (R. BUNSEN). Figure 16.1 shows heating, producing ﬂashes of a tried-and-tested experimental arrangement. The gas is conﬁned at light (sonoluminescence, see the left under the pressure p of a mercury column. At the upper end Comment C9.12). This also of the container is a small opening in a thin metal disk. leads us to suppose that the gas phase disappears rapidly. For demonstration experiments, we replace the small opening at the top of the tube by the porous wall of a clay cylinder (Fig. 16.2). At its bottom, a water manometer is connected to the cylinder, and a wide beaker is placed over it. Then for example hydrogen is blown into the beaker. The pressure within the clay cylinder rises sharply. The

Figure 16.1 The comparison of molecular masses, after R. BUNSEN. One measures the time in which the mercury (shown black in the figure) in the left-hand section of the tube rises from the lower mark to the upper one. shows ). ;however ). Reason: 16.3 .Everygas 9.24 14.9 . molecules diffuse h Video 16.1 2 onto a glass plate. It un- h ) Video 16.1 Diffusion through a porous clay Demonstration of diffusion using molecules diffuse in large numbers into the clay cylinder, 2 , it is dropped from a height 16.4 Temperature changes accompanying volume changes 2. more rapidly out of the cylinderthem than can the diffuse air inwards. molecules which replace Given the great importancement of with diffusion our ‘steel-ball processes, gas’ a is quite model appropriate. experi- Figure here, it is divided inthermore, the on center both by sides athe there wall artificial are thermal with oscillating motions. a pistons small Withlimited which static opening. to maintain pictures snapshots; Fur- in they aeffect give book, of only we this are a demonstration experiment pale in impression action of ( the lively plate and rises up to well above its original height warms on compression and cools on expansion (Sect. dergoes an elastica reflection second, and smaller flies glassthe up plate rising is again; ball moved at isflies downwards reflected the by back on hand. same down a time, withseveral Now, wall times, an which then increased the is ball velocity. moving is towards allowed This to it; game fly it up is past repeated the second glass the setup, which we have already encountered in Fig. On expansion, the molecules are reflecteding from away, a and wall this which reduces isis their mov- velocities. moving On inwards compression, into theare wall the reflected container, from so itcan that experience be an the clearly increase molecules demonstrated in which Fig. their with velocities. a single This ‘steel-ball molecule’. In cylinder about four times faster than theAfter slower several air seconds, molecules the can diffuseis hydrogen out. flow removed. is stopped Veryconverted and into quickly, a the reduced the pressure. beaker overpressure The confined H in the clay cylinder is Figure 16.2 two steel-ball model gases. Top:there are At initially the only left, small ‘molecules’,the at right only large ones.in Bottom: the separating The wall opening hasdiffusion been has unblocked begun and (see also Figure 16.3 reason: H ian ROWN , diffusion, os- Heat as Random Motion 16.14 16 principle is explained in Fig. mosis and the B “Model experiments on diffusion and osmosis” http://tiny.cc/xhgvjy With an apparatus whose Video 16.1: can diffuse through, but not the large balls (a “semipermeable membrane”). motion are illustrated with the model. The openings in the separating wall are large enough that the small balls 432 Part III 16.2 The Recoil of Gas Molecules Upon Reflection. The “Radiometer Force” 433

Figure 16.4 A model experiment on warming of a gas by compression (HARALD C16.3 SCHULZE ) C16.3. HARALD SCHULZE (1900-1997), student in the experimental physics lectures of the author. The experiment is mentioned already in the ﬁrst edition of the “Thermo- 16.2 The Recoil of Gas Molecules Upon dynamics” lectures (1941). Reﬂection. The “Radiometer Force”

Figure 16.5 shows schematically a glass bulb which can be evacuated. It contains a small plate P, for example of aluminum foil or mica. The plate is mounted on a leaf spring and this serves as a force meter. – If one produces a temperature gradient between the two surfaces of the plate at a low gas pressure, then a force F acts on the plate in the direction of decreasing temperature. This phenomenon is called the Part III radiometer effect. This somewhat misleading name is due to the fact that the temperature difference is usually produced by irradiating one side of the plate with light. Especially suitable for demonstration experiments is the “light mill” or CROOKES’ radiometer (Fig. 16.6). In it, four mica platelets, blackened with soot on one side, are mounted as the blades of a pinwheel. The wheel has a point and socket bearing at its hub, allowing it to rotate freely in a horizontal plane, as seen in Fig. 12.50. When irradiated with light, it always begins to rotate in the direction indicated

Figure 16.5 A schematic arrangement for detecting the radiometer effect, which results from the recoil of the reﬂected gas molecules. For quantitative measurements, a torsion band should be used instead of the leaf spring F shown here.

Figure 16.6 A horizontal section through C16.4. Sir WILLIAM a“CROOKES radiometer” (W. CROOKESC16.4, CROOKES (1832–1919). 1874) (Video 16.2) He studied gas discharges and discovered the element thallium.

Video 16.2: “CROOKES radiometer (light mill)” http://tiny.cc/qhgvjy F mmHg –2 the light from the 10 away towards –3 10 with increasing gas pres- ). The details would take Pressure in the radiometer bulb 16.7 –4 ^ =10 0.01

∙K

0.005 F decreases 2 Area of blades ∙Temperature difference ∙Temperature blades of Area rgo few collisions with each other;

cm millipond force Radiometer depends in a characteristic way on the gas F thus ). In the low-pressure range, the force increases , N). 6 16.7 C16.5 10 33 hPa, 1 mil- The depen- : (Fig. 1 81 : p 9 ESTPHAL D us too far afieldticles here. or – thin The fiberscarbon radiometer are particles effects extremely on in strange. small thetiny focus suspended For spiral-helical par- of example, orbits the small for condenser suspended hours; of(photophoresis). the a orbits projection themselves lamp form follow closed rings At higher gas pressures,longer long the compared mean tophenomena free can the occur, path and dimensions the of force of the the gas radiometer. molecules is Then, no other sure (dashed segment of the curve in Fig. dence of the radiometer force on the gas pressure in the low-pressure range. A temperature difference is present between the two surfaces of the platelet. (Original measurement by W.H. W Figure 16.7 pressure instead, they are reflectedwalls only of by the the bulb. radiometerature When blades than the the and blackened shiny by surface surface,a the has the higher a molecules higher average are temper- reflected velocityreflected from molecules than it produce from with a theon recoil cooler the on warm shiny both side than sides; surface. on but the it The cooler side. is Thus, greater the resultant force proportionally to the pressure.of the In gas this moleculesradiometer. range, are The the large molecules compared mean unde to free the paths dimensions of the points in the direction of decreasing temperature.tional The to force is the propor- frequency ofto the the collisions gas with pressure. the blade, and therefore light source. The radiometer force is then directed source. Radiometers rotate evensides. when The the essential light point istwo is the coming surfaces difference from in of all light thetween absorption the blades; by two the sides it and produces thus causes a theThe radiometer temperature force. radiometer gradient force beby the arrow. Its rotationalthe frequency light. increases with the intensity of In spite of itsradiation, name, in the particular radiometer notble light. effect with has the This nothing can tinywhich be to is radiation most blackened do pressure with simply with of soot shown on by visi- the using side a which sheet faces of mica the old-fashioned units: 1 mmHg lipond EINRICH H (1882–1978), , 207 (1920). 2 ILHELM Heat as Random Motion ESTPHAL 16 C16.5. W from 1920 professor of physics at the University of Berlin, from 1935 at the Technical College in Berlin; author of several textbooks. See also W. Gerlach, Z. Physik W 434 Part III 16.3 The Velocity Distribution and the Mean Free Pathof the Gas Molecules 435 16.3 The Velocity Distribution and the Mean Free Path of the Gas Molecules

We now know two methods for determining the velocities of gas molecules experimentally (Sects. 9.8 and 16.1). Both methods are based on applications of momentum conservation and yield only average velocities. However, the distribution of the velocities around this average value can also be determined experimentally. To this end, one makes use of molecular beams.InFig.16.8, a small block of silver Ag is vaporized from an electrically-heated molybdenum crucible. The highly-evacuated glass vessel is equipped with two slits, A and B. They select a sharply bundled beam from the silver atoms which are flying in all directions out of the crucible. This beam collides with the cooled wall W. There, the atoms form a sharply- bounded, highly reflective spot. Its shape corresponds to the path of the beam shown as a dashed line in the figure. Above the second Part III slit B, the glass walls of the vessel remain free of precipitated silver. To measure the velocities of the silver atoms, the whole apparatus is mounted on a rapidly-rotating turntable, whose rotation axis is perpendicular to the plane of the paper in Fig. 16.8. We then have the same situation as described in Chap. 7 for the measurement of the velocity of a bullet. The silver atoms are deflected to one side by the Coriolis force, and from the degree of deflection, we can compute their velocities (Sect. 7.3, Point 5). Carrying out this measurement with a similar apparatus yields the results as shown in Fig. 16.9 for nitrogen at two temperatures: A distribution of the velocities over a wide range is observed.

The distribution over a wide range of velocities can be represented by the “distribution law” derived by MAXWELL. It gives the fraction dN=N of

Figure 16.8 The production of molecular beams. ‘Ag’ is the silver which is being C16.7 evaporated from a Mo crucible . C16.7. OTTO STERN used the experiment in Fig. 16.8 to determine the average thermal velocity of Ag atoms (Zeitschrift f. Physik 2,49 and 3, 417 (1920)). The velocity distribution as in Fig. 16.9 was measured with a more advanced apparatus using Cs atoms, described in his Nobel lecture (1946) and by J. Estermann, O.C. Simp- son, and O. Stern, Phys. Rev. 71, 238 (1947). D f ) ’s u 16.3 (aver- (16.9) ( 509 m/s). (16.10) or most (16.8) D of the ve- s AXWELL 22 than the f mean : u rms 1 u .M / 22 D : u 1 293 K (20 °C), d 3 2 C D D q u u d . T : 2 rms : : 2 u f mu m kT N and u 1 2 m RT M frequently-occurring 500°C RT 3 u M 2 13 08 times higher than the mean : e : s 1 3 2 s 1 u Á D rms D D D u f kT u m kT f m 2 kT 22 13 m 3 u : : 2 2 1 1 : p Velocity Velocity r 28 g/mol; thus at 2 rms r 20°C u u is still lacking. This is the name given to 4 D D f p D D C16.6 u 500 1000 1500 2000m ,and m f f m D rms u u u u is thus higher by a factor of M N N d corresponds to the most rms u is given by oduced the rms (“root-mean-square”) value 0

16.9 1

0.5 1.5

+10 m/s +10 and between velocities u u u

. – The differences between the most probable, the mean, and

2 The velocity distribution of gas molecules using the example Fraction of N of Fraction molecules with molecules m mean free path velocity: u probable distribution law age) velocity velocity Taking the arithmetic mean of all the velocities, we obtain the This rms velocity most probable velocity A detailed examination ofcurves this in equation Fig. shows that the maximum of the It is thus somewhat higherInitially, we than intr the most probable velocity. the rms velocities are thus of no practical importance. locity, deﬁned by the equation the molecules whose velocities lie between The velocities are marked by arrows. 417 m/s and the root mean squared velocity is Our molecular picture of a gascept is of now the nearly complete. Onlythe straight-line the path con- of a molecule between two successive collisions Figure 16.9 of nitrogen (molar mass ,3rd OLTZMANN -B Heat as Random Motion AXWELL 16 C16.6. Often called the M distribution. For its derivation, see e.g.: W. Nolting, Statistische Physik ed., Vieweg Braun- schweig/Wiesbaden (1998), Exercise 1.3.7. English: see https://en.wikipedia.org/wiki/ Boltzmann_distribution 436 Part III 16.4 Molar Heat Capacities in a Molecular Picture.The Equipartition Principle 437 with other gas molecules. – Nitrogen, to mention one example, has a mean free path of l 6 108 m at 0 °C and 1013 hPa, roughly 20 times larger than the average molecular separation under these conditions. Experiments for determining l will be described in Sect. 17.10.

16.4 Molar Heat Capacities in a Molecular Picture. The Equipartition Principle

The molar heat capacities of ideal gases can be understood in C16.8 a molecular picture as follows (A. NAUMANN, 1867): We write C16.8. ALEXANDER the corresponding defining equations (14.12)and(14.13), as well as NAUMANN, Gießen, Ann. der Eq. (14.32) for the two molar heat capacities Chemie 142, 265 (1867); see Â Ã also the book by the same au- 1 @U thor: “Lehr- und Handbuch CV D ; (16.11) n @T der Thermochemie”,Verlag Part III Â Ã Vieweg (1882), Chap. 9. 1 @H NAUMANN was not aware C D D C C R : (16.12) p n @T V of the rotational degrees of freedom and interpreted For an ideal monatomic gas, that part of the internal energy U which their contribution to the is due to random motion is mainly the kinetic energy Ekin of the linear specific heat as molecular motions of the molecules; briefly, their kinetic energy of translational vibrations (“heat of atomic motion. A single molecule makes a contribution of motions”). – The rotational degrees of freedom (zero 3 3 nRT for single atoms, two for Ekin D kT D : (16.2) 2 2 N diatomic and three for polyatomic molecules) became A gas with N molecules or the amount of substance n gives the con- understandable only after tribution 3 RUTHERFORD showed in U D NE D nRT : (16.13) 1911 that the masses of kin 2 atoms are concentrated in Therefore, from Eqns. (16.11)and(16.12), we find their atomic nuclei! (see the footnote on the next page). 3 5 Cp CV D R ; Cp D R ; D 1:67 : 2 2 CV

A monatomic molecule has three degrees of freedom (of translational motion). That means that its velocity along its linear path (translation) consists in general of three components, one in each of the directions of three-dimensional space. Each one of these three degrees of freedom of a single molecule contributes the kinetic energy

1 1 nRT E D kT D (16.14) kin 2 2 N

(k D BOLTZMANNconstant D 1:381023 Ws/K,R D 8:31 W s/(mol K)); V . C = p (16.16) (16.15) C 14.1 diatomic : D 33 : 1.67 1.40 1.33 1 : D 40 : of a gas containing p V 1 n C C : D ), we find for V ; R R p V R 3 2 5 2 C Vol ume 3 C R C nRT 4 : 2 1 16.12 ; D 31 W s/(mol K)) principle of statistical equipar- : D p R 8 nRT C 2 7 kin 2 5 D )and( D R ; p R R NE D R ,givenby p 5 2 7 2 Molar heat capacities C at constant Pressure 4 R n C U D 3 16.11 2 D tri- and polyatomic gases ; 3 N kin,N R R ; 2 E 2 NH 2 5 O ; 3 4 CO ; D . This contributes two additional degrees of free- 2 2 CO, HCl C Hg vapor 2 V H CH noble gases 3 C Examples D summarizes these results for several gases. They agree Molar heat capacities ( V C 16.1 : . Thus a diatomic gas has all together five degrees of freedom. molecules, we find for its kinetic energy: The rotation around the long axis itself need not be considered, since its moment monatomic diatomic polyatomic Type of molecule As a result, from Eqns. ( gases A diatomic moleculearound has two axes, the each perpendicular shape toof the of the other dumbbell and a to dumbbell. the long axis It can rotate dom. These new degrees ofto freedom the are translations; considered to this be is equivalent called the tition They contribute a kinetic part toing the the internal amount energy of of substance a gas contain- Table 16.1 well in general with the measured values as shown in Table Table N of inertia is vanishingly small. 2 or, as the contribution of an amount of substance Tri- and polyatomic moleculestheir rotations. have Together with three their three degrees degreeslation, of freedom one of for thus trans- freedom finds for for Heat as Random Motion 16 438 Part III 16.4 Molar Heat Capacities in a Molecular Picture.The Equipartition Principle 439

5 R 4 2 2.0∙10 H W s v 2

C kmol∙K Ar, He 1.5 3 R 2 1.0

1 R 0.5 2

Molar heat capacity 0 0 100 200 300 K Temperature T

Figure 16.10 The molar heat capacity of hydrogen as a function of the temperature (R is the gas constant D 8.31 W s/(mol K)). For comparison, the results for two monatomic gases are also shown. In the dashed region, H2 is liquid and solid (cf. Fig. 14.5).

In solids, the molecular building blocks have fixed rest positions. They cannot participate in progressive motions. Instead, the atoms can vibrate around their rest positions. For the kinetic energy of these Part III vibrations, there are three degrees of freedom. By themselves, they 3 C16.9 would give Cp D 2 R . To maintain thermal equilibrium, in the C16.9. For solids, there is solid we must add an equal amount of potential energy of vibration. practically no difference be- 3 3 tween C and C , as long This second energy contribution leads to Cp D 2 R C 2 R D 3R.This p V explains the limiting value of Cp 3R which is found for most solids as nonlinear effects can be built out of many individual atoms. (This is the rule of DULONG and neglected, i.e. as long as the PETITC16.10, cf. Fig. 14.5.) temperature is not too high (T Ä 300 K). Since measure- These successes gave experimental support to the equipartition prin- ments are always carried out ciple. But in any case, it must be considered to be only an idealization at constant pressure, only Cp of the limiting case, permissible in the range of high temperatures. is given here in the text. This can be seen from the measurements given in Fig. 16.10.They refer to the molar heat capacity of a diatomic gas (H2) at various tem- 5 peratures. At high temperatures, CV has a value of 2 R;butasthe C16.10. A.T. Petit, P.L. Du- 3 long, Ann. Chim. Phys., 2nd temperature is lowered, it decreases and finally reaches a value of 2 R, i.e. the value expected for monatomic molecules. – Interpretation: As Series, Vol. 10 (1819), p. 395. the temperature decreases, the rotations gradually come to rest; only translational motions remain, as for monatomic molecules. – At this point, we can go no further with the methods of “classical physics”. We instead must turn to quantum mechanicsC16.11. C16.11. The same is true of the lattice vibrations in We summarize the essentials of Sects. 16.1 through 16.4: The ideal solids. For a discussion of the gases gave us the opportunity to interpret the state variables tem- temperature dependence of perature and an important part of the internal energy in a graphic the specific heat as seen in manner. The directly measurable state variables temperature and Fig. 14.5, see e.g. R.O. Pohl, pressure arise from the random thermal motions of the molecules Am.J.Phys.55, 240 (1987). (Sect. 16.1). They are found to be statistical averages over an enormous number of individuals (molecules or atoms). We can make only statistical statements about the individual molecules. According to the equipartition principle, we can say: On statistical average, each : ). ). 6 ) Oin 2 16.2 .The ,made ). OH) and ). 3H 16.14 2 osmosis. Fe(CN) ( 16.6 , i.e. pene- 4 6 16.12 CH RAUBE production re- 3 16.11 sK Anex- Ws/K. T 23 Fe(CN) .There,weseetwo 4 10 ; the surrounding so- ORITZ artificial 38 16.13 semipermeable : , a pressure difference will alf-hour, plant-like formations kT 2 1 D will be described in Sect. ) onto the bottom of a cuvette filled constant, 1 2 kin k temporarily E . Its preparation is simple: One adds for exam- cell membranes. The best OLTZMANN pressure difference between the solution and the sol- . Osmotic phenomena can also be observed when dif- , 1748). In this case, also, the pressure difference is living steady is the B OLLET k This phenomenon is what today is exclusively called temporary copper ferrocyanate 1 liter of water). In the course of a h will grow from the crystals up towards the surface (Fig. Experiments of thisgenerically kind described on and the their basis of quantitative Fig. extensions can be of closed at the bottoma with dish the containing membrane pure(I. from A. water. a N pig’s The bladder membrane into will bulge outwards perimental determination of following section has the goal of preparing us for that task. lution becomes more concentratedsinks due to to the the bottom loss offorming of the visible this container striations as water, (shadow a projection; and result cf. of Fig. its greater density, Then a skin-like bubble ofIt copper ferrocyanate quickly forms bulges on out the due surface. to its uptake of water With a suitable experimentala preferred setup, direction. this For puffingtals example, up of one takes ferrous can throw place chloride a in (FeCl few small crys- with a potassium ferrocyanate solution (30 g K ple a droplet of concentrated copperof sulfate a solution onto dilute the solution surface of yellow potassium ferrocyanate (K fusion between a solution andthis, the we pure can solvent make takes place. the wall To show or membrane only trable by the solvent andleads to impenetrable a by the solute. Diffusion then vent. Semipermeable barriers are realized in theirfold most forms complete as and mani- arise. This phenomenon is best known with two gases (Fig. 16.5 Osmosis and OsmoticOsmosis Pressure originally meant “diffusion through poroustwo membranes”. substances If are separated byfuse a at wall different through rates, which they then can dif- molecule makes the followingfor contribution each of to its the degrees of internal freedom at energy a sufficiently U high temperature T A similar experiment can also beple: carried out We immerse with a two glass liquids. vessel Exam- filled with alcohol (CH mains the membrane described in 1867 by M where Heat as Random Motion 16 440 Part III 16.5 Osmosis and Osmotic Pressure 441

Figure 16.11 Pufﬁng up of a membrane in a solution as a result of osmotic pressure. The bubble which is hanging under the surface of the solution appears bright in the shadow projection. At the lower end of the striation which is sinking downwards, we can see a vortex ring.

Figure 16.12 Osmotic pressure can produce plant- like formations Part III Figure 16.13 Volume increase of a solution due to osmotic pressure

chambers, separated by a semipermeable partition W. The chambers are each equipped with a piston of cross-sectional area A.They contain a solvent (shaded), e.g. water, and the left-hand chamber in addition contains dissolved molecules of a solute (black dots). This arrangement is not in equilibrium: Both pistons are moving to the left; the left-hand piston is being pushed out and the right-hand piston is being pulled in. , ; F and This c 16.13 liquid- and de- d After either 3 and c , right side): The . of the manometer , left side). If larger h 15.9 on a spring force me- in the middle for this , so that a mechanical 16.14 os 16.14 p the membrane, pulling the press is possible only when water D ); it can thus be treated like a rigid A 9.5 = ). Little steel balls represent wa- F rest position on a spring. In both cases, a force (Sect. 16.14 flow through pull , while the water level on the right falls 0 can be measured in two ways (Fig. . Such cases can be readily demonstrated ˛ os to p ˛ of the figure). . In it, the piston is replaced by a free surface, e at its lowest point. The solution in the left-hand movable . – The simplest form of manometer is still a . In both, the final displacement W The dissolved molecules behave qualitatively like a gas (Part 0 ˇ 16.13 . This osmotic pressure pushes out the left piston and to os p manometer ): Either we let the left-hand piston ˇ b in Fig. Water has a sizeable tensile strength of the springs has beenincrease deformed by of a the sufficient solution amount, the stops, volume and equilibrium has been established. By the addition of springs, eachinto of a the two pistons has been converted from connection between the two pistons. Compare also Sect. 3 Explanation: column manometer and the force ofWe the could thus spring use by either the one weight of of the the setups column sketched of in liquid. Parts ter, or the right-hand piston Part results which hinders the motion of the left-hand piston thereby increases the volume of the solution in the left chamber. volume increase by the osmotic pressure from the right chamber can right-hand piston inwards. The osmotic pressure ter molecules, while larger ballsouter represent walls the are solute vibrating molecules.motion pistons; The of they the model produce molecules. thebers The random partition has between thermal holes the like two cham- apass. sieve through A which spiral the spring small provides ‘molecules’ a can “semipermeable” partition. If onlypartition remains small in molecules its are center position present, (Fig. the In many cases which arepartition of is practical itself importance, the semipermeable by a model experiment (Fig. solute molecules are added topressure the pushes left-hand the chamber, partition their to “osmotic” the right (Fig. indicates the osmotic pressure to be measured. One can also combine the setups sketched in Parts their thermal motions produce asure” pressure, called the “osmotic pres- d left-hand chamber increaseslarger its molecules. volume due Indistribution to both of the the images, small pressure one molecules,or of can the e.g. right-hand the all begin chamber; they of with will always themlibrium an reproduce distribution in the arbitrary after same the equi- some left-hand time. chamber rises from sign the chambers to have theThen same we diameter arrive as at the a manometer simpleable tubes. U-tube, which partition is divided by a semiperme- Heat as Random Motion 16 442 Part III 16.5 Osmosis and Osmotic Pressure 443

Figure 16.14 A model experiment showing the origin of osmotic pressure. On each side of the axle a, we can see the outer coil of a spiral spring. The motion of the semipermeable partition is damped by an oil-filled shock absorber (not shown) (Video 16.1). Video 16.1: “Model experiments on diffusion and osmosis” http://tiny.cc/xhgvjy This model experiment explains for example the behavior of red blood cells in pure water. They puff up due to their semipermeable, elastic skins. Finally, the skin bursts. To prevent this, one must never replace serious blood losses by injecting pure water into a blood vessel; instead, a “phys- iological solution” must be used, which has the same osmotic pressure as that found in the interior of the red blood cells (pos D 7 bar, corresponding 3 C16.12 to a NaCl solution with a concentration of c D 0:16 kmol=m ) . C16.12. Correctly known as Part III “normal saline” (see S. Awad, The essential point in all osmotic phenomena is the volume increase S.P. Allison, and N. Lobo of the solution. It always occurs when the solute molecules cannot Dileep (2008), "The history leave the solution, but the solvent can enter it. This condition can be of 0.9 % saline", Clinical Nu- fulfilled even without a visible semipermeable partition; one can for trition (Edinburgh, Scotland) example use a vacuum as a “semipermeable” region. This is the case 27 (2), pp. 179–188). Every e.g. in an isothermal distillation. molecule of NaCl adds two ions to the solution, so that In Fig. 16.13 ,Partf , we see an evacuated vessel containing two the total ion concentration is measuring cylinders. The left one contains the solution, e.g. LiCl 0.32 kmol/m3. 0.9 % refers in water; the right cylinder contains the solvent, e.g. pure water. to the mass concentration, Initially, the two cylinders are filled to the same height; the liquid 9 kg/m3 NaCl in water (see surfaces were at ˛ and ˇ. In the course of a few days, the volume of also Comment C16.14.). For the solution increases, and the levels become different (a foolproof comparison: Seawater con- demonstration experiment!). Their final stationary height difference tains about 3 % NaCl. is denoted by h. Then the pressure equivalent to a liquid column of height h is equal to the osmotic pressure pos,i.e.

pos D h%Sg (16.17)

%S D density of the solution; g D acceleration of gravity/:

Interpretation: Above the two liquid surfaces is saturated vapor. It consists only of molecules of the solvent, but the vapor pressure pS above the solution is lower than the vapor pressure p0 above the pure solvent (demonstration experiment in Fig. 16.15). As a result, in Fig. 16.13 ,Partf , more water molecules strike the surface at ˛ than can evaporate from it; that is, water is distilled from ˇ to ˛.

When does this process of distillation come to an end? Answer: The vapor pressure p0 of water corresponding to the temperature of the observations applies to the vapor directly above the water surface. As the column becomes higher, the pressure decreases, according to the barometric pressure ) ) S p the )). 9.20 as are ( 14.21 boiling 0 (16.20) (16.18) ( (16.19) 13.7 ˛ of the solu- S p : , and furthermore, S S p p S the speciﬁc heat (latent D p v , especially Eq. ( 0 l h , the higher terms can be p p 0 ::: m S 13.4 of the solution with the vapor T at some point becomes equal : % M ; m 0 C S gh 0 S RT h os 0 os p T is only slightly greater than 1, so p % p S p RT M 1 % 0 0 p S 0 % m p S % 0 . Then we obtain RT T , the vapor pressure % p 0 0 % g M e = v S % ˛ x 0 l e 0 p S p 0 S =% % D D D p p D os D 0 0 p S x D 0 S h p ln p p p p p os ) as just derived provides a convenient indirect D p m ln RT h is the boiling point and S M % 0 above the water surface, it is reduced to 16.19 T h D os . Then for the osmotic pressure, we ﬁnd the simple p increases, the pressure 0 ), we take T h of the pure solvent, and compute the osmotic pressure using The vapor pressure of a solu- and ). In fact, one does not usually measure the vapor pressures 0 16.17 p S C16.13 of the solution and the solvent, but rather the corresponding density of the water vapor above the water surface). 0 16.19 p is the density of the solution; for rather dilute solutions, it is D S 0 % % neglected.) Direct measurements ofconsuming. Equation the ( osmotic pressure are tedious and time- method. We compare the vapor pressure Furthermore, we treat the watergas vapor and to set a good approximation as an ideal (The approximation is valid, since that in the series expansion ln Eq. ( points T ( As the height pressure and relation density of the solvent. heat) of vaporization of the solvent; cf. Sect. from Eq. ( thus obtaining or formula. At a height ( to the pressure above thetion; surface then at the sameevaporating number from of it molecules in fall aend. given onto time. the The surface volume at increaseMaking then use comes of to this an result,motic we pressure wish and to the derive vapor the pressure. relation between We set the os- tion is less than thatupper of stopcock the is pure used solvent. tofrom The pump the out apparatus, the while air thebe lower opened one to can equalize thesides. pressures on both Figure 16.15 - ) )is 16.20 16.20 LAUSIUS )) and involves relation for )toEq.( ). 19.19 Heat as Random Motion 16.19 16.19 16 LAPEYRON C intuitively clear. The mathematical derivation of ( proceeds via the C C16.13. The transition from Eq. ( are however equally valid as those used to arriveEq. at ( the vapor pressure curve (cf. Eq. ( some approximations which 444 Part III 16.6 The Experimental Determination of BOLTZMANN’s Constant k from the Barometric Equation 445

All the direct and indirect measurements of osmotic pressure lead in the case of dilute solutions (of the order of one-tenth mole=liter) to a surprisingly simple result: For the molecules of the solute, we can apply the equation of state of an ideal gas4 This formulation is often called “VAN’T HOFF’s law”:

posV D nRT or pos D cRT (16.21)

(n D amount of substance dissolved in the volume V of solution, that is n=V D concentration cC16.14. For c D 0:1 kmol/m3 and T D 273 K (0 °C), C16.14. Here, a new quantity : 5 the osmotic pressure is pos D 2 24 10 Pa 2bar.) is deﬁned, the (molar) concentration c D n=V,where This occurrence of the same equation of state under quite different n is the amount of substance conditions is very instructive. We can see that the equation of state of the dissolved material is in the end based on the statistical laws governing the thermal mo- (solute) and V is the vol- tions of a large number of particles, in particular on the fundamental ume of the resulting solution relation (unit: kmol/m3 or mole/liter). 1 Ekin D kT : Alternatively, the mass con- 2 centration can be used:

This holds even for macroscopic objects and for systems which are c (mass) D M=V,where Part III not chemically uniform, such as for example dust-like suspended par- M is the mass of the solute ticles in liquids and gases. (unit: kg/m3 or gram/liter). The conversion factor is the molar mass Mm of the solute: c .mass/ D c .molar/ Mm. 16.6 The Experimental Determination In chemistry and medicine, of BOLTZMANN’s Constant k a practical unit is also used: the ‘percent solution’. It is from the Barometric Equation the mass (in gram) of solute dissolved in 100 ml of solu- Everyone knows what happens in a liquid which is clouded by sus- tion, quoted as a dimension- pended particles: In the course of time, it clears up; the particles less number, ‘% solution’; “drift down to the bottom” of the container. In this process, the e.g. 5 g of salt (NaCl) dis- larger particles form a clearly-deﬁned layer, while the ﬁner ones are solved in 100 ml of aqueous in a diffuse cloud which gradually becomes thinner. – Interpretation: solution is a ‘5 % saline (salt) The particles are pulled downwards by their weight (reduced by their solution’. static buoyancy!). But the random thermal motions hinder their sinking downwards5 The result of this competition is a distribution of the particles over the whole depth of the liquid, just as the molecules of the air are distributed over the depth of the atmosphere.

4 Therefore, one often uses the osmotic pressure to determine the molar mass Mm of a dissolved substance. One obtains from Eq. (16.21) M T Mm D R V pos

.M=V D density %S of the solution/:

5 The BROWNian motion (cf. Sect. 9.1). Every suspended particle undergoes a rapid, random series of collisions with the invisible molecules of the liquid. Within the time t between two such collisions, each suspended particle has on ) ) ) 16.2 16.2 ( 9.20 ( the density of . 0 % /: 9.33 m. The instantaneous 0 gh 0 p kT directly, then insert it into Eq. ( % 3 2 the pressure and u 0 m in height. The series of these D p e , 2 h D m each. mu is far too short to permit a measurement of their h 0 1 2 t p p of 10 h m and a density of kg; and its “effec- is the mass of the suspended particle 16 m . . k . The mass of a single kg. 10 3 shows an example of suspended gummi gutta partic- 17 The density distribution of , we have already discussed the distribution of the air 25 : 10 9 . It represents four horizontal in water, with a diameter of 0.6 16.16 17 is the pressure at the altitude : 1210 kg/m h 2 p ERRIN the gas at the altitude zero (sea level)). ( C16.15 D D The particles are grains ofwith gummi a gutta diameter of 0.6 % metric pressure formula. It was found to be given by: This qualitative agreement is rathertative convincing; evaluation however, will the be quanti- decisive. In Chap. molecules in the gravitational ﬁeld of the earth, described by the baro- images show the distribution of thewhich particles are in four each horizontal separatedimages layers by is 10 in relativelythrough the good model agreement gas with atmosphere, i.e. a with Fig. longitudinal section particle is 1 layers at height intervals Figure 16.16 suspended particles in water. Thising draw- is based on photographsJ. taken P by tive” mass after subtracting its buoyancym is velocities. Otherwise, one could measure and thus determine average the energy Unfortunately, the time interval Figure les ERRIN Heat as Random Motion 16 mastix. a natural resin, which isever how- poisonous. J. P used a different resin, called C16.15. Gummi gutta is 446 Part III 16.7 Statistical Fluctuations and the Particle Number 447

Now we replace the ratio of the pressures by the ratio of the number densities of the molecules. We write

p N ; Number/Volume at the height h h D V h D : (16.22) p0 NV;0 Number/Volume at the height zero

Furthermore, we relate the pressure and the density of the gas via the equation of state for ideal gases. We write this in the form

RT kT p0 D %0 D %0 (14.21 and 14.23) Mm m

(k D BOLTZMANN constant, Mm D molar mass of the gas, m D mass of one molecule) and obtain the barometric pressure formula in the form

mgh NV;h D e kT : (16.23) NV;0 Part III

This equation contains two unknowns, namely m and k.Ifhow- ever the mass m is known (it can be found e.g. from the particle diameter and density), then one can determine k by counting the particle numbers N as a function of the height h.Thiswasﬁrst carried out by J. PERRIN in 1909)C16.16. The “effective” mass of C16.16. See J. Perrin, the individual suspended particles was m D 2:17 1017 kg (cf. the “Atoms”, 2nd ed., Van legend of Fig. 16.16). In air (the mass of a nitrogen molecule is Nostrand & Co., New York 4:65 1026 kg), the pressure p and the density % decrease by half (1923), Chap. 4. – JEAN for each 5.4 km of altitude increase. In the case of the suspended PERRIN (1870–1942), Nobel particles, this same decrease occurs over a height increase of only Prize 1926. 4:651026 : 6 : Another method of determin- : 17 5 4 10 mm 0 01 mm D 10 m. Compare Fig. 16.16. 2 1710 ing k is described at the end A concentration gradient of the suspended particles can be produced in of Sect. 17.5. several other ways. Frequently, the weight is replaced by centrifugal force (Sect. 9.12, Point 1). With today’s materials, centrifugal accelerations of up to the 106-fold of the earth’s acceleration of gravity can be reached (rotational frequency 1800 s1 with a circumference velocity of around 900 m/s at a radius of 8 cm). Then instead of g in Eq. (16.23), we can insert 106 g. In this way, even large molecules can be given a density proﬁle like that of suspended macroscopic particles (“ultracentrifuge”).

16.7 Statistical Fluctuations and the Particle Number

The upper part of Fig. 16.17 shows a snapshot of our steel-ball model gas (Sect. 9.7, exposure time 105 s). The whole volume is divided into 16 partial volumes by lines drawn through the container. In the lower part of the figure, the same partial volumes are shown, each one with a number N giving the number of molecules that it contains , 2 6 4 " –5 25 –2 : 1 5 9 (16.24) –3 +1 . 1 8 (16.26) (16.25) 9 1 = ≈ 8.8 25 to be con- +5 +1 1 N 2 ) V 9 N 12 –3 +4 16 Δ 4 9 ≈14%; ≈12% 11 +2 +3 8, but the individual , along with those of ≈ 8; ( 2 gas molecules, for the 8 8.8 N 1 8 0 0 +1 = N N 2 ) ,thatis 16.17 2 : : ) N 1 7 1 N is –1 –1 N V kT Δ V ( ( 1 2 N : 790193 37 9105139 5 9 Average: Average: = p p 2 –5 25 –3 ε 1 N 1 2 1 2 2 2 ) ) N N N N D N N Δ Δ 2 D D (Δ (Δ " Mean value pot kT 1 2 E of the individual value from the mean N are also shown in Fig. , then its potential energy increases by N V The ex- by The mean value of the squared deviations is equal to the V Deviation ) D . We take the average value of the squared deviations, i.e. 2 " / 16.25 Averaged over time, this must be equal to fined in a cylinder whichThis is piston closed can at be one considered endparticipates to in by act the a statistical as fluctuations freely due a movable to verythe piston. the large gas. thermal molecule. motions within As such,If it the piston duringvolume its random back-and-forth movements decreases the We present here aa proof large for volume of the such density a fluctuations gas, in we an imagine ideal a gas. partial volume In N The values of . and find after aresult: sufficiently large number of such experiments the values exhibit considerable fluctuations around this mean,the defined equation by Figure 16.17 density of a gas,rates, for etc. the fluctuations over time of radioactive decay This relation, which we have found hereerally, empirically, e.g. holds quite for gen- the volume occupied by the In words: inverse of the average number of individual objects involved perimental derivation of Eq. ( at that moment. The average value of Heat as Random Motion 16 448 Part III 16.8 The BOLTZMANN Distribution 449

Now from differential calculus, we have

dp p D V dV

or, after insertion into Eq. (16.26),

kT .V/2 D : (16.27) dp=dV

The equation of state for the ideal gas (Eq. (14.23)) gives us the denomi- nator: dp NkT D ; dV V2 and thus Â Ã V 2 1 D : (16.28) V N The left-hand side of this equation is the relative ﬂuctuation of the volume containing the N molecules. We can now introduce the number density of the molecules, that is NV D N=V.WehaveNV.t/ V.t/ D N D const, so that (from the product rule for differentiation): Part III

NVV C NVV D 0 (16.29)

or N V V D ; (16.30) NV V and ﬁnally Â Ã Â Ã N 2 % 2 1 V D D (16.31) NV % N .% D mass density/:

16.8 The BOLTZMANN Distribution

We apply the barometric pressure formula:

mgh NV;h D e kT : (16.23) NV;0

The product mgh has a simple physical meaning: it is the difference E of the potential energies of a molecules in the gravitational ﬁeld at two altitudes separated by the height difference h. We thus obtain

NV;h E D e kT (16.32) NV;0

(NV;h D number density of the molecules at the height h, NV;0 D number density of the molecules at the base height. Molecules with the index h have more energy than those at the base height by an amount E.) ). ) refers 16.3 E . 16.33 (16.35) (16.33) (16.34) (16.36) collide elas- ) will have to 2 . E refers to the work ondition, if follows 16.32 E and ), we present below 1 E is the separation energy /: 0 2 16.32 E E : /; /: . 0 which fulﬁlls Eqns. ( 2 2 0 2 N : E E / E / . . E 0 E 1 N N . ˇ C E / / e . 0 N 0 1 1 1 0 E N E E N . . conductivity of a non-metallic elec- D D N N velocity distribution (Sect. D / 2 / , derived here for a special case, holds 2 E E E . . const const C N N 1 of a light quantum of frequency / . This is the case for the trial function D D 1 E ! N N E . h per molecule for the reaction. E in some arbitrary force ﬁeld N OLTZMANN distribution . Then we have from energy conservation .Thenwehave 0 2 / -B E For all processes which take place in thermal equi- )that E . refers to the heat of vaporization per molecule. N simultaneously and E ) 0 1 16.34 from right to left. We denote the number of molecules with the by E is the energy refers to the kinetic energy of the molecules. E

N E E AXWELL 16.35 OLTZMANN reaction energy from left to right in this equation must be equal to the number of tran- ! another straightforward derivation: We assume that two molecules with the energies energy tically during their thermalenergies motions; after their collision, they have the Owing to the great general importance of Eq. ( N sitions Now we have to search for a function In statistical equilibrium (quasi-stationary state), the number of transitions We take the two constants to be equal;is that justiﬁed is later a by plausible the assumption, success which offrom our results. Eq. With ( this c and ( function of an electron in the material. The spectral energy distribution of theThen radiation from a ‘black body’. of an electron from its binding site. The electron emission of a glowing body. Then The dependence of the electrical tron conductor on the temperature. Then The dependence of the equilibrium ofcentrations a of chemical the reaction reactants on (the the law con- of mass action). Here, to the Within the scope of this introductorysible text, a applications few of examples of this the very pos- general equation ( sufﬁce. It can be used for example toThe describe: dependence of theture. vapor pressure Then of a material on its tempera- The M Then quite generally. librium, it gives the ratio ofdiffer the by numbers of the molecules energy whose energies This B Heat as Random Motion 16 450 Part III 16.8 The BOLTZMANN Distribution 451

It converts Eq. (16.35)into

ˇ. / ˇ. 0 0 / 2 E1CE2 2 E1CE2 ; N0 e D N0 e

and this, when Eq. (16.33) holds, is an identity. Furthermore, it follows from Eq. (16.36)that N.E1/ ˇ. / D e E1E2 : (16.37) N.E2/ Finally, the comparison with Eq. (16.23), our special case for the barometric pressure formula, gives ˇ D1=kT. We thus obtain quite generally

. / E E N E2 2 1 D e kT : (16.38) N.E1/

Electronic supplementary material The online version of this chapter (doi: 10.1007/978-3-319-40046-4_16) contains supplementary material, which is available to authorized users. Part III rsn ihntetolqis rdcdfreapeb local by example are for variations density produced downwards- and directed upwards local liquids, be will that there two Then differences. imagine temperature the we within case, present second the In ifso;temta nemxn ftetotpso oeue is molecules of molecular types their two of the result a of simply intermixing com- mutual true become represents will the case liquids This diffusion; two solution. homogeneous the a out, weeks, into washed mixed many pletely become of course gradually the will in layers and two the between surface ihtelqi fhge density higher of liquid the with ifrn u icbelqisaearne ntolyr c.Fig. (cf. two layers that two imagine in arranged us are let liquids miscible First, but – different mixing. involving phenomena other il hs r nedpolm fpatcliprac,bttheir but importance, mate- practical satisfying. this very of of not still problems some is over indeed treatment quantitative skim heat are to and These make conduction want to heat may rial. of Beginners then topics – and related diffusion, the transport. about of remarks treatment few quantitative a the of pects Conduction Heat and Diffusion Processes: Transport obgn ene odmraetecnetof concept the demarcate to need we begin, To Mixing and Diffusion 17.2 motions thermal for picture molecular in the times both (Sect. processes, with diffusion with connection twice dealt already have We Remarks Preliminary 17.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © ial atrs hstp of type This patterns. nizable otiuint h nemxn ftetolqis ncomparison, in liquids; magnitude orders (being of two insignificant practically the become may of diffusion genuine intermixing the to contribution a hslte feti xmlfidi hr ae o,w mgn that imagine we Now, case. third is a in convection exemplified is effect latter This stirrer 16.1 ,w produce we ), currents .I h rsn hpe,w att ics oeas- some discuss to want we chapter, present the In ). slower forced ihntelqis ntal omn clearly-recog- forming initially liquids, the within sn oigsldbde eg h ldsof blades the (e.g. bodies solid moving Using : ). ubln flow turbulent olsItouto oPhysics to Introduction Pohl’s reconvection free eo.Teinitia The below. hra motions thermal . ae considerable a makes diffusion O 10.1007/978-3-319-40046-4_17 DOI , l hr boundary sharp lly . lal from clearly 9.1 ), 17 453

Part III : ) 17.1 13.1 ( : : a ; l V N ) of the solution V 17.1 D of dissolved molecules V N x of molecules that diffuse through the area N @ @ N Vo lu m e @ ; behind the partition, all the molecules that The partition serves only to avoid disturbing a and reproduce it schematically in Fig. ; V , and thus experimentally determine the “molec- Number N t @ 16.2 ’s First Law and the Diffusion D , is allowed to diffuse through a porous partition of V 2 N ICK The derivation of Eq. ( Constant . On both sides of the partition and within its channels, l within the time We measure the number A diffuse through are removed by someare arbitrary blown mechanism, away e.g. by they ana constant air gradient stream. in the Then number within density: the partition, there is On the frontdensity (left) constant at side of the partition, we maintain the number thickness air serves as “solvent”. convection currents. As usual, we define thetient number density of the molecules by the quo- Figure 17.1 17.3 F We refer to Fig. In order to observepress pure, convection genuine in diffusion by itsof itself, two a we suitable forms, must experimental freefluids sup- arrangement. and with We forced, a can lower by fordensity, carefully density example making avoiding allow to use the “float” occurrence oftions. on local Most other temperature simply, we varia- fluids can with introducethe one solid a of phase. higher the molecular species in Agas,e.g.H Transport Processes: Diffusion and Heat Conduction 17 454 Part III 17.3 FICK’s First Law and the Diffusion Constant 455 ular current”:

@N @N DDA V : (17.1) @t @x

In words: The current of diffusing molecules is proportional to the gradient of their number density (FICK’s ﬁrst law). The constant of proportionality D is called the diffusion constant (here for H2 diffusing in the air of the channels). This law holds in general, not just for the particular geometry described above. So much for the empirical facts. The molecular picture leads us to an interpretation and permits the calculation of the diffusion constant D in simple cases. The diffusing molecules are continually undergoing collisions with the molecules from their surroundings (the “solvent”). Each of them

is subject to a time-averaged force F in the direction of the diffusion, Part III which moves it against the frictional resistance of its surroundings at an average velocity u. The corresponding frictional work is performed with the power WP D uF (5.33) and is returned to the surroundings as kinetic energy. – For later considerations, we deﬁne the quotient u D (17.2) F as the mechanical mobility.

If the mean free path is short compared to the diameter, then it follows for example for spherical molecules from STOKES’s formula for frictional motion (Eq. (10.7)) that:

D .6R /1

(R is the radius of the molecules, and is the coefﬁcient of viscosity of the surroundings, i.e. of the solvent).

Figure 17.2 represents a thin layer of the solvent perpendicular to the direction of the diffusive motion. The cross-sectional area of the layer is A, and its thickness is x. It contains N D NVAx dissolved molecules (black dots). Each one of them is subject to the force F. This force can be replaced by an osmotic pressure p D .p1 p2/ which presses against the surface element A=N. Then we ﬁnd the relation: A 1 p F D p D : (17.3) N NV x ) ) N lists 17.1 , ( (17.6) (17.8) (17.7) (17.4) t 14.23 ( (17.5) 17.1 ’s ﬁrst law, ICK ) yields /s. Table ; 2 17.2 kT

: V : ) to yield F x : ; N : : V D x kT V V N V x V 1 17.8 D N kT V N N V of the partition. Then we have kT N t ) into Eq. ( N N tAuN D A V DA ’s law D N )and( 17.4 1 kT D A D ICK D D implies that within the time V N u

p D N 17.6 D t u N has the dimensions m u D u D )and( p D F 17.3 Diffusion constant The mechanism of F molecules diffuse through an area The diffusion constant This “diffusion velocity” Finally, we combine Eqns. ( we obtain or, for the diffusion velocity, or, with the abbreviation These “charge carriers” acquiretric a ﬁeld preferred is direction when applied an during elec- their diffusion, and they thus carry an some measured values. Often, the diffusing particles are electrically-charged molecules. It leads to which we had already obtainederations: above from purely empirical consid- Figure 17.2 For the osmotic pressure, the ideal-gas equation holds: Inserting equations ( Transport Processes: Diffusion and Heat Conduction 17 456 Part III 17.4 Quasi-Stationary Diffusion 457

Table 17.1 Some examples of diffusion constants and diffusion paths (Molecule) diffuses at a temperature with the diffu- and a single molecule of (ıC) Âsion constantÃ D moves from its original m2 position in one day (ac- s cording to Eq. (17.16)) by a distance of 5 H2 in air at atmospheric 0 6:4 10 3.3 m 5 O2 pressure 0 1:8 10 1.8 m Urea in water 15 1:0 109 13 mm Salt 10 9:3 1010 13 mm Cane sugar 18.5 3:7 1010 8mm Gold in molten lead 490 3:5 109 25 mm Gold in solid lead 165 4:6 1012 0.9 mm 8 H2 in a KBr crystal 680 2:3 10 6cm Potassium 650 5:2 108 9.5 cm as color centersa a Color centers: See Vol. 2, Sect. 27.14. Part III electric current. In this way, for example ionic currents and electron currents can arise in liquids, in gases and in solids. In favorable cases, these directed diffusion processes can be followed directly by eye(seeR.W.Pohl,Elektrizitätslehre, 21st ed. (1975), Sects. 16, §6 and 25, §22; or cf. the image on the Wiki site for R.W. POHL, https:// en.wikipedia.org/wiki/Robert_Pohl).

The mobility e of the charge carriers is not referred to the force, but rather to the electric ﬁeld E D force F=charge e. We thus obtain the electrical mobility u ue D D D e (17.9) e E F (e D charge of the carriers, e.g. in ampere second (A s); as deﬁned in Eq. (17.2)).

17.4 Quasi-Stationary Diffusion

The application of FICK’s first law presupposes that we know the gradient of the number density, i.e. NV=x. It can be readily determined for a stationary state (Fig. 17.1). This may also be the case to a good approximation for many processes which are only approximately stationary (quasi-stationary states). An example of this type is sketched in Fig. 17.3. A solid body Y contains N molecules of type I in the volume V; their number density is thus NV D N=V. Think of a solid-state solution, e.g. of thallium atoms in a KBr crystal. Into this solid body, we suppose that N molecules of a gas diffuse inwards from the left, for example Br2 molecules. They are supposed to unite at the diffusion front with N molecules of type I and therefore to drop out of further diffusion processes. By what distance x does the diffusion front move forward within a time t? ; I (17.11) (17.12) (17.13) (17.10) molecules of type and treat the process x x d /: A V : : : . The concentration of dif- ; N t x V V d d x x N D Dt N A 17.1 const (17.14) V V V D relative to N N N N x D D D 2 molecules and obtain together with D t x V D D d x d N N t 2 2 V t N d x x N d )theresult at the left of this layer, and at the right, after ‘const’ is a pure number ,thereared . x V d N )tothe 17.11 A 17.1 )and( Linear concentration gradient for which has already been chemically converted takes over x 17.10 thickness the function of thefusing partition molecules in is Fig. as if it occurs practically at a fixed location. As a result, the layer of We apply Eq. ( thus, we have Eqns. ( the layer, i.e. at theagain diffusion find and as reaction an front, approximatetion it expression gradient for is the zero. diffusion concentra- Thus we The solution of this differential equation is and thus, as the answer to the question in italics above, Figure 17.3 This process can still betionary. treated That to is, a we very good can approximation neglect as d sta- Within the volume diffusion accompanied by chemical reaction Transport Processes: Diffusion and Heat Conduction 17 458 Part III 17.5 Non-Stationary Diffusion 459

This result has been demonstrated with the example mentioned C17.1 above, the diffusion of Br2 into a Tl-doped KBr crystal . The C17.1. The crystal takes previously brown layer x becomes clear, since the TlBr molecules on a brown color on be- formed are colorless. – Equation. (17.14) plays an important role for ing heated in K vapor surface reactions on metals, i.e. when they “tarnish”. (E. Mollwo, Ann. Physik, 5th Series, Vol. 29, p. 394 (1937)). 17.5 Non-Stationary Diffusion

In our two examples of applications of FICK’s ﬁrst law, the number density of the diffusing molecules at the front of the diffusion path (at a distance l from the left-hand surface in Fig. 17.1 or the coordinate x C x in Fig. 17.3, respectively) was held constant at the value zero. In general, the number density NV of the molecules on both sides of the diffusion region being considered is variable over time. The process is then no longer stationary; the spatial distribution of the diffusing molecules varies in the course of time. The increase in the number density N in a region between x and x can be obtained V 1 2 Part III from the difference in the numbers of molecules ﬂowing in at x1 and out at x2.

If we again assign a positive sign to a particle current moving in the positive x-direction, then for the rate of change of the number density NV in the volume V between x1 and x2 we ﬁnd ( ˇ ˇ ) @ @ ˇ @ ˇ NV 1 N ˇ N ˇ D ˇ ˇ : @t V @t @t x1 x2

Setting V D A.x2 x1/ and making use of Eq. (17.1), we obtain from this

@N @2N V D D V : (17.15) @t @x2

This differential equation is called FICK’s second law.

We also give an example of a non-stationary diffusion process here; however, we will not derive it. In our example, at the time t D 0, the number density in a whole region is taken to be zero. At the front side of this region, it has the value NV;a, and this value is held constant during the entire process of diffusion. How does the distance x between the location of a certain concentration NV;x and the entry point x D 0 change? Answer: Once again, we have

x2 D D const : (17.14) t

The conclusions implied by this equation are represented graphically in Fig. 17.4: The distributions of the number densities at various times remain similar to each other. They can be made identical by a suitable choice of the time-axis scales. x ), /: 17.14 (17.17) (17.16) tion takes place Ws/K). 23 by using Eq. (10.7) 16 10 D √t , heat can be transported 38 : 1 const 13.3 D R : x= ). We then obtain k D kT 2 ough “radiation”. The necessary 3 4 17.5 dient. Heat conduc D from the entry surface D x 2 t the viscosity coefﬁcient of the ﬂuid x 2 t x

constant ( . In this case, for the constant in Eq. ( t 1 1234567 = of a given particle from an arbitrary starting point Distance t x ian motion, one does not observe diffusion as a mass

OLTZMANN

1.0 0.8 0.6 0.4 0.2 . =0 at Concentration

x ROWN

, behave as the square roots of the diffusion times, e.g. as Concentration at distance at Concentration x a ; V The diffusion gradient as a function of time. The distances N Conduction and Heat Transport is the particles’ radius, R convection . From this equation, wethe value can of also the make B an experimental determination of phenomenon, but instead asprogress an of individual a process. certain One concentration,sures cannot how but the follow only distance the of single particles. One mea- we ﬁnd the number 2. We thus have gradually increases with time (Stokes’ law) and Eqns. (17.2) and ( If the particles are spherical, we can compute In the case of B traversed by a particularinitial given value value of the number density, e.g. 40 % of its 1:2:4. in the interior of materialsular via motions different in mechanisms, such gasesmotion as in and molec- solids. liquids,heat or In conduction elastic due gases to waves and processes on andtinguished liquids, the from electronic as molecular scale the for must usually be diffusion,forced predominant dis- “genuine” heat transport by free and 17.6 General Considerations on Heat As we have already mentioned in Sect. Figure 17.4 either through “conduction” orprecondition thr is a temperature gra Transport Processes: Diffusion and Heat Conduction 17 460 Part III 17.6 General Considerations on Heat Conduction and Heat Transport 461

Figure 17.5 Convective heat transport via convection cells, each of which is formed by a flow of hot liquid from the hotplate upwards and of cooled C17.2 liquid back downwards (BÉNARD cells (1900) ) within a liquid layer C17.2. E.L. KOSCHMIEDER: (e.g. in liquid paraffine). The cells are deformed by mutual pressure into “BÉNARD Cells and Taylor mainly hexagonal shapes, sometimes with a regularity which approaches that Vortices”, Cambridge Univer- of a honeycomb. In each cell, the liquid is rising in the interior and sinking sity Press (1993). Part III on the outside of the cell. The flow is completely stationary. If it is disturbed by stirring, a new pattern is formed within a fraction of a minute (actual size).

Figure 17.5 shows an example for heat transport by convection. A hot metal plate is covered with a layer of liquid about 3 mm thick; above it is the cool room air. The liquid contains suspended fine aluminum particles which make its free convection visible. They exhibit a complicated, honeycomb-like pattern. Heat transport by forced convection can be found for example in the cooling systems of automobiles. An important and instructive application of heat transport mainly by convection is exhibited by the countercurrent heat exchanger.–In the laboratory, one sometimes needs to change the temperature of a flowing substance temporarily, e.g. in order to accelerate a chemical reaction within a fluid or to purify a liquid by distillation. Then the scheme which is sketched in Fig. 17.6 can be used: At the left, heat is applied to the flowing substance by a Bunsen burner flame below the flask, while at the right, heat is removed by a water-cooled condenser. Such a setup is convenient but wasteful. All of the heat power added at point a is lost again at b and carried off by the cooling water. Such an arrangement would be unacceptable for large-scale technical applications; but it can be avoided, in the ideal limiting case completely. This is the function of the countercurrent heat exchanger, C17.3 developed in 1857 by WILHELM SIEMENS . Its principle is indi- C17.3. W. SIEMENS:see cated in Fig. 17.7. At the left, the liquid whose temperature is to be Comment C15.6 in Chap. 15. varied flows in, for example water at room temperature T1;below,in – An important technical the round boiling flask, it has a temperature T2, say 353 K (80 °C). application of the coun- At the upper right, water at room temperature T1 again flows out. tercurrent exchanger is in In principle, one needs to add heat to the flowing liquid only at the the liquefaction of air by beginning of operation; in this example, by heating the water in the the LINDE process (see boiling flask to 80 °C. From then on, further heat input is theoretically Sect. 15.5). , C17.4 n without adding limiting case, the countercurrent anger can functio without . As a result, continuous addition of heat 1 Schematic of a countercurrent heat Producing temperature . Secondly, the flow within the tubes must be turbulent 17.6 For this reason, this trickreduce is heat used losses, by for many example when warm-blooded theirwater animal feet or species remain ice in to for contact with long cold is periods. brooding – an Example: egg. Thefor male He around Emperor has two penguin to months who stand without any on nourishment! the ice in the south-polar weather They can, however, be minimized by using very long, helically coiled tubing. any heat energy atthe all. tubing are Firstly, unavoidable losses due to heat conduction along exchanger; the temperature change ofstance the is flowing accomplished sub- with minimal energy losses Figure 17.6 is necessary, but atFig. a much lower power level than in the setup of 1 to temporarily change the temperature of aa flowing continual substance high without power input. with only tiny temperature gradients.the At upwards-flowing (outgoing) every level water alongthe is the downwards-flowing only (incoming) tube, slightly water adjacent warmer to than it. In reality, no countercurrent exch in order to ensure goodhowever heat requires exchange. a Maintaining power a inputment. turbulent to flow a pump or some similarSummarizing arrange- briefly: Inexchanger the fulfils an ideal important technical function: It makes it possible Figure 17.7 unnecessary. The upwards-flowing waterheat in up the to outer the tube downwards-flowing watertubes gives in are its the sufficiently inner long, tube. this When “temperature the exchange” takes place variations in a flowing substance energy conservation . 10.4 Transport Processes: Diffusion and Heat Conduction 17 were already described in Sect. C17.4. Turbulent flows 462 Part III 17.7 Stationary Heat Conduction 463 17.7 Stationary Heat Conduction

Genuine heat conduction can be readily observed in solid bodies. We suppose that in a time t, a (thermal) energy Q passes through an area A, driven by a temperature gradient T=x. Then we obtain for the heat current:

Q T DA : (17.18) t x

In words: The heat current is proportional to the temperature gradient. The proportionality constant is called the thermal conductivity coefﬁcient. It depends strongly on the temperature. This is illustrated in Fig. 17.8 with three examples. – In copper, a typical metal, heat transport is accomplished almost entirely by the (conduction) electrons; in crystalline quartz, an insulator, it is due completely to high-frequency elastic waves (quantized sound waves or phonons). Part III Both the electrons and the elastic waves are scattered by other elastic waves (electron-phonon and phonon-phonon scattering). The frequency of occurrence of such scattering processes decreases with

102 W cm2∙K/cm Cooper drawn and tempered 10 λ

Quartz, crystalline 1

10–1 Thermal conductivity coefficient Thermal conductivity coefficient

10–2 Quartz, vitreous

–3 10 1 10 100 1000 Temperature K

Figure 17.8 The dependence of the thermal conductivity coefﬁcient on the temperature ), up- /s). .In 17.9 2 is the 17.18 (17.19) T 17.4 localized (units e.g. m is its speciﬁc heat. c t : 2 T x 2 @ @ =const √ x direction (i.e. one-dimensional), c x % from the heated end x 2 8 min. ’s second law for diffusion. For heat D t ICK temperature conductivity 0, we suppose that the temperature T @ 0.5 @ D Distance t = t 0 5 10 15 20cm °C

400 300 200 100 at one end of the rod; thereafter, we observe the tem- 1 Temperature is called the T Raw data from a demonstration experiment showing the time c All these topics belong among the problems of solid-state % . ). The distance travelled by a point at a particular temperature is , at the time is the thermal conductivity coefﬁcient deﬁned by Eq. ( C17.5 17.9 is the density of the conducting material, and We will give onlyIt one is example analogous of to non-stationary the heat diffusion conduction. process illustrated in Fig. Here, The ratio % same at all points withinto a the value metal rod. Then it is suddenly increased Fig. physics. conduction which is limited to the perature distribution along the rod as a function of time. Figure Figure 17.9 17.8 Non-Stationary Heat Conduction Non-stationary heat conduction canequation, be analogous to described F by a differential decreasing temperature. At very lowbance temperatures, of a heat different transport distur- predominates, namely scattering by lattice defects and by thedisorder surfaces in of crystallites. quartz Given glass, thesmall complete for example, its heat conductivity is very evolution of a1 temperature m gradient length, (in withoutper an any image iron thermal insulation. rodproportional of to The 8 the mm setup square diameter root is of shown and the in time. the it is given by , 991 74 Transport Processes: Diffusion and Heat Conduction Rev. Mod. Phys. 17 C17.5. See e.g. R.O. Pohl, Xiao Liu, and E. Thomp- son, (2002). 464 Part III 17.9 Transport Processes in Gases and Their Lack of Pressure Dependence 465 shows the results: The distance x from the entry point x D 0tothe point at which a certain temperature Tx is observed increases proportionally to the square root of the elapsed time. The temperature distributions observed at different times remain similar to each other. They can be made to appear identical by a suitable choice of the time scales.

17.9 Transport Processes in Gases and Their Lack of Pressure Dependence

In gases and liquids (i.e. ﬂuids), the relation between diffusion and heat conduction is intuitively clear. In the case of diffusion, we are dealing with the statistically-ordered forward motion of the molecules. Heat conduction can be described concisely as the diffusion of an excess of kinetic energy of the molecules.InFig.17.10, we suppose that the wall at the left is at a higher temperature than the gas in the container; convection is excluded. Then the adjacent Part III gas layer will be the ﬁrst to be heated, i.e. its molecules will acquire an increased kinetic energy. This distinguishes them from the molecules in other layers. No distinguishing feature of any kind can be maintained in a statistical process with a large number of individuals (molecules). The distinguished molecules must therefore give up a part of their excess kinetic energy in collisions with the other molecules. Thus, the excess kinetic energy gradually diffuses into the gas layers further to the right. In a completely analogous manner, we can understand another phenomenon which depends on molecular motions but which we have thus far not explained: Internal friction (Sect. 10.2, see in particular Fig. 10.2). In Fig. 17.11, we suppose that the left wall is moving upwards with a velocity u. The molecules in the adjacent layer acquire

Figure 17.10 The mechanism of heat conduction in gases

Figure 17.11 The mechanism of internal friction in gases di- mu , a pressure ˇ . There, the spac- and and its surrounding a of the molecules. L ˛ . 10.4 20 hPa. We then pump a large fraction or viscosity can be concisely described as the Pa). During the downward motion of the ball as it falls 5 20 hPa. 10 , an inner cylinder is rotating within an outer cylinder. The lack of dependence of the internal friction (viscosity) on 17.12 15 mm). The difference in the outer diameter of the ball and the inner internal friction diameter of the tube is aboutpressure 0.01 mm. ( The tube contains air at atmospheric The following experiment is even moreupper striking: end We put of a a steel ball precision into glass the tube which is standing vertically (diameter pressure. The spacing between the rotating cylinder internal friction. In Fig. guishing feature cannot bemolecules; maintained thus, in the aally upwards-directed transported statistical into excess the ensemble momentum layers of further ismove, to albeit the gradu- right more and slowly, causes inthat them the to same direction asdiffusion of the an wall. additional momentum We component see Just like diffusion andcan heat be conduction, strongly enhanced internal by convection, frictionvection. especially by (viscosity) We turbulent have con- already seen this in Sect. The relation between diffusion, heat conductionin and gases internal friction becomes veryphenomena clear are through independent of aThis the common surprising gas feature: pressure fact All over can these a be wide most range. simply illustrated for the case of Figure 17.12 a preferred direction of motion (upwards)wall, through collisions and with therefore this have an additionalrected momentum upwards. component This is indicatedone-sided in excess the momentum distinguishes ﬁgure the by molecules small ofnearest arrows. the the This layer wall from all the others in the container. This distin- housing is exaggerated in the drawing for clarity. Their spacing is about 1 mm, aparting from a is segment reduced tothe roughly two 0.2 cylinders mm. is set Duringto in rotation, its motion the in viscosity. air the Then, between direction between of the the two rotation, due regions difference is built up, here of the air, 80 % orNevertheless, more, out the of manometer the chamber continuesdifference containing to the of cylinders. show the same pressure Transport Processes: Diffusion and Heat Conduction 17 466 Part III 17.9 Transport Processes in Gases and Their Lack of Pressure Dependence 467

Figure 17.13 A simple demonstration experiment to show the lack of dependence of the heat conductivity of a gas on its pressure. The heat current ﬂows out of the hot water bath through the gas layer in the double-walled vessel into the diethyl ether ((CH3CH2)2O) and produces a corresponding current of ether vapor. The strength of this current is shown by the height of an ether ﬂame which is burning at the top of the vessel. Over a large range, it is independent of the pressure of the gas layer between the vessel walls.

through the tube, the gas within the tube must ﬂow around the ball through a very narrow, circular slit. Its viscosity produces a large resistance: The ball does not “fall” with an accelerated motion, but instead it “sinks” (after

a brief initial period) with a constant velocity (Sect. 5.11). At this velocity, Part III it covers a distance s (e.g. 60 cm) in a time t (e.g. 30 s). If we reduce the gas pressure p, the time required for the ball to sink initially remains the same. Only at a pressure of p 0:16 105 Pa does it begin to sink noticeably faster; at p 1:3 Pa, it ﬁnally approaches free fall.

The lack of dependence of the heat conductivity of a gas on its pressure is likewise relatively easy to demonstrate. Details are shown in Fig. 17.13. So much for the facts. Now we consider their explanation on a molecular basis: The number of molecules diffusing in a given time through a given area, the amount of their additional momentum or of their excess kinetic energy are all proportional to their number density NV. They are in addition proportional to the mean free path l of the molecules, i.e. to the distance they travel on the average between collisions (Sect. 16.3). NV increases in direct proportion to the gas pressure, but l decreases in indirect proportion to the pressure; their product is thus independent of pressure. Therefore, in gases, every type of diffusion (or “transport process”) does not depend on the pressure.

Hydrogen has a very long mean free path; under standard conditions (0 ıC; 1013 hPa), it is l D 1:4 107 m. As a result, hydrogen has a very high heat conductivity (cf. Fig. 17.14).

At very low pressures, the concept of the mean free path l begins to become meaningless: The mean free paths of the molecules become larger than the dimensions of their container. The molecules can then bounce back and forth unimpeded between opposite container walls. The momentum or energy transferred to and from the walls becomes smaller as the density of the gas decreases. This is the principle of the Thermos bottle, a container which has double walls separated by ) at A ). 17.24 makes it 17.8 Pa, the l ), ( 2 onduction, free 17.22 , where we dealt with the 9.8 remains dark; it is cooled by the as used there, we refer here to 2 here denotes the mean free path. l 9.25 ition of the mixture. This is the reason ; , coming from the left and the right. / x l which is about four orders of magnitude , W/(cm K) between the outer wall at 300 K C 7 x C17.6 . and / and air. In addition to genuine heat c l 2 at the position x A . A simple demonstration experiment for comparing the heat Path )) can be obtained from fairly simple considerations. We . We consider the molecules which pass through a cross- ities of H 17.26 17.15 lower than the thermal conductivity of vitreous quartz at 10 K (Fig. thermal conductivity of these “superinsulated”tainers Thermos is or of cryogenic the con- order of 10 But even these radiationvacuum losses can space be is reduced! ﬁlledplastic, e.g. with To Mylar, accomplish a typically this, 30 large layers/cm,other the number which by are of separated an thin from electrically-insulatingSince each layers material, of the for aluminized temperature examplesmall, Nylon difference the netting. radiative between heatoverall transport neighboring between heat Mylar them is transfer layersthickness, also through is very and small. this thus can The stackductivity be between is represented the inversely outer, ascold proportional warm a surface. surface mean to of apparent its Even the thermal with container con- and a its residual inner, gas pressureand of the some inner 10 wall at 20 K convection also occurs.are Two heated identical by platinum the wiresand same (connected electric is current. thus in very series) The hot, wire in while air the glows wire bright in yellow, H high thermal conductivityconductivity of depends the on hydrogen. thewhy compos heat In conductivity mixtures is ofcomposition often of gases, used a the gas in mixture. heat technicalcan – be applications readily The demonstrated to basic with monitor aspects the of above the the setup. different processes the positions Two other areas are drawn in to the left and the right of the area 17.10 Determination of the Mean Free The relationship between theecules, of momentum three and of diffusion energy) and phenomena the mean (of free path mol- and ( pressure of a gas. InsteadFig. of Fig. sectional area an evacuated space. Heat transportonly to through radiation. and from its interior can occur Figure 17.14 possible to determine this important quantitydifferent experimentally methods. by three The necessary formulas (Eqns. ( conductiv proceed in a similar manner as in Sect. Ex- , KIN W. E ACK , Oxford University Transport Processes: Diffusion and Heat Conduction 17 Low-Temperature Measure- ments Press (2006), p. 57 and Fig. 2.4, p. 58. perimental Techniques for C17.6. J 468 Part III 17.10 Determination of the Mean Free Path 469

Figure 17.15 The derivation of Eq. (17.20)

On average, the molecules which will pass through A from the left and from the right suffer their last collisions before reaching A at these two adjacent areas. The number densities NV of the molecules and their velocities u thus remain constant within the two shaded volumes. From the left, in a time dt, a number

1 . /. / ; dN1 D Adt 6 NVu xl of molecules arrives at A, and from the right, a number

1 . /. / : dN2 D Adt 6 NVu xCl Part III 1 We have encountered the factor 6 previously in Sect. 9.8. The resulting molecular current in the x-direction is then C17.7. We now apply the

dN A A d.NVu/ general equation (17.20) D Œ.N u/. / .N u/. / D 2l ; dt 6 V xl V xCl 6 dx to the particular case of molecules with their random or thermal motion. We cannot specify a microscopic velocity here; only a statistical dN l d.N u/ average over the six spa- DA V : (17.20) dt 3 dx tial directions and the wide range of magnitudes can be given. It was introduced in Note that this equation is quite general and could be applied to any Sect. 9.8 as the “rms (root- ensemble of moving particles, macroscopic or microscopic. We will mean-square) velocity” urms, apply this general equation to particular situations involving molecu- and more precisely defined lar motions in the following: in Eq. (16.3). The average molecular velocity referred 1. Diffusion of molecules. The temperature is the same everywhere, and to here and used in the next thus u, the average molecular velocity, is constantC17.7. For the diffusing sections is this statistical ve- molecular currents, we obtain locity. It is an average over dN l u dN dN the microscopic, random ve- DA V DDA V ; (17.21) dt 3 dx dx locities of the molecules, and we denote it here for short which is FICK’s first law with the diffusion constant by u, to keep it separate from macroscopic velocities such as u?,referredtointhedis- l u D D : (17.22) cussion of viscosity, or the 3 phonon velocity u mentioned in Comment C17.8. ) ) as in 10.2 ( 17.18 ( (17.23) (17.25) momen- (17.26) (17.24) m . Each molecule ). They thus have ? x u d d V , we find: /: 17.10 AN c x u T 3 l % d d u C17.8 (indicated by small arrows in l ; fk : x T number of degrees of freedom, V 1 3 d ; d ? ? as in Fig. D m x N u u x D u d A V specific heat / D 2 1 d N A f ?

energy current A ( , and all the molecules together trans-

D fk u 3 u l V 3 c Q l to be: x D D ; fkT N umu d D

1 2 D u V F t 6 Q l macroscopic flow velocity d A F N d D . d D t Q density A d in the direction of this velocity. For the d l 3 D ? p .% D constant). For the ), ? t p d d 5.41 of Transport Processes in Gases ,wehave . This is typically a . Perpendicular to the direction of their diffusion, the molecules , u OLTZMANN 17.11 17.11 B Diffusion of energy: heat conduction Diffusion of additional momentum, i.e. internal friction or viscosity D Fig. 2. have an additional velocity component Fig. an excess momentum tum current and, assuming a homogeneouslocity velocity to gradient and generalizing the ve- with the “thermal conductivity coefficient” or, using Eq. ( or we find the viscosity coefficient transports the additional energy k port in this way a thermal energy 3. 17.11 The Mutual Relations Up to now, we havependent of treated one the another. various This transport isHowever, processes quite when correct as in we inde- a look firstthe approximation. more different closely, transport phenomena wefer are find four mutually examples experimentally of interrelated. that these interrelations: We ofis c ttice as elastic u replaces the ), where u )) holds also for 17.8 ) that propagate Transport Processes: Diffusion and Heat Conduction 17.26 14.5 17 ). average molecular velocity at the velocity through the crystal la waves (then the specific heat contribution due to lattice vibrations (see Fig. the heat conductivity in electrically-insulating solids (see Fig. u C17.8. This expression (the right-hand term in Eq. ( 470 Part III 17.11 The Mutual Relations of Transport Processes in Gases 471

Figure 17.16 The production of a temperature difference by diffusion. The smaller image below shows the brass slit valve which separates the two chambers I and II. The thermocouples 1 and 2 are made from silver foils with welded- on wires of steel and Constantan.

1. Diffusion in gases produces temperature differences andtheyin

turn lead to heat conduction. In Fig. 17.16,chamberI contains hy- Part III drogen, and thus a gas with a small molar mass, Mm D 2g/mol. Chamber II contains carbon dioxide, with Mm D 44 g/mol. Both of these gases are at the same pressure and temperature; 1 and 2 are thermocouples connected in opposition. – By a slight rotation around the long axis of the chambers, we can open a slit valve which separates the two chambers (cf. small image below!). Then the two gases can mutually diffuse into one another. After we open the valve, for around 30 seconds, a temperature difference of about 0.6 °C is established, with chamber II at a lower temperature.

Explanation: The small H2 molecules quickly penetrate into the CO2 via isothermal diffusion, so that the number density NV and the pressure p temporarily drop in chamber I. In order to restore their original values, the gas in chamber II expands adiabatically, thereby compressing the contents of chamber I and performing external work. As a result of this work, the gas in chamber II cools; the temperature thus exhibits a gradient, increasing in the direction II ! I,inthe same direction as the diffusion of the heavier molecules (CO2). 2. Temperature differences produce a pressure difference in gases (KNUDSEN effect).InFig.17.17, a portion of room air is conﬁned within a porous clay cell. Inside the cell, there is an electric heater. As a result, the temperature in the tiny channels of the clay walls is higher near the inside of the walls than on the outside. A glass tube which extends below the cell into a beaker of water allows the air in the cell to expand and escape to the outside (bubbles!). We observe a continual stream of escaping air: Room air is constantly pulled through the porous clay walls into the heated cell, and as a result, the pressure within the cell is higher than outside it.

The explanation starts with the condition for a stationary state. The number of molecules passing through a cross-sectional area A in the time dt is NVuA dt. Assuming that the same number enter and leave the channels ) ) 16.1 ( 14.23 (17.27) ( (17.28) effect which we NUDSEN : ) with ; 2 9 2 ; / / u 2 T V kT . p kT N 2 3 p . The K V N D D D 2 1 D 1 u / / p u m 1 T V . p 1 2 N . p is combined with the ideal gas law, C17.9 ) 17.27 effect Porous clay cylinder for demonstrating NUDSEN In gas mixtures, temperature differences produce concentra- yielding after some simple manipulations: where the indices 1sides and of 2 the refer porous to wall,This the respectively. equation ( quantities on the hot and the cool That is, when thewall, temperature the pressures is also different become different. on the two sides of the porous a uniform cross-section, this leads to: and (from the kinetic theory of gases, Chap. in the cell walls in a given time, and that the channels on average have enriched in the cooler region; the heavier molecules thus move in the discussed above undermolecules. Point It was 2 simply convenient to requiresture describe of it only gases, there namely using one room a air. mix- speciesWe of could gas leave out theof porous a wall, pure usewithin gaseous a the mixture substance, gas of and mixture. gases maintain instead Then a the temperature molecules gradient of higher mass will be Figure 17.17 3. tion gradients (thermodiffusion). the K . is 2 )is A t A ( d = D 2 .The ; 2 )is 1 1 17.27 / N . A 1 has de- uA u and exit at V V 17.28 1 N N D . T 1 . These currents ; . The mean free t h 2 c ), taking d u u = 1 plays no role within N D l Transport Processes: Diffusion and Heat Conduction 17.27 2 ; . They enter the chan- h 17 continuity condition 2 c Eq. (16.1), making use ofideal the gas equation (14.23). simple manipulations of air molecules outside the cell at (2) are in equilibrium at the (lower) temperature T a temperature the channels. There, the molecules are rapidly ther- malized to the (higher) inner channels, their entering molecular current d creased slightly. Since no molecules are lost or gained within the The rest of the calculation leading to Eq. ( nels with the net average (one-dimensional) velocity is equal to the exiting current d Eq.( the net cross-sectional area of the channels). This gives u C17.9. In fact, Eq. ( (1) with the somewhat higher velocity are given by Their density path 472 Part III 17.11 The Mutual Relations of Transport Processes in Gases 473

Figure 17.18 Separation of a gas mixture by thermodiffusion in a “separation tube”. A wire, tightly stretched in the middle of the tube, is heated until it glows in a mixture of CO2 and H2 (partial pressures 0:37 bar and 0.13 bar; the CO2 should be ﬁlled ﬁrst!). In about 5 minutes, the hydrogen becomes enriched in the upper part of the tube, and prevents the wire from glowing there due to its good heat conductivity. (For the demonstration setup, the tube length is 1 m and its inner diameter is 1 cm. The wire should be well centered and the long axis of the tube exactly vertical). One can also use a mixture of argon and bromine vapor. Then, the enrichment of bromine at the lower end of the tube causes it to liquefy there.

direction of decreasing temperature. This phenomenon is called thermodiffusion.ItwasappliedbyK.CLUSIUS very successfully for the separation of molecular mixtures, in particular mixtures of different Part III isotopes. His “separation tube” consists of a long, vertically-mounted glass tube with an electrically-heated wire along its central axis. The warm mixture of gases rises near the center of the tube, while cooler gas sinks near the outer wall. The molecules of greater mass diffuse preferentially radially outwards and are carried down by the gas flow at the perimeter of the tube so that they are enriched near the bottom. Figure 17.18 shows a demonstration setup. Thermodiffusion also occurs when one of the “molecular species” consists of larger particles. For example: Warm air rises above a radiator, and between the radiator and the cooler wall, there is a temperature gradient. Dust accumulates in front of the wall, so that it becomes soiled with streaks of dust. – When a pot is used for cooking over an open fire, small particles of soot from the hot flame gases drift towards the bottom of the pot and cover it with a layer of carbon black.

The explanation of thermodiffusion also follows from Eq. (17.20). We must simply take into account that NV and u in Fig. 17.15 are, in a precise description, somewhat different on each side of the surface A when a temperature gradient is present along the x-direction. – In Eq. (17.20), we 2 replace the number density NV by 3p=mu (this expression follows from Eq. (9.14) with the density % D NVm). Then we obtain dN pl d.1=u/ DA : (17.29) dt m dx Now using 1 3 mu2 D kT (16.1) 2 2 we find after some straightforward manipulations dN lp dT DCA p : (17.30) dt 2T 3mkT dx Thus, a molecular flow results, and it moves in the direction of increasing temperature. This flow is stronger for the lighter molecules in a gas . Centrifugal 6 bar, we can p . At this point, , there is an ori- D b b p vortex tube: R. Hilsch, provides a means to regulate ILSCH H -H ANQUE , 208 (1946)) 1 shows a “vortex tube”, above as a section along its Vortex tube (R 17.19 Pressure differences in gases produce temperature differences. mixture than for heavier ones.molecules In are the enriched stationary towards the state,are hotter therefore, enriched side the towards and lighter the the cooler heavier side. molecules Z. Naturforschung Figure 17.19 4. Figure length, and below in cross-section at the location air enters tangentially into the tube at a high pressure force ensures that thetube pressure than is at higher its just center inside axis. the To the walls right of of the position fice of about 2 mm diameter. A valve the relative strength of the airflowsto which the are moving left. to thewhile The right the and airstream stream exiting which at exits the the left is tube warm. at With the right is cool, readily obtain a temperature differencethe of right-hand 40 end °C. of Within a the short tube is time, covered by a thick layer of frost. Transport Processes: Diffusion and Heat Conduction 17 474 Part III Entropy, Function State The .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © 18.1 Figure are occur, the differences in temperature – no which in – processes limit magnetic ideal and electrical mechanical, All Processes Reversible 18.1 nFig. in initial Their previously. taken back- have restored, ‘run be they to can that state made path be the principle along in wards’ could processes such that means n ftecmoet novd oeexamples: Some involved. components the of any eto qaisai” ti lutae nFig. weight in the and illustrated spring is con- helical stretched the It a explain to “quasi-static”. serve can of process reversible cept a of example third storage A temporary a as only energy. serves in potential it changed for process; not medium this is in plate way steel lasting The any motion. upwards an into converted some requires state tial sequence. periodic a The in over and over state initial electrical its an reproduces or mechanical A tece spring stretched a of expansion static hspoesms rce eysol,adtu rcial without time. practically any thus at and slowly, restored very be proceed thus must can process state initial This The direction. other the eycoet qiiru;ti sacmlse yalvrsystem small lever arbitrarily a an Then by between accomplished variable. difference is continuously is this leverage equilibrium; whose to close very refall free 5.10 ihishl,teaclrtddwwrsmto a be can motion downwards accelerated the help, its With . h quasi- The faselbl slkws eesbe u etrn t ini- its restoring but reversible, likewise is ball steel a of F without and S uiir apparatus auxiliary olsItouto oPhysics to Introduction Pohl’s F G asn a causing a rcptt oini h n or one the in motion precipitate can oscillation permanent F G ae lc eesby it reversibly; place takes ..ahr te lt as plate steel hard a e.g. , fams r etalways kept are mass a of 18.1 O 10.1007/978-3-319-40046-4_18 DOI , hneo tt in state of change h force The . reversible .This F of 18 475

Part III , ; A also take place quasi-statically Briefly, we define , we see a cylinder . may . Then the piston can 18.3 B temperature differences quasi-static .InFig. brought back to their initial states quence of equilibrium states. take place quasi-statically if they are to vice versa Arbitrarily small . must , take place extremely slowly, i.e. C reversibly and Quasi-static shows the reversible, quasi-static expansion of a gas. The Reversible evaporation A : All reversible processes are characterized by three fea- ; in this case, heat must be given up to the surroundings. Both 18.2 C S either rise very slowly and convert all the liquid into vapor, Case and therefore sion of a liquid into awith gas or a piston. Below the pistonpor) is a between liquid, the with piston its gaseous anddown phase the by (va- a liquid weight; surface. above the TheThe piston, piston the pressure is cylinder of pressed has been the evacuated. so piston that can the pressure be ofvapor adjusted the pressure by piston of is choosing the practically the liquid equal weight substance, to Case the saturation quasi-statically. They be considered reversible. Examples: Figure variable leverage has to be adjusted to fitAs the a particular second gas. example, we consider the reversible, quasi-static conver- Figure 18.2 expansion of a working substance, for example compressed air Figure 18.3 in this case, heat energyelse must be it supplied can from the descendCase surroundings. very Or slowly and condenseprocesses, all the gas to liquid, acceleration. Such a processa is quasi-static called process as a se Processes in which temperature differences occur Summary tures: Reversible processes can be suffice to push the process in one or the other direction. The State Function Entropy, 18 476 Part III 18.2 Irreversible Processes 477

(if necessary using auxiliary apparatus) by simply following their path backwards. Restoring their initial states requires no net expenditure of energy, and they do not cause a permanent change of state in any of the bodies involved.

18.2 Irreversible Processes

The converse of reversible processes are the irreversible processes. Among them are in particular diffusion, the expansion of gases through a throttle or a nozzle, or into a vacuum, external and internal friction, plastic deformation of objects, heat conduction (when the temperature differences are not vanishingly small), energy input by radiation, as well as all those chemical reactions which do not run their course with inﬁnite slowness. Irreversible processes are characterized by three features:

1. All irreversible processes proceed spontaneously in only one di- Part III rection. This corresponds to our daily experience. The molecules of a perfume which are diffusing into the air in a room never return voluntarily to the open bottle from which they came. An object whose motion has been slowed by air friction will never be accelerated again by the air molecules so that it regains its original velocity. A portion of the air outside a house will never give up its internal energy to heat our apartment or even to fire up the boiler of a steam locomotive. When a stone falls on the floor, it suffers an inelastic collision there and comes to rest. We never experience the converse of that process: No one has ever observed a stone that one day rose back up spontaneously. These possibilities could readily be reconciled with the First Law of thermodynamics, but the molecules never realize them in practice. They can be depended on to divide up a large fortune (of energy, for example); but they will never agree to transfer that fortune voluntarily to a single, distinguished individual (the perfume bottle, the stone etc.). 2. In all irreversible processes, work is wasted, i.e. the possibility of obtaining useful work which exists is always defaulted upon. Instead of useful work, the processes give us only warmer objects. Examples: A certain amount of air is confined within a cylinder equipped with a movable piston. The air is heated while the piston is held fixed. Thereafter, it gives up energy by conduction and radiation until it has again cooled to room temperature. During this cooling process, work is wasted, and the chance of obtaining useful work is lost: We could have given the heated air the opportunity to push out the piston and to perform work until its expansion had cooled it back to room temperature. – In the first case (cooling by heat conduction), the additional heat energy delivered from the fuel used to heat the air is divided up among the enormous number of individual air molecules in the room and is no longer available for performing macroscopic the system, permanently ound”. outside however, there is an , by replenishing the 1 the system is . It is supposed to disturb the ho- outside 18.4 : We must not be dealing with a “closed system”, . We could indeed restore the original state after systems, irreversible processes lead to permanent it, and excess heat must be given up to the outside. For shows a gas that is being expanded irreversibly without The irreversibility of temperature closed 14.9 outside In The gas in the left-handthe part gas of at the the containerconserved. right is is to The to be become gas madea cooler. warmer, at steam while the Overall, engine, thermal left while energyits could the will exhaust cool then be steam. gas be at usedHow the does to our right inventor heat proceed? could He the be boresthe a used boiler two hole to in of halves the condense of partition separating theof container fine and hairs. closes His it plan is on the one following: side the with velocities a of the weir molecules made are The existence of irreversible processesIt is has a been fact confirmed completely bySuch verified experience. by an the inventor efforts might ofimagine for the many example setup unhappy try sketched inventors. in tomogeneous Fig. trick temperature distribution the within molecules; a gas we without consuming could work. S That is why it is preferable to use the word “irreversible”, which is derived from cylinders (work of acceleration!), istemperature simply through internal warmed friction back in to the ambient right-hand cylinder. We let a stone which haskinetic been energy lifted on up impact, warming fall the back ground tomation through earth work and of and friction. defor- waste If its itit had could been have connected been to allowed aform to suitable useful apparatus, work sink in slowly the toclock process; the and think of ground of its and the chimes. to weights of per- a3. pendulum changes of state an irreversible process has run to completion performing work. Wenecting could the have two placed pressure cylinders athe and turbine pressure obtained equalization. in work from the Instead, it theafter tube during air being con- in cooled the by right-hand cylinder, the expansion through the valve between the changed. i.e. the work must befrom input to the objects whichexample, the make turbine up mentioned the above would system wards have to as be driven a back- pump,or the with stone the would necessary havesuch to work cases, be provided lifted fuel from back is up outside; and burned using thus or muscle the power. food state In is of consumed some body work which was wastedessential during limitation the process – 1 Latin, rather than the literal translation “impossible to turn ar Figure 18.4 work; in thework), second it case can be (cooling made by use of. expansion whileFigure performing equalization The State Function Entropy, 18 has been completely verified by the efforts ofunhappy many inventors.” “The existence of irreversible processes is a fact confirmed by experience. It 478 Part III 18.3 Measurement of the Irreversibility Using the State Function entropy 479

statistically distributed. Only the fastest molecules coming from the right should be able to slip through the weir; the slower ones would rebound from it. Then the fastest molecules, with their higher kinetic energies, would pass over into the left-hand part of the container. There, they will have to share their energy with all the other molecules present; but at least the average kinetic energy would increase on that side. Its temperature would rise, while the temperature at the right would fall. – What is the problem with this “invention”? Answer: The BROWNian motion of the hairs of the weir. They would have to be so ﬁne that they could be moved aside by individual, fast molecules. However, if they were that ﬁne, they themselves would act as large “molecules” in the statistical distribution of the thermal motions. The weir would open and close statistically.Often enough, it would be open just when an undesirable molecule with only a small kinetic energy arrived at the partition. Thus, on the average, nothing would be gained and both halves of the chamber would maintain the same temperature.

18.3 Measurement of the Irreversibility

Using the State Function entropy Part III

Completely irreversible processes occur frequently2. Figure 18.5 shows an irreversible expansion as discussed in Sect. 14.8 in connection with an experiment using the steel-ball molecular model. Completely reversible processes, in contrast, are ideal limiting cases; all real processes are only partially reversible, they always contain irreversible contributions. These facts made it necessary to characterize reversible processes and to measure the degree of their irreversibility. This goal was met by discovering a new state function, called the entropy. One can arrive at this new function by the following route: In Fig. 18.6, a gas consisting of an amount of substance n at a constant temperature T1 can be allowed to expand in a reversible, quasi-static manner. Its volume is allowed to increase from V1 to V2, while the pressure decreases from p1 to p2. The work performed by the gas

Figure 18.5 A steel-ball model of a gas; at the top before, and at the bottom after a small expansion from a net volume V1 to V2 (cf. Sect. 18.4)

2 For example, all the motions treated in Sect. 5.11 are irreversible. In such motions, temperature differences occur due to internal and external friction. ) D rev / and Q 1 . / ,and 1 (18.1) 1 14.35 . .Then ( rev T 2 /: rev Q . For this T Q and path 1 V 1 2 p p : 2 1 ln V V : uniquely connected with nR ln from the whole system in 2 1 1 universal gas constant V V T rev not nR D , via a different ; but nevertheless, ln Q 1 R D ; D T nR / 2 1 2 2 . V V T 2 and T rev ln 2 D Q V / . 2 . nR D 1 rev / T 1 determined only by the initial and final states; Q . 1 Reversibly absorbed heat T rev D not Q 1 Temperature during the heat uptake W state variable The definition amount of substance (the gas) . 1 reversibly before the expansion starts by using a suitable D T n the same initial and final states are different, because the “path” was different! – In contrast, . reversibly from the large water bath, thus keeping its tempera- / 18.6 2 . 1 S W rev the same final state, namely Q At the end of the expansion, we put back reversibly the heat we carry out a slowand isothermal for the expansion heat at that this is lower taken temperature, up, we now find only Fig. purpose, we extract a quantity of heat is the same in both cases, namely however, the quotient Figure 18.6 of a reversible process auxiliary apparatus, and thereby reduce its temperature to the expansion; it is We can show thistaining by carrying out the expansion of the gas, main- ture constant. This heat which is taken up is however it it thus not a which we had previouslyperature removed, and thus restore the initial tem- during this isothermal process is At the same time, the gas takes up an amount of heat energy Therefore, we have the same initial state, namely The State Function Entropy, 18 480 Part III 18.3 Measurement of the Irreversibility Using the State Function entropy 481

This quotient is independent of the path taken by the process; it is thus a state function. We can use this state function as a measure of the irreversibility. It has been given its own name, entropy3. For the potential energy of a body or its internal energy, the zero point is always arbitrary; we can measure only changes or differences in these quantities. The same applies to the state function entropy: Its zero point is also arbitrary. The special case that we have just considered, a reversible expansion at constant temperature, simply makes a contribution to the entropy already present, since of course the ideal gas in our example had already taken up thermal energy or released it at some temperature or other. Therefore, we ﬁnally deﬁne the entropy increase by

Q S D S S D rev : (18.2) 2 1 T

In Fig. 18.6, the expansion was performed reversibly. The “system” Part III consisted of a large water bath and a cylinder containing the confined gas. The gas thus took up heat quasi-statically (Qrev.1/), while the water bath gave up heat quasi-statically .Qrev.1//. According to the defining equation (18.2), in this reversible process, the entropy of Q . / the gas increased by rev 1 and that of the water bath decreased by T1 Q . / rev 1 . For a reversible process, we thus have T1 X Q rev D 0 : (18.3) T Thus, a reversible process in a closed system produces no change in its entropy. We can henceforth use Eq. (18.3) for such a system as the defining feature of a reversible process. A completely reversible process is an ideal limiting case which can- C18.1. Here, we are dealing not be attained in practice. All real processes are more or less irre- with entropy production. versible. Then for a closed system, we findC18.1 Corresponding to the observations of irreversible X Q processes described in rev > 0 : (18.4) Sect. 18.2, their entropy can T only increase. The magnitude of this entropy production As an example, we choose the irreversible process of heat conduc- is the measure of the irre- tion. It is likewise presumed to occur in a system composed of two versibility of the process. – In the limiting case of parts. The thermal energy Qrev is given up by the system at the higher reversible processes, the temperature T1 and is taken on at the lower temperature T2.Inthis entropy production is zero, so process, the entropy of the hotter body decreases by Qrev=T1, while that the entropy of the system remains constant, i.e. it obeys 3 The name was coined by R. CLAUSIUS in 1865 from the ancient Greek for a conservation law (compare “turning towards”, referring to what he had previously called the “transformation contents”. Sects. 18.5 and 18.6). ) x S in = 1 18.2 some- ( (18.5) D , once 1 1 1 ,weini- 2 w .Thesum V -th fraction D 2 x . This means 2 T 1 = of the observed w V rev is the Q entropy increase 1 : I I V just as clearly as other 2 3 N Ã Ã Ã x x 1 1 x 1 are: .Inthevolume Â Â Â irreversibility x 1 . We obtain = 1 entropy V rev N 2 D D T D V x Q V 1 1 1 D D D or in 2 S 1 W ;w ;w ;w V V , but only with the probability 1 1 1 1 1 1 2 is thus positive. This V D D D S ,i.e. denotes how much more probable it is to find instead of in 2 2 2 2 and imagine how the corresponding model ex- 1 2 w w w V V D =w 2 1 in the molecular picture. In this picture, we will be 18.5 T w = . For two molecules, the probabilities of finding both T 1 = D rev observations, on statistical average we find it V -th partial volume, i.e. within the volume rev Q x x molecules, Q N 2 within the volume T ratio W = S rev quite generally. In order tothe justify ratio that, we wish to clarify the role of and for three molecules, and for The that with the volume tially assume that only a singlewith molecule absolute is certainty, present. and thus It with can be the located probability heat conduction phenomenon. of the system is a unique measure of the where Q of the large volume within the all the molecules in periment would function. The small volume state variables and functions, i.e.energy and the enthalpy. temperature, This pressure, level of internal as intuitive a understanding is rule possible only for the simple caseWe of refer ideal to gases. Fig. able to “visualize” the state function 18.4 Entropy in a MolecularWe have Picture so far derived theless, entropy we only will for apply a the special defining equation case. Neverthe- that of the cooler body increases by an amount molecules simultaneously in The State Function Entropy, 18 482 Part III 18.5 Examples of the Calculation of the Entropy 483 or ln W D N ln x : (18.6)

With N D nR=k (Sect. 14.6)andx D V2=V1, we obtain V k ln W D nR ln 2 V1 or, together with Eqns. (18.1)and(18.2),

S D k ln W : (18.7)

The increase of the entropy which accompanies the expansion of an ideal gas can thus be reduced to the ratio of two probabilities. We also require the value of the universal constant k D 1:38 1023 Ws/K. An increase in the entropy means a transition to a state of higher probability.InFig.18.5, ﬁnding all of the gas molecules in the partial V volume 1 is not impossible, but rather only extremely improbable. Part III That already holds for the relatively few molecules of the model gas, and a fortiori for the enormous number of molecules in a genuine gas. The relationship between entropy and probability was recognized by LUDWIG BOLTZMANN (1844–1906). For this reason, the constant k carries his name.

Think of a mixture of ice and water at 0 °C. In the ice, the molecules are arranged very regularly in the form of a crystal lattice; this is a very improbable state. In the water, they are in a more probable state, with a large degree of randomness. As a result, the entropy of the water is considerably greater than that of the same amount of ice. Nevertheless, a thermally insulated block of ice does not spontaneously convert even a part of itself into water. That would lead the whole system to an extremely improbable state; a portion of the ice would have to cool below 0 °C in order to provide the necessary heat of melting for the rest. That would reduce the entropy of the whole closed system: The entropy of the ice would have to decrease more at a temperature below 0 °C than the entropy of the water at 0 °C would increase through the addition of heat. Another example is perhaps intuitively clearer. The letters which make up this text are in an extremely improbable state; they therefore have a much lower entropy than if they were simply jumbled up in a box with no kind of order. Nevertheless, the letters of this text will certainly not spontaneously convert themselves into a disordered heap; such a transition would have to pass through an extremely improbable intermediate state: Many of the letters would have to accumulate a very great thermal energy and – as “molecules” – use it to jump over their neighbors.

18.5 Examples of the Calculation of the Entropy

Working through examples and applications is always the fastest way to become accustomed to a new physical concept and to be able to . T we : (18.9) (18.8) (18.10) R 18.6 20 : ; is heated at 1 : ) T D M j ; is thus taken up T 13.1 / j pj f . c Ws j mol K rev Ml T j to a final temperature Q i X 10 T R . Its melting point is D M /: f l j 64 S : n D X 2 Ws : kg K Ã 18 g/mol, the molar entropy in- f 3 D W s/kg (see Table ; T 10 D 5 Ml . Let us assume that an object has n 10 kg Ws A substance of mass Ws = C 22 D : mol K CD lting object increases by the amount 3 1 M 31 W s/(mol K) T / S : 34 : 2 22 10 8 . 2 D 3 D T 2 D 8 S D rev p2 : m D T M c S R Q f on melting M n 11 . l . C D 200 g/mol), the corresponding numbers are ; f T C C18.2 l D 18.3 1 / 1 , reversible processes; that follows directly from its T m . ; p1 273 K 1 M c rev T 1K D Â : Q T and a specific heat of melting M 234 M D D D S S T quasi-static crease becomes Then the increase in the specific entropy is Numerical example for water at standard atmospheric pressure (1013 hPa) For mercury ( or, with the molar mass Entropy increase on heating. Entropy increase S . In this process, small amounts of heat are transferred at gradually f T amass by the object at practically thethat melting case, temperature, the i.e. entropy reversibly. of In the me The process of meltingslightly takes higher place in temperature. in surroundings The which latent are heat only increasing temperatures, and thus reversibly. Forof the the entropy substance increase heated, we therefore find will treat the firstto applications measure of the the stateploy calculated function entropy, values. we – will In always have order to em- definition in Sect. 1. On reversible melting, the entropythe of same the amount as surroundings it decreases increases by in within every the reversible object process that in melts. aentropy Thus, closed remains as system, unchanged. the – overallalso amount The in of corresponding all conclusion the holds following examples. 2. use it productively. To this end,tion we entropy will first for calculate several the important state func- cases, and then in Sect. constant pressure from an initial temperature (ac- ). Thus, in en- ,butrather 18.3 exchange increase production The State Function Entropy, 18 we are not dealing with true entropy companied by a transfer of heat). C18.2. The tropy in all the examples treated in this section always refers to the substance considered, which exchanges entropy with its surroundings via reversible processes. For the overall system, the entropy remains constant, in accord with Eq. ( with entropy 484 Part III 18.5 Examples of the Calculation of the Entropy 485

Table 18.1 Specific state variables for water along its vapor-pressure curve. The reference point for the enthalpy and the entropy is chosen to be 0 °C. Liquid Saturated vapor Temp- Vapor Vol ume V Enthalpy H Entropy S Vol ume V Enthalpy H Entropy S erature pressure Mass M Mass M Mass M Mass M Mass M Mass M Â Ã Â Ã Â Ã Â Ã Â Ã Â Ã Â Ã (°C) N m3 Ws Ws m3 Ws Ws 105 105 103 105 103 m2 kg kg kg K kg kg kg K 17.2 0.02 0.001 0.724 0.255 68.3 25.33 8.71 59.7 0.20 0.001 2.495 0.829 7.79 26.09 7.91 99.1 0.98 0.001 4.149 1.298 1.73 26.71 7.37 151 4.9 0.0011 6.364 1.851 0.382 27.47 6.83 211 19.6 0.0012 9.043 2.437 0.101 27.97 6.36 310 98 0.0014 13.984 3.345 0.0185 27.26 5.61 374 221 0.0037 20.625 4.313 0.0037 22.07 4.61 or, in the limit of infinitely small increments and with constant specific heat, Part III Zf dT Tf S D Mcp D Mcp ln ; (18.11) T Ti i and, with Mcp D nCp (see Eqns. (13.5)and(13.6)),

Tf S D nCp ln : (18.12) Ti

Numerical example for water on heating from its melting point up to its boiling point at standard pressure (1013 hPa). Its heat capacities are nearly constant over this temperature range:

Ti D 273 K ; Tf D 373 K ; Ws Ws c D 4:19 103 or C D 75:5 ; p kg K p mol K S Ws S Ws D 1:31 103 or D 23:6 D 2:84 R : M kg K n mol K

Table 18.1 gives some additional values of S=M at various temperatures. These values play an important role in technical applications. To simplify the calculations, the speciﬁc entropy of liquid water at 0 °C (273 K) and standard atmospheric pressure (1013 hPa) is arbitrarily set equal to zero. We make use of this convention in quoting measured values, and denote the entropy thus deﬁned by S.

3. Entropy increase on evaporation: PICTET-TROUTON rule. Let a liquid have the mass M and the specific heat of vaporization lv.It is evaporated at constant pressure (its saturation vapor pressure) and the corresponding temperature T. Then for the increase in entropy, we have Mlv S D : (18.13) T . ). 1 T 2, it 26 > : . (18.14) ! 2 2 : R T : D R 18.1 16 v 1 l : Ws kg K 13 3 D 10 : at 0 °C is converted 2 3 p p 37 Ws 373 K and : R mol K ln 2 7 1 D and proceed in two steps 2 : T water ) takes up heat at constant D 109 (Pt. 3) T n 13 , quite similar values for the nRT D 18.7 n D on vaporization S = Ws n kg K S 3 T d ) to the state 2 (temperature T p 10 C or 3; and then along the path 3 1 C . T / 3 at 100 °C, the molar entropy of the water ! 1 Z 06 : C18.3 ). When liquid ), Ws n kg K ); then we find R 6 R 3 D C 84 diagram in Fig. 64 10 vapor 14.3 : : S (Pt. 2) 2 14.35 ’s rule 31 V on heating : - 06 : 1 p . 6 D D )and( is possible only if a quantity of heat equivalent to the D Calculating the entropy of ideal S 2 S M ROUTON S T M n 18.12 W s/kg (see Fig. 6 10 Numerical example For water at standard pressure (1013 hPa), Entropy changes associated with changes of state of ideal gases S This latter quantity is calledpor. Values the for specific other temperatures entropy can of be the found saturated in va- Table from the state 1 (temperature First, the gas (amountpressure of along substance the path 1 receives work at constantperature temperature. The constancy of the tem- 4. We look at the gases work is given up by theEqns. gas ( to its surroundings. Then, making use of Figure 18.7 evaporation of many other substancestent are of obtained. “T This is the con- When liquid water isentropy converted is into thus water about vapor,into five the liquid times increase water greater in than (2 its into when saturated ice water isincreases converted by or, referred to its mass, For the molar entropy increase ”, (1883). Grundriss (1876) and AIME Chemical Educa- , https://de.scribd. ICTET ROUTON (2001), p. 55. Available The State Function Entropy, 6 ILNIAK 18 English: see J R.T. T the-Ratio com/document/71366734/ Frederick-Thomas-Trouton- The-Man-The-Rule-And- tor online at published independently by R. P and E. Wicke, “ der physikalischen Chemie Akad. Verlagsges., 10th ed. (1959), Chap. 2. Apparently C18.3. See e.g. A. Eucken W 486 Part III 18.6 Application of Entropy to Reversible Changes of Statein Closed Systems 487

Now, T3 D T2, p3 D p1. Then we ﬁnd for the increase of the molar entropy S T2 p2 D Cp ln R ln : (18.15) n T1 p1

The entropy of an ideal gas thus increases with increasing temperature and decreases with increasing pressure. The entropy of an ideal gas accordingly consists of two parts: The ﬁrst part depends on the temperature, and the second part depends on a geometrical constraint, namely the available volume, which determines the pressure and the number density. On isothermal compression, the entropy of an ideal gas becomes smaller. – For the derivation of this equation, we could have carried out the transition from state 1 to state 2 along some other arbitrary path, e.g. in the two steps 1 ! 4and4! 2. The entropy is a state function; it is independent of the path along which a change of state is accomplished.

18.6 Application of Entropy Part III to Reversible Changes of State in Closed Systems

When reversible processes occur adiabatically, i.e. without any exchange of heat with their surroundings, then the sum of the entropies of all the bodies involved remains constant; the processes are isen- tropic. This constancy of the entropy in reversible, adiabatic processes is often employed. We begin by showing in Fig. 18.8 some adiabatic curves in the p- Vm diagram (Vm D V=n) of an ideal gas for reversible expansion: At the lower end of each adiabatic curve, the constant value of the associated molar entropy S=n is noted. Formation of fog or clouds on adiabatic expansion. Water vapor with a vapor pressure of p1 is expanded adiabatically, and its pressure decreases to p2. What fraction y of the water will precipitate as fog droplets? This case plays an important role in meteorology – think of an upwardly-directed current of warm air.

Before the expansion and cooling, the vapor pressure p1 is associated with the temperature T1. At this temperature, the water vapor of mass M has the entropy S1. During the expansion and cooling process, a fraction y of the vapor is condensed to liquid water (fog droplets). The mass of the vapor is reduced to M.1 y/, and at its temperature T2, it retains the entropy .1 y/S2. Furthermore, water droplets of mass My are formed. At the temperature T2, this liquid 0 water has the entropy yS2. Setting the entropies before and after the condensation equal, we ﬁnd

. / 0 : S1 D 1 y S2 C yS2 ) T - p ) or : : 18.13 ( (18.16) ,orthe 14.3 V n - = p Ws Ws kg K kg K V S/n 3 3 10 10 is (Fig. ). 2 Ws T 91 71 kmol∙K : : 7 8 4 3 D D m 10 kmol 1 2 : M M S S 2.5 Diagram with 1.67 0.23 : : M Exercise 18.1 2 2 v v /; l kg T l T Ws ; C18.4 =885 K Molar entropy C /; 6 ı T 1 C S 1 D ı : 10 095 : 295 K 590 K 8 V/n : 0 M 17 0 , or the molar volume 2 OLLIER . 2 S 45 59 D : M liquid . S 2 = y 3K : V D D 2 v K y l 333 K 290 1260 K or M S vapor D D 1 2 840 Molar volume T T K H-S ; ; 627 560 K 0102030 5 15 Pa 10

Adiabatic curves for a diatomic, ideal gas as curves of constant

5 Applications. Supersonic Gas Jets Pressure Pressure 200 hPa 20 hPa p 10 D D 1 2 p p i.e. 9.5 % of the saturated vapor has been condensed to fog droplets. The specific heat of vaporization (latent heat) of water at Result: Numerical example for water vapor S Combining these two equations, we obtain entropy. The reference pointstandard for atmospheric the pressure entropy (1013 has hPa) (cf. been chosen to be 0 °C and 18.7 The Thus far, we have depicted the states of substances only in Figure 18.8 Furthermore, we have diagrams. Their ordinates representshows the pressure, the while specific the volume abscissa 1 S , D 1 p 0 T 1013 hPa D as end points, 2 0 p T , 2 p (Eq. (18.15)), then 2 The State Function Entropy, S and 18 1 as starting values, use respectively, to determine 273 K and T convert to specific entropies. and C18.4. Determined using Eq. (18.15). Taking 488 Part III 18.7 The H-S or MOLLIER Diagram with Applications. Supersonic Gas Jets 489 temperature T. One could however equally well use other pairs of state variables to construct such diagramsC18.5. C18.5. For some examples of MOLLIER diagrams, see As one of the many possibilities, in Fig. 18.9 we show an H-S di- http://www.engineersedge. agram for the case of air. On the ordinate, the specific enthalpy is com/thermodynamics/ plotted, i.e. H=M, and the abscissa represents the specific entropy, enthalpy_entropy_mollier. i.e. S=M. The values on the ordinate are computed from Eq. (14.27), htm while those on the abscissa are found with Eq. (18.15). In both cases, Supersonic jets are described the temperature dependence of the specific heats has been taken into for example in https://www. account (see, for example, Fig. 16.10). fas.org/sgp/othergov/doe/lanl/ In an H-S diagram, the adiabatic curves are straight lines which run pubs/00326958.pdf parallel to the ordinate axis. The isotherms, drawn here for several temperatures between 130 °C and C50 °C, are straight lines only at low pressures, and then they are parallel to the abscissa axis. – In a p-V diagram, the isobars and the isochores would be straight lines, while in the H-S diagram, these lines of constant pressure and of constant volume are curved.InFig.18.9, only a few isobars are shown for pressures in the range .0:01 200/ 105 Pa.

The H-S diagram plays an important role in the adiabatic changes of Part III state of ﬂowing substances. One can determine the technical work which can be obtained from a change of state without any computations; only the values on the ordinate need be read off. In the following, we treat an example of an application which is equally important in basic physics and in technology. It concerns the adia-

Pressures 105 Pa p=200 100 40 10 4 1 0.4 +5∙104 W s +50°C kg +20° α 0 M –30° 4

/ Mass –5∙10 0.1 H β –80° –10∙104 Enthalpy –130° 0.01 –15∙104 –1500 –1000 –500 0 W s/kg∙K Entropy S/ Mass M

Figure 18.9 A portion of the H-S or MOLLIER diagram for air (from Zeitschrift des VDI 48, 271 (1904)). R. MOLLIER (Professor in Dresden) introduced the enthalpy as the quantity on the ordinate of state diagrams in 1904. The curves are isotherms and isobars, respectively. The values of the enthalpy and the entropy are referred to standard conditions, 0 °C and an air pressure of 1013 hPa. ) 2 is p V 2 = dia- p S of the M - (18.17) (18.18) A H D can be set . The dif- ). See also % Q Pa. Then the b. The values 5 18.9 . The adiabatic a. In all cases, In our example, ˛ 10 . 14.42 18.10 max Pa and a temperature : 10 : 18.10 in Fig. 5 2 D ˇ kg Pa has been assumed. – 10 Ws 2 5 Mu Pa, the highest obtainable 4 p 1 2 5 40 : 10 by the point : 2 10 D 10 D . 6 1 Au Mu 40 : Nevertheless, one cannot achieve p 18.9 % ) shows the final or muzzle veloc- 2 1 tech 9 t 40 . It is given by the product of these can be read directly off the D W u D 1 D 1 D D H 1 p 18.17 vacuum 1 M 1 H p 760 m/s. This maximum value is obtained 438 m/s. H 2 along the vertical axis gives the decrease in M H D 2 D of the air, to the cross-sectional area ˇ depend on the initial and final pressures? 2 H ). We can for example assume that the air in the within a pressure cylinder, where the pressure u % u H 1 max and p u 18.9 ) and setting the work of acceleration equal to the ˛ 340 m/s at 20 °C). D of the air that flows out is proportional to the time of ( 14.11 M The jet velocity can be considerably greater than the speed of sound , to the density t S final state of the air is marked by the point The enthalpy difference Result: sound c lower. During the expansion,and imparts the kinetic air energy (in performs oneresulting work direction!) jet to of velocity itself. acceleration How does the gram for air (Fig. pressure cylinder is at a pressure of of 20 °C. Its state is marked in Fig. expansion proceeds to a final pressure of muzzle velocity is with an initial pressure of when the air flows out into a is held constant.forming The a air jet, is and allowed enters to a flow space out where through the a ambient nozzle, pressure Inserting this value into Eq. ( ference between the specific enthalpy during the expansion. It is In an adiabatic process, no thermalroundings. energy Therefore, is in exchanged with the the equation sur- for the First Law, to zero. For the workusing of Eq. acceleration ( of the flowing air, we then find, kinetic energy of the air in the jet, velocities greater than a certain limiting value u can be obtained, and they are plotted in Fig. batic flow of a gas“supersonic which jet”. is escaping from a pressurized container:For this a example, we chooseto the gas a to high be pressure air. The air is compressed On expansion, the density of the air (i.e. the quotient In a similar fashion, the flow velocities for other final pressures ity of the air jet to be a constant initial pressure of decreases. This is shown for our example in Fig. shown here were calculated with the aid of Eq. ( nozzle and to the flow velocity four quantities, that is Footnote 1 there. The mass flow The State Function Entropy, 18 490 Part III 18.7 The H-S or MOLLIER Diagram with Applications. Supersonic Gas Jets 491

Figure 18.10 The su- m personic ﬂow of a gas s 800 a jet from a nozzle. All u three curves apply 600 to an initial pressure 5 400 of p1 D 40 10 Pa. % The values of u and 200 are those found after the Flow velocity expansion. kg m3 b

ρ 40

20 Density

m2 kg/s c –4

u 3∙10 1 ρ = I 2∙10–4 γ –4 Area A

1∙10 Part III Mass current 0 10 20 30 40∙105 Pa p p p 5 Pressure drop ( 1– 2) for 1=40∙10 Pa

With the mass current Mass M of the ﬂowing air I D ; (18.19) Time t we obtain A 1 D : (18.20) I %u

This quantity is plotted in Fig. 18.10c. We consider its content in detail, making use of the other two parts of the figure. We find the following: In Fig. 18.10a, the curve of the flow velocity hardly deviates from the ordinate axis up to about 70 m/s. Thus, the density curve for low velocities in Fig. 18.10b corresponds to a single fixed point, namely the point on the ordinate axis: The density % is thus constant up to about 1=5 of the velocity of sound (Sect. 10.1). At “low” velocities, gases behave like non-compressible fluids: The quotient A=I decreases in Fig. 18.10c with increasing values of u. – However, at high velocities, the situation is quite different: Here, the density % decreases rapidly with increasing velocity. As a result, in Eq. (18.20), the increase in u is compensated by a decrease in %, so that the quotient A=I becomes constant over a certain range (in Fig. 18.10c). Later, the decrease in % begins to predominate over the increase of u, and the quotient A=I once again rises. At its minimum (Point ), the flow velocity is equal ) S - H Then, 14.56 ( remains I at the location . 1 p at every point. Then ), the point of small- 53 I : of the nozzle must be 0 A D T 18.11 : 2 1 p m R M Ã has been attained 1 2 C DE s Â ; therefore, the mass current D D 2 1 , then additional “downstream” reductions p p 2 nozzle for pro- sound p . If we want to obtain higher muzzle veloc- c , and thus the maximum of the mass current, is I = AVA L A ). The cross-sectional area AdeL Example of a simple nozzle, not suitable for produc- 18.12 At the narrowest part of the nozzle, the flow velocity is still temperature of the adiabatically expanded gas at the point of the , 1845–1913, Sweden) velocity of sound D T For air, this occurs at an external pressure of The minimum of reached when ( smallest cross-section). S AVA L of the smallest cross-sectionin in the external the pressure nozzle by a sufficient reduction If a simple nozzle is used (as shown in Fig. in the pressure havewith no the further velocity effect. ofvelocity sound, They and and penetrate can therefore upstream propagate to cannot the at overcome point most the of smallest flow cross-section. ducing supersonic gas jets (C. G. P. ing supersonic velocities Figure 18.11 Figure 18.12 L This can be derived in thequalitatively: general If case, the velocity but of it sound can also be understood to the est cross-section falls togetherin with the the muzzle of muzzle a of simple nozzle,to the the the flow nozzle. velocity velocity can be of atities, sound most equal then the nozzlepoint must (Fig. be enlarged conically after its narrowest diagram. equal to the velocity of sound adapted to the area required by the mass current the gas can exit the muzzle with the full velocity predicted by the the same as before, without the conical extension of the nozzle. The State Function Entropy, 18 492 Part III Exercise 493

Exercise

18.1 Using the adiabatic curves shown in Fig. 18.8, determine the corresponding molar entropies and compare them with the values given in the figure. (Sects. 18.4 and 18.6. Hint: Refer to Eq. (18.15).) Part III okn usac ae nteha energy heat the on takes substance working following. A the in shown a be in will engine” This the statically. “heat permit a instead via and equalized versible be radiation, such to work and effects difference irreversible of conduction temperature all amount heat eliminating maximum friction, by ideal as performed the be obtain could could that we contrast, In ihu h eito famcie eprtr ifrne are differences temperature machine, a by of equalized mediation the Without perma- fuel. is of supply which the quantity is only consumed the nently reproduced; and periodically substance and working edly flowing a of process means this by region cool a ok h eodLaw Second The Work. into Heat Converting 1 to- engines everyone engine heat to steam known All well the are engines, day. engine, heat combustion mechanical of internal types performing the important and for most differences The temperature order work. of in use innovation technological make a to as developed Law were Second engines Heat the and Engines Heat 19.1 eesbepoess(nSect. (in processes reversible .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2017 K. Switzerland Publishing International Springer © chapter. this reading before stu atd ri te od,a potnt opromuseful perform to opportunity an (Sect. utilized words, not other work is in both, work in or irreversible; wasted, are thus processes is these of Both radiation. and energy olrsror hnteecs etenergy heat excess the Then reservoir. cool eevi ttehg temperature high the at reservoir tp hntewrigsbtnehsflwdt h olreservoir cool the to temperature flowed lower has the substance at working the When step. h te nemdaesesi h ahn’ prto aeplace ( take Eq. operation from machine’s case, the that all in In that steps reversibly. assuming intermediate work, mechanical other to the converted completely be can h nrp hne szr,i.e. zero, is changes entropy the ti ugse htterae fam reader the that suggested is It Q anr hti,altesesi h rcs utocrquasi- occur must process the in steps the all is, that manner; rev . 2 periodically / hra processes thermal ( < ) gi na stemladrvril tp othe to step, reversible and isothermal an in again 0), Q mediate olsItouto oPhysics to Introduction Pohl’s 18.2 h nta tt ftemciei repeat- is machine the of state initial The . rev T T 18.2 1 2 . 1 tgvsu h mle muto heat of amount smaller the up gives it , npriua h xmlsgvnudrPit2) Point under given examples the particular in , / laiehm rhrefwt h rpriso ir- of properties the with herself or him- iliarize C ). h rniinfo o einto region hot a from transition the ln,ie hog etconduction heat through i.e. alone, Q T rev T 1 18.3 2 na stemladreversible and isothermal an in . 2 / ,w e httesmo all of sum the that see we ), D 0 Q : Q rev rev . 1 . O 10.1007/978-3-319-40046-4_19 DOI , / 1 / C ( > Q )fo hot a from 0) rev . 2 / D repeat ( 18.3 re- W 1 ) 19 495

Part III is per- (19.1) W than lift- (19.2) consumed, The Second empirical re- heat engine C19.1 ). Nevertheless, / 1 of a . (1822–1888). ideal : rev

14.11 ideal 2 Q

T D is output quasi-statically. 1 . It states that, “It is not and also the work / / T 2 2 . 1 . does nothing more LAUSIUS T rev ideal rev C /

Q 1 Q . D LANCK is input quasi-statically, and the or the amount of pressurized, stored of an ideal heat engine. The com- / C rev ) yields / 1 / 1 . . Q 1 W . ’s considerations were still based on ideal rev rev UDOLF

), both 19.1 rev Q Q reduced Q performed to heat ) and our knowledge of its general applica- / D 19.2 0 D ARNOT )and( / C 1 19.2 ideal .< . .

W . rev 18.3 W of the gas is Q C19.3 C19.2 ) is a quantitative statement of the Second Law of , from Eq. ( 2 thermal efficiency 19.2 density T ARNOT D C 1 T . The discussion above clearly shows this. The ‘law’ is based on In the isothermal expansionby of the a gas gas, is the indeed entire converted to amount work of (see heat Sect. taken up this does not contradict the Second Law,work, because in for addition example to performing the working: of The lifting a weight, something else is happen- air is decreased. ADI defines the Law limits the conversion of heat into work. For The First Law states thatto the a sum of change all of the stateperimentally energies remains by which constant. contribute converting This work can completelyusing be into friction. demonstrated heat, ex- for – example The reverse process is not possible: possible to construct a machine which ing a weight and coolingamount of a heat” heat bath by withdrawing the equivalent lower temperature at which the heat the assumption of ainterpretation of “heat Eq. substance” ( (“phlogiston”). The modern bility are due in the main to R The highest theoretically-possible efficiency Equation ( thermodynamics. Its essentialS content was formulated in 1824 by bination of Eqns. ( formed are equal tothe zero. Second Law This which fact is is due the to basis P of a formulation of thus independent of all theThe details essential of point is its only constructioneliminated; and that then all operation. the possible irreversible determiningperature processes factors be at are which simply the the higher heat tem- The Second Law of thermodynamics is also a purely sult the successes of technologyengines. and its experience in constructing heat The ratio of work and 0and W > (1796– W Q . 2 )), here D Q 2 ARNOT Q 0. 14.6 C Réflexions sur la W D 1 ”, Paris (1832). 0. The energy balance is Q W Converting Heat into Work. The Second Law is positive, while D are negative.) < 1 D C 19 2 1 Q 1832): “ puissance motrice du feu et les machines propres a développer cette puissance or Q Q ( C19.3. This is the reasonthe why Second Law is oftenmarized sum- by the statement: “It is impossible to construct a perpetual motion machine of the second kind”. That would be a machine which does nothing other than convert heat to mechanical work with 100 % efficiency. C19.2. S. C C19.1. The negative sign of the output quantities is due to the formulation of the First Law (Eq. ( Q For a heat engine, W then Q 496 Part III 19.2 The CARNOT Cycle 497

19.2 The CARNOT Cycle

The considerations in Sect. 19.1 stand and fall with the possibility that a working substance can take up an amount Qrev.1/ of heat energy reversibly and can release a smaller amount Qrev.2/ reversibly.

The fundamental possibility of such processes can be demonstrated with a machine which carries out a “CARNOTcycle” (Fig. 19.1). It makes use of a gas which is conﬁned in a cylinder, as shown in Fig. 18.2. This cylinder is ﬁrst brought into thermal contact with a hot reservoir or heat bath (T1 in Fig. 18.6). The gas expands isothermally and quasi-statically by taking up the heat Qrev.1/ reversibly from the heat bath and performing mechanical work with it. The cylinder is then thermally isolated from its surroundings and the gas is further expanded adiabatically until it reaches the temperature T2 of a cooler heat reservoir. During these two expansion steps, the gas performs all together the work W1. In the third step of the cycle, thermal contact is made with the cooler heat bath. The gas is compressed isothermally and quasi-statically, thereby giv- inguptheheatQrev.2/ reversibly to the cooler heat bath. In the fourth step, the cylinder is again thermally isolated and the gas is compressed Part III adiabatically until it reaches its initial temperature T1. Then the initial state of the system has been restored. During the two compression steps, the work W2 must be performed on the gas. The net output of work is Qrev.1/ C Qrev.2/ D.W1 C W2/ DW. At each changeover between isothermal and adiabatic volume variations, the adjustable lever arm shown in Fig. 18.2 must be set to the appropriate length.

The decisive point in the CARNOT cycle is that the temperature of the working substance (the gas in the cylinder) is equilibrated to the temperature of the respective heat bath with which it is brought into contact. This requirement can be fulﬁlled by a slightly different cyclic process in the STIRLING engine, which can be more readily constructed. It is based, as we shall soon see, on the use of a regenerator.

Figure 19.1 CARNOT cycle p

T 1 T 2 V 2 T )ex- )and and b 1 periodic 19.2 T heat stor- ). d , and will employ 19.2 at the temperatures and is moving the piston to the 1 19.2 T Engine ,the in Fig. c ). This means that during the displacement II , the air has expanded isothermally by taking and a a and TIRLING I was previously used for small-scale industrial pro- , the motion of regenerator d ( The operation of C19.4 . , moving the working substance (usually air) alternately engine. At is at its reversal point, , the displacer shoves the air towards the cooler heat bath. ), respectively, are attached to the left and the right ends and b 2 D b T > TIRLING 1 displacer . It contains channels along its length. This drum is pushed back T (Video 19.1) the piston is reversed (rest points) right. At displacer while at gives it up to the gas which isThe flowing to four-step the operation left (Part ofparts this of the heat figure. engine At up is heat illustrated at in the the higher four temperature of the gas, thedrum latter takes flows up through heat the from channels the in gas the flowing drum. to The the right (Part of a cylinder.D In the cylinder,and besides forth the within piston thecrankshaft there cylinder (not is by shown), means ato with of the drum a a piston. phase connectinga shift The rod drum of to fulfills around the atowards 90° the double hot relative function: and the First, coolage heat medium it bath. acts Second, as it acts as a Figure 19.2 ( 19.3 The S This engine it several times in order to demonstrate the content of Eq. ( process. We will explainuse its of the construction semi-schematic and illustrations operation in by Fig. making perimentally. The two heat baths cesses and as ain toy. an especially It clear demonstrates way:tween the a that hot essentials is, and a the of cool mediation a reservoir of by heat a heat engine working transfer substance be- in a aS ,aScot- TIRLING TIRLING S Converting Heat into Work. The Second Law 19 OBERT “Operation of a S Video 19.1: tish clergyman, in 1816. not only by the displacer drum, but also by awithin wire the net cylinder. using a lecture-demonstration model. There, the function of the regenerator is performed engine” http://tiny.cc/gigvjy The operation is explained C19.4. It was invented by R 498 Part III 19.3 The STIRLING Engine 499

On the way, it is cooled to the lower temperature T2 while flowing through the channels in the displacerC19.5. At c, the piston is moved C19.5. This step is an to the left by the flywheel (stored mechanical energy!) and the air is isochoric change of state compressed isothermally at the lower temperature T2, giving up heat. (Sect. 14.9, Point 3). The At d, the displacer shoves the compressed air back towards the hot cyclic process here is com- reservoir. On the way, it is warmed up to T1 by flowing through the posed of two isotherms and channels, taking up the stored heat from the regenerator. Then the cy- two isochores. The important cle can begin anew: The compressed air can again take on more heat point however (just as in at the higher temperature, expand isothermally and push the piston to the CARNOT cycle) is that the right, performing work. After 1/4 of a rotation, the initial state a the expansion occurs at the has again been reached. higher temperature and the compression at the lower In the ideal limiting case, the work performed by the engine with this temperature. cyclic process would be

T1 T2 W D Qrev.1/ : (19.2) T1

Here, Qrev.1/ is the heat energy which is taken up at the higher temperature, causing the gas to expand. This heat uptake is isothermal in Part III the ideal case. Then we haveC19.6 C19.6. These equations hold under the assumption that the V2 p1 working substance is an ideal Qrev.1/ D nRT1 ln D nRT1 ln (14.35) V1 p2 gas, which is the case for air near room temperature. The

(n D amount of substance (air), R D universal gas constant, p1 (V1)and statement of the Second Law p2 (V2) are the pressures (volumes) before and after the isothermal expan- (Eq. (19.2)) is however inde- sion). pendent of this assumption; it holds quite generally. Combining Eq. (14.35)andEq.(19.2) yields

p1 W D nR.T1 T2/ ln p2 or W D const .T1 T2/: (19.3)

In words: The work performed by the STIRLING engine is determined only by the temperature difference T1 T2. This assertion can readily be conﬁrmed in a demonstration experiment. Figure 19.3 shows a STIRLING engine in silhouette. The upper half of the cylinder is held at the constant temperature of C20 °C (293 K) by circu- lating water, and the lower half is alternately heated by a glycerine bath at C220 °C (493 K) and cooled in liquid air at 180 °C (93 K). In both cases, the temperature difference is the same, namely 200 K. Indeed, the machine runs in both cases at the same speed of rotation and thus performs the same amount of work in equal times (here the work serves only to overcome the friction of its bearings).

A more modern engine would use a fixed regenerator. It connects two cylinders in which the pistons move with an appropriate phase shift. ) air 14.48 .Inthe ( 2 V /: 1 : 14 of the steam in Table engines are of great . 4 : 0 must be considered. The 1 / 1 Ã km(!). Therefore, if the steam 18.7 ). After combustion, the tem- 1 2 .The V V specific enthalpy ;. Â ) with 19.4 1 19.2 T for air 19.3 I . D internal combustion (Fig. step, the corresponding velocity would be 2 1 T V . While pushing out the piston, the working 1 T single (Video 19.2) ) sketched in Fig. II D adiabatic exponent Verification of Eq. ( serve as input and output open- engine. The crank 2 moves the D Rs the cylinder, at its upper end. The working substance is (height of the water level above the turbine) for water turbines TIRLING process, it cools to a temperature of Apart from steam engines, around 1.5 km/s. For thisseveral reason, stages steam connected turbines in are series. subdivided into The working substance for turbines remainster today vapor; in in most special cases wa- using cases, a the circuit steam of stages Naevaporated, or are is biphenyl preceded passed vapor. by The through water a aunsaturated vapor, stage “superheater”, after vapor, i.e. being a it pure isare gas. converted employed. to Temperatures of up to about 500 °C substance expands adiabatically up to the cylinder volume Figure 19.3 end, denoted by displacer drum ( tube ends ings for the room-temperature watermaintains which the temperature of theof upper the cylinder. half Its lower halfcooled is by in liquid this air, case and is thus the cooler with a small component (less thanbustion 21 products. mole – percent) of Assume gaseousabove the com- volume the of piston the to combustion chamber be technological importance. In their case,within the heat energy is produced The remainder of the heatis produced, which given is up not to converted to the work, surrounding air along with the exhaust gases. Its head corresponds to the decreasea of the steam turbine.corresponding In to modern a turbines, water it headwere expanded of can in 122 be a up to 0.33 kWh/kg, 19.4 Technical Heat Engines Technical heat engines dotant not operate steam reversibly. engines today Thethe are most dependence of steam impor- the turbines. gasaccount. density In on The their its points pressure construction, treated must be in taken Sect. into perature rises up to aS differ- ). TIRLING Video 19.1 engine” between the two halves Converting Heat into Work. The Second Law TIRLING 19 http://tiny.cc/gigvjy aS Video 19.2: “Technical Version of eration, see also ( engine. The engine shown here is about 100 yearsFor old. a demonstration of its op- only the temperature ence of the cylinder determines the operation of the S http://tiny.cc/kigvjy In the video, it is shown that 500 Part III 19.5 Heat Pumps (Refrigeration Devices) 501

Figure 19.4 Regarding the efﬁciency of an internal combustion engine

temperature decreases in this process from T2 down to the outside temperature. In order to avoid taking average values for the temperatures, we insert T1 and T2 into Eq. (19.2) and obtain as the largest theoretically-possible efﬁciency Â Ã 1 T1 T2 V1 ideal D D 1 : (19.4) T1 V2

The smaller the ratio V1=V2 (its inverse is called the compression ratio), the cooler the exhaust gases and the higher the efﬁciency.

The required quantity of air and fuel can be introduced into a small Part III combustion chamber only by compressing them strongly. If the piston compresses a fuel-air mixture (NIKOLAUS OTTO, 1876), the ratio V2=V1 8 cannot be exceeded if we wish to avoid premature ignition. This corresponds to an efﬁciency of ideal D 57 %. If only the air is compressed by the piston and the fuel is then injected into the combustion chamber (RUDOLF DIESEL, from 1893), then we can increase the compression ratio to V2=V1 16. This corresponds to ideal D 67 %.

OTTO and DIESEL engines have roughly the same temperature in their combustion chambers, T1 1900 K. But the DIESEL engine, with V2=V1 16, can eject its exhaust gases at a temperature T2 which is lower than in the OTTO engine, with V2=V1 8. The practically achievable efﬁciencies are for the OTTO engine 30 %, and for the DIESEL engine 35 %.

19.5 Heat Pumps (Refrigeration Devices)

In Fig. 19.3,weshowedaSTIRLING engine as a readily understandable heat engine. At its upper end, heat was applied, while the cooler reservoir was below. Based on this particular experiment, we can establish a general scheme which is applicable to every heat engine (Fig. 19.5, left). It represents the ideal limiting case of complete reversibility. A working substance moves periodically between two containers I and II at different temperatures. It mediates the transfer of heat from the warmer container I to the cooler container II.The working substance takes up the heat Qrev.1/ at the higher temperature T1. At the lower temperature T2, it gives up the smaller quantity of . be ,for II I engine 10 K has and I D heat engines 2 T TIRLING 1 T ? We surely can! We refrigeration devices ,theS II previously performed by 0) (cf. Comment C19.1). 19.6 . The cyclic process of the < W ( W at the cost of heat pump I . . I is reached. is 2 to v T re Th II D , for example the room air. As heat pumps in the I , right). Compare it to the scheme of all 1 heat pump TIRLING 0). The difference between these two heat energies T (Video 19.3) , but instead as a < .) 19.5 ( AS The is cooled, and the upper half is heated correspondingly. Q engine then proceeds in the reverse direction, i.e. compres- / 2 . II (Fig. rev Q heat engine , for example the interior of a household refrigerator, relative to TIRLING the machine and allow ita to run backwards. Then it no longer acts as need only supply the quantity of work S reversed again, can we warm In the scheme shown,a the weight lifted work to is a certain storedthe height. as temperature The the process difference comes potential to hasport, energy an end i.e. been of when when equalized by the energy trans- a thermocouple narrower sense, they have the task of heating an insulated space already been produced. By applying“pumped mechanical up” work, from heat has been This experiment leads directlypumps to the idealized scheme for all heat Can the temperature equalization between the heat baths is driven by ancylinder electric motor; in theAfter process, a the short lower time, half a of temperature the difference of is employed to perform useful work Figure 19.5 We first show this experimentally. In Fig. sion at a higher temperature and expansion at a lower temperature. heat Figure 19.6 engine used as a heat(refrigeration pump device) to the left; this needs no further explanation.Often, heat pumps areAs ‘refrigerators’, used under they the haveII the name task ofits cooling surroundings an insulated space (refrigeration device) as the reverse of a heat enginehave (we left off the index on the heats http://tiny.cc/ ( ). Converting Heat into Work. The Second Law 19 demonstration model as in Video 19.1 gigvjy “Heat pump/refrigerator” http://tiny.cc/1hgvjy The experiment is carried out with the same lecture- Video 19.3: 502 Part III 19.5 Heat Pumps (Refrigeration Devices) 503 example a living room, relative to its surroundings II, for example the free atmosphere outside; or, as “air conditioners”, of cooling a similar space relative to the outside air. Depending on their mode of use, their efficiency has to be appropriately defined. We do this again for the the ideal limiting case of complete reversibility; the required work is then minimal. For a refrigeration deviceC19.7, we have C19.7. the energy balance for the refrigeration device or Qrev.2/ (Heat taken up at the low temperature) .T2/ heat pump is given by ideal D W (Work expended) Q2 C W DQ1 . (Q2 and W are input to the Qrev.2/ D ; device, Q1 is the heat output. Q . / Q . / rev 1 rev 2 Compare Comment C19.1). or, with Q . / T rev 1 D 1 ; (18.3) Qrev.2/ T2

T2 ideal D : (19.5) T1 T2 For the heat pump,weﬁnd Part III

Q . / (Heat taken up at the high temperature) .T / D rev 1 1 ideal W (Work expended) Q . / D rev 1 Qrev.1/ Qrev.2/ or,withEq.(18.3), T1 ideal D : (19.6) T1 T2

Technical details would lead us too far afield. We will have to be content with a few remarks: 1. From Eq. (19.5), we obtain the basic rule of refrigeration technology: In order to cool an object to a low temperature T2,theworking substance should never take up heat at a temperature lower than T2. The lower T2, the less efficiency, according to Eq. (19.5). In short: “In short: You shouldn’t You shouldn’t cool champagne with liquid air. cool champagne with liquid air”. 2. Gases are not suitable as working substances for refrigeration devices and heat pumps. The volume of the gas cannot be changed rapidly and isothermally, i.e. practically without temperature changes, because the heat exchange with the surroundings is too slow; furthermore, high pressures are required for effective cooling. For this reason, one preferentially uses substances for which under the given conditions the liquid and the gas phases exist in equilibrium, e.g. CO2, ammonia, freon. Their volumes can then be readily changed isothermally by a phase change, i.e. evaporation and liquefaction. 3. A numerical example of Eq. (19.6): A residence is to be heated by using a heat pump. The heat input to the machine is to be taken from (cf. / 2 T ELVIN ELVIN 273 K. .This without 1 1 T D T : . 2 2 7 T : 14 D 293 K, , later Lord K . We need only fix one 20 to an arbitrarily-chosen D 293 2 of complete reversibility, 1 T D T and carried out by heat en- or HOMSON 1 273 of two temperatures, i.e. T T 293 ient. A more physically acceptable 293 difference ILLIAM D 2 . Then the other temperature is uniquely T 2 1 T T 1 of the two temperatures at an arbitrarily chosen value, we T the heat from outside the house. In our example, temperature difference for the ideal limiting case D one ) ) contains no material constants. We can therefore de- ideal Free and Bound Energy of Temperature ). Practically, it is identical to the temperature scale as 19.6 19.2

). pump in 14.6 14.6 independent of all such materials properties Instead of fixing method would be theergy to following: We should use the electrical en- Today, our living spaceis is very often heated convenient, by but electric ineffic radiators. That From Eq. ( we find was first recognized by W about 7 % of theThis otherwise means necessary that electric withwe an power could bring would energy around suffice! consumption 14 of kilowattincreased hours one of use heat kilowatt of into hour, our heatour rooms! pumps energy The resources. would be highly desirable to conserve gines. It is however also possible to convert heat into work numerical value, e.g. determined by the thermalchine. efficiency of In a principle, completely we reversiblea have ma- machine only in to measure order the to efficiency of obtain such the unknown temperature could choose such a value for the (Sect. 19.7 Pneumatic Motors. Thus far, we have treatedmaking the use conversion of of a heat into work based on 2 Equation ( (1824–1907). For that reason, the absolutethe temperature scale scale (that which is, contains no negative values) is named after K 19.6 The Thermodynamic Definition the air outside. Atature an of outside temperature 20 °C of is 0 °C, to an be inside maintained. temper- Then defined by good-quality gas thermometers. Sect. or the other of the two temperatures fine a measurement procedure foris the temperature with its aid which Converting Heat into Work. The Second Law 19 504 Part III 19.7 Pneumatic Motors. Free and Bound Energy 505 a temperature difference, that is in an isothermal process. A clear example of this is a pneumatic motor which is operated isothermally. We repeat from Sect. 14.11: The isothermally-operating pneumatic motor is a machine which converts the heat that it takes up from its surroundings into work. Its efficiency is ideally 100 %. The conversion is accomplished by the expansion of compressed air. Now we add a new point: The expansion increases the entropy of the compressed air. Its entropy increase is:

Qrev C19.8 S D and thus Q D T S : (18.2) C19.8. This equation in its T rev generalR form is: This entropy increase characterizes the permanent change which Qrev D T dS . the working substance (compressed air) undergoes when it performs work isothermally. We will show the effects of this permanent change in the general case, i.e. not limited to the expansion of compressed air.

The First Law: Part III 9 > Q DW C U => increase of energy output (14.6) heat input the internal > as external work ; energy leaves it completely undecided as to how the thermal energy input is to be divided between the two terms on the right. This is determined only by the Second Law. Inserting Eq. (18.2) into Eq. (14.6), with

Q D Qrev ; we obtain W DT S C U (19.7) or, for an isothermal process, i.e. a process occurring at constant temperature, Wisoth D .U T S/: (19.8) The brackets contain only quantities of state. Therefore, their content is also a quantity of state (a state function)3. It is called the free energy F; thus, F D U T S : (19.9) The free energy is smaller than the internal energy; their difference

U F D T S (19.10)

3 Only differences,suchasU, S, F, etc. are measurable. If one determines numerical values for U, S, F, etc., they always hold only for a particular reference temperature, which must be quoted, as for example in Table 18.1. .In (19.13) (19.11) (19.12) (19.14) which can be isoth W : : U S : C T D T F d . . rev 4 ), we obtain the equation (named T D isoth T Q d D ), only reversible processes were presup- W of the ideal gas remains constant on d 19.7 D isoth rev isoth T 18.2 U W Q W d D )or( isoth T ), it follows that W d is not wasted, but rather it is determined in bound energy d isoth S 19.8 or on the maximum work W 18.2 F T ) for the maximum work which can be obtained from ) by applying it to a very small temperature difference . Then we obtain T d of the Free Energy 19.2 D compressed-air cylinder as an energy storage medium 2 . T ELMHOLTZ 5 1 Maximum work, because in Eq. ( The internal energy of a material can be compared with the assets of a company; performed by an isothermalfrom process. Eq. ( We can obtain this inﬂuence T It plays an important role in physical chemistry. for H Inserting this into Eq. ( which thus represents the maximuman work isothermal that and can reversible be process obtained from From this, with Eq. ( such a way thatwork it is not available for the performance of additional The bound energy an isothermal process: For many applications, one requires theon inﬂuence the of free the energy temperature the free energy with itsassets “liquid” (e.g. assets, plants and the and bound property). energy with its “non-liquid” posed. 19.8 Examples of Applications 1. A 4 5 is also referred to as the the particular case of compressedple, air, since the the situation internal is energy especially sim- Then Converting Heat into Work. The Second Law 19 506 Part III 19.8 Examples of Applications of the Free Energy 507 isothermal expansion and therefore, U D 0. We thus obtain from Eqns. (19.11)and(19.8)

Wisoth D F DT S : (19.15)

Here, we have from Eq. (18.15) p S D nR ln 1 : (19.16) p2

Example A steel cylinder of mass 64 kg and volume 42 liters at a pressure of p1 D 190 105 Pa (D 190 bar) contains 330 moles of compressed air at room temperature. On isothermal expansion to p2 D 1 bar, its free energy is reduced by

Ws 190 F D293 K 330 mol 8:31 ln mol K 1 (see also Exercise 15.1). The logarithmic factor has the value ln 190 D 5:25. Then F D4:2 106 Ws1:2kWh: Part III An electrical storage battery with about the same mass (70 kg) reduces its free energy on complete discharge by around 2 kWhC19.9. C19.9. For comparison, we mention that the free en- 2. Entropy elasticity or rubber elasticity. For the majority of solid ergy of fuels (butter, coal) materials, e.g. the metals, the elastic forces which accompany defor- amounts to about 10 kWh/kg. mation arise from the change in their internal energies. – The situation is quite different for ideal gases. The setup sketched schematically in the upper part of Fig. 19.7 permits the measurement of the elastic force Fel which is produced by a column of air conﬁned in a container. If the sliding carriage is moved isothermally to the left, the length of the air column will be changed by l < 0. The force Fel remains practically constant. The air is compressed and, from Eq. (19.15), work is performed on it:

Fell D F DT S

For the elastic force, we thus obtain

S F D T : (19.17) el l

The elastic force Fel is therefore proportional to the temperature. It arises simply because the entropy of the air is reduced on isothermal compression. As a result, we speak of its entropy elasticity. The entropy of the air, according to Sect. 18.5, Point 4., is given by

S T2 p2 D Cp ln R ln : (18.15) n T1 p1

If the air column at the top of Fig. 19.7 is shortened adiabatically by l, the entropy of the gas remains constant, but its two contributions , el 2 on F p , to . Sup- top 1 , it will l p , the elastic , bottom). at a constant decreases to a good ap- 19.7 el T bottom F is a galvanometer for temperature G of the band. But now the entropy . This can be readily understood: ) is again applicable. We ﬁnd that materials, one can also observe entropy elasticity, ) both change: On adiabatic compression from entropy The demonstration of entropy elasticity with a compressed col- extension 19.17 solid of a rubber band is usually proportional to 18.15 dependence of the melting temperature on the pressure l the gas is increased by adiabatic compression. Among the ﬁrst term increases at the cost of the second. The temperature of length proximation. When the band isbecome adiabatically warmer. stretched by Fromenergy these of rubber-like two materials facts, ission nearly it and independent follows volume. of Therefore, that their here exten- only the also by internal the the elastic force is determined umn of air and ameasurements; stretched the rubber force band meter ( is shown only schematically. At the e.g. in rubberEquation and ( the rubber-like polymers (Fig. (monomers) are joined like themolecules. links In of a a disordered chainprobable pile, into arrangement they long form (high threadlike a entropy).the matted wad When threadlike as the the molecules band most some are is extent; their stretched, pulled arrangement becomesprobable apart more (lower ordered entropy). and and As thus become achains less model, parallel by we macroscopic to can chains simulatemagnetic. the of polymer If ca. we 10 cm letsimulate them length, lie the whose on links thermal a are glass motions,wad (with plate they high and entropy), will vibrate like the the form plate polymeric a molecular to chains. When disordered, a matted rubber bandmains is constant, adiabatically it relaxed, will cool. so that its3. entropy The re- isothermal Rubber-like materials belongstances. to In the these materials, group identical of and relatively high-polymer small sub- molecules Figure 19.7 acts to the right,force against acts the to motion the of left). the carriage; at the in Eq. ( pose that a substance at constant temperature and pressure changes Converting Heat into Work. The Second Law 19 508 Part III 19.9 The Human Body as an Isothermal Engine 509 its volume by V on melting. Differentiation of the molar work of expansion W DpVm with respect to T yields

dW dp D V : (19.18) dT dT m Inserting this equation into Eq. (19.13), we obtain from the Second Law the so-called CLAUSIUS-CLAPEYRON equation:

dT T D Vm : (19.19) dp Lf

It shows the pressure dependence of the melting temperature. (Lf is the molar latent heat of meltingC19.10; cf. Comment C13.10). – C19.10. A similar equation For some substances, the melting temperature increases with increas- applies to the vapor pressure. ing pressure. Examples: Wax and CO2 (Fig. 15.10). In their case, The derivative dT/dp then = > > dT dp 0. Then, according to Eq. (19.19), V 0 must also hold, holds along the vapor pres- Part III i.e. these substances must expand on melting. sure curve, Vm refers to the difference in molar vol- For other substances, the melting temperature decreases with increas- umes of the gas (vapor) and ing pressure. Example: Water (Fig. 15.11). In this case, dT=dp < 0. the liquid, and the constant Then, from Eq. (19.19), V < 0 must hold, i.e. these substances Lv is the molar latent heat of must contract on melting. Both kinds of behavior are observed ex- vaporization. perimentally.

19.9 The Human Body as an Isothermal Engine

Energy is input to the body by the oxidation of our foodstuffs. For example, we ﬁnd for 9 Butter ::: 9:1 > > Oatmeal ::: 4:2 => kWh Rice ::: 3:9 > kg Bread ::: 2:3 > ;> Potatoes ::: 0:9

At rest, maintaining the life functions of an adult human requires C19.11. For electrical energy, a power of about 80 W. That is, a human body requires an energy this corresponds today in input of around 2 kWh per day. When the person is performing me- Germany to a consumer cost chanical work, the energy input must be increased up to 3 to 4 kWh of around 400 Euros or 440 $. per day; for heavy workers, this can be as much as 6 kWh per day. In the U.S., the average con- On the average, a human requires an energy input of only about sumer cost would be around 1300 kWh per year (commercial value around 3 $!)C19.11. 130 Euros or 145 $. . 2 liters Work per- C)! So only ), the internal ı bout a half hour ) 19.2 19.5 465 K (192 5 °C, a heating boiler in D will be consumed in the ideal 1 el T W The online version of this chapter (doi: . It requires an oxidation step, has a low . It can be replaced in a 2 293 K (20 °C), from Eq. ( seconds D 2 T production of muscular work is a possible explanation. ). 5.2 At an outside temperature of per minute). Aroundcarrying 1/5 of out this mechanical power,high-level work i.e. activity, the (in about ‘muscular 300 contrast storage W,the is battery’ to order has available of isotonic an for 100 energy work). kilowatt- reserve of For brief efficiency and produces quite aContinuous lot of athletic heat. activity, isotonicallyinput of or about 1.4 in kW (corresponding motion, to requires an oxygen a consumption of power 4 In a careful observation,the work one performed has bythis to muscles. process consider During is two one similarenergy of different to is them, converted processes discharging to force a mechanical in work. is storage Thereach generated; battery: efficiency 90 of %. this Stored Then process a can chemical secondrecharging process follows; the figuratively speaking, it battery. is like proceed This only in second the presence process, of O in contrast to the first, can A small fraction, whichbe decreases converted to strongly mechanical work. withcan increasing By expend using demand, this a can energy(Sect. mechanical reserve, a power human of several kilowatts for a few seconds after complete exhaustion, accompanied by an uptake of 15 liters of O isothermal formed reversibly is an ideal,not but necessarily even worth this striving ideal for. is, like some others, In that case, aroundis 60 converted to to heat! 80 % Work, forates of example heat. the climbing a (In chemical mountain, these energy gener- figures,requirement of of the our 2 basal kWh food metabolism per of day the at body, complete i.e. rest, its is not included.) an case of completely reversible operationheat of input the to the pump, boiler if amounts the to required 1 kJ? (Sect. body temperature would have to be Exercise 19.1 Electronic supplementary material 10.1007/978-3-319-40046-4_19) contains supplementary material, whichable is to avail- authorized users. The work performed by ourwork muscles is is by just no means asperformed reversible. irreversible reversibly Their as is that much of too technical ponderous heat and engines. slow. Work The thermal efficiency ofthrough training, our it muscles can ismuscles be cannot increased in up possibly general to beent around ca. working temperature 37 20 as %; %. of heat Therefore, engines. the At an ambi- a house ispump. to How much be electrical energy maintained at a temperature of 40 °C by a heat Converting Heat into Work. The Second Law 19 “Work performed reversibly is an ideal, but even this ideal is, like some others, not necessarily worth striving for”. 510 Part III Table of Physical Constants

Some important physical constants (CODATA values from Dec. 2014) 11 2 2 Gravitational constant D 6:6674 10 Nm /kg 12 Electric field constant "0 D 8:854 188 10 As/Vm 7 Magnetic field constant 0 D 12:566 370 10 Vs/Am 1=2 8 Velocity of light in vacuum c D ."0 0/ D 2:997 925 10 m/s 1=2 Wave resistance of vacuum Z D . 0="0/ D 376:73 Ohm Relative atomic mass of the proton .A/p D 1.007 277 u Relative atomic mass of the neutron .A/n D 1.008 665 u 27 Proton mass mp D 1:672 621 10 kg 8 Rest energy of the proton .Wp/0 D 9:382 720 10 eV 31 Rest mass of the electron m0 D 9:101 383 10 kg 5 Rest energy of the electron .We/0 D 5:109 99 10 eV Ratio of proton mass/electron mass mp=m0 D 1836.152 Elementary electric charge e D 1:602 177 1019 As 11 Specific charge of the electron e=m0 D 1:760 366 10 As/kg Boltzmann’s constant k D 1:380 648 1023 Ws/K D 8:617 325 105 eV/K Planck’s constant h D 6:626 070 1034 Ws2 D 4:135 667 1015 eV s 2 2 10 Smallest orbital radius of the H atom aH D "0h =m0e D 0:529 177 10 m Bohr magneton B D 0he=4m0 D 1:165 407 1029 Vsm 2 15 Classical electron radius rel D 0e =4m0 D 2:820 419 10 m 4 = "2 3 : 15 1 Rydberg frequency Ry D e m0 8 0h D 3 289 842 10 s 4 = "2 3 : 1 Rydberg constant Ry D e m0 8 0h c D 10 973 731 6m 12 Compton wavelength C D h=m0c D 2:426 310 10 m 2 Sommerfeld’s fine structure constant ˛ D e =2"0hc D 1=137:036 Electron’s velocity u in the smallest H orbit D Velocity of light c

511 Solutions to the Exercises

I. Mechanics

1.1. After 100 years, each day will be 1:5 103 s longer than today; then after one year, it will be 1:5 105 s longer, and tomorrow, it will be 1:5 105=365:25 s, i.e. 4:106 108 s longer. Then in 100 years, the clocks will be ahead by 4:106 108 s .1 C 2 C 3 C :::C 36 525/ D 27:4s.(The sum can be solved analytically and its solution can be found in many mathematics textbooks).

1.2. T D 0:05 s

2.1. Downriver, at an angle of 53.13ı to the bank.

2.2. s D 34:3m

2.3. t D 17:35 s; h D 293 m

2.4. a D 2:475 m/s2; F D 49:5N

2.5. a) t D 1s;b)t D .=4/ s

2 2 2.6. ar D 2:10 10 m/s

ı 3.1. F D 63:1N;Ftot D 42:75 N to the east (˛ D 90 )

ı 3.2. a) F D 0:431 FG=.0:959 sin C 0:413 cos /;b) D 66:7

3.3. v D 86:9cm/s;Ekin D 0:196 Nm; FF D 0:189 N

2 4.1. u0 D 0:628 m/s; a0 D 3:95 m/s

: 2 =. / 4.2. TM D 3 16 s; mM D grM 10G (G D gravitational constant)

2 2 4.3. The new angular velocity is !1 D l !=.l l1/

4.4. z D 5086 km

4.5. With a tangential velocity of 1:5 105 m/s, the model earth could circle the sun on a stable orbit. Its “year” would be exactly as long as a real year! Solutions to the Exercises 513

5.1. W D A%g.H h/2=4

5.2. W D 0:136 kWh; Power P D 8:16 kW p 2 p 5.3. h2 D h1 C . 2g.h h1/ p=m/ =2gI v D 2gh2

5.4. a) v D 10 m/s; b) v D 5m/s

5.5. a) From the conservation of energy and momentum, we find: v2 D 2.m=M/v1=.1 C m=M/ 2.m=M/v1 and v D 2v1=.1 C m=M/. b) After the first impact, the balls swing outwards until their kinetic energy is completely converted to potential energy. Then they swing back again, so that the first impact is played out again in reverse. c) When the large ball collides with the small one, the latter flies away with the velocity 2v2 (this can be seen especially clearly in the frame of reference of the large ball). The large ball loses a velocity of 2v2.1 C M=m/, that is less than 1 %. This result is verified by comparison of the two amplitudes.

5.6. h D m2v2=.2g.M C m/2/

5.7. v D 2:97 m/s; 13.36 J still remains from the original 1350 J.

5.8. This follows from the conservation laws for momentum and energy, together with the Pythagorean Theorem.

5.9. Mp D 1:718M0 or 63 % of the total initial mass.

5.10. dM=dt D Mt.ao C g/=ur (g D acceleration of gravity).

6.1. a) m2 D 1:875 kg; b) 7/18 of the length of the beam, as measured from m2; F D 441 N.

6.2. One can either use the equation of motion to compute the acceleration of the cylinder axles, pdv=dt D .2=3/g sin ˛, or else use the law of energy conservation to ﬁnd the ﬁnal velocity, v D 4gh=3. Both expressions contain neither the mass (or the density) nor the radii of the cylinders. The two cylinders thus arrive at the bottom simultaneously.

6.3. W D 976 J

6.4. l D 2a=3 p 6.5. a) At one of the two points at a distance of a=.2 3/ from the center of gravity; b) at one of the two points at a distance of a=6 from the center of gravity.

2 2 6.6. 0 D 0:05 kg m ; S D 0:03 kg m

6.7. a) D 0:00579 kg m2 (make use of Eq. (6.12)); b) d D 0:28 mm (here, the torsion coefﬁcient changes by 2mgd).

6.8. ' D 2mr2=. C mr2/

6.9. Under the assumption that the end of the pencil does not slide immediately off the edge of the table, the pencil will begin to rotate around its point with the angular momentum ! (where 514 Solutions to the Exercises

. = / 2 v = D 1 3 ml ),p which is equal to the orbital angular momentum m l 2 at this point. It follows that ! D .3=2/ 2gh=l,or D 5:3 rotations per second. This result could be checked simply by using a video camera.

6.10. See Eq. (6.16)

6.11. The gyroscope is also rotating in the moving frame of reference, but without precession. For each of its volume elements, the rotation can be decomposed into a vertical velocity component parallel to the vector of the angular velocity !P and a horizontal component. The latter leads to a CORIOLIS force on the volume element, which causes a torque to act on the axle of the gyroscope. The sum of all these torques leads to an overall torque which is parallel to M but opposite in direction. Its magnitude would have to be found from a computation; here, we content ourselves with the statement that it would have the same magnitude as M. In the rotating frame of reference, the sum of all the torques along the axle of the gyroscope is thus zero.

7.1. a) FC D 2mu! D 50 N. b) While the bullet is moving within the barrel of the pistol, it moves along a circular orbit with a continually increasing radius. The distance s along this orbit is given by s D R!t and R D ut. It then follows that s D u!t2, and thus the acceleration is d2s=dt2 D 2u! and the force is FH D 2mu! D 50 N, so that it has the same magnitude as FC, but the opposite direction.

7.2. For the observer in the rotating frame of reference of the swivel chair, the pendulum is moving on a circular orbit with the velocity u D !R. This requires that a radial force m!2R directed towards 2 the center of rotation be acting. This force is composed of the centrifugal force FZ D m! R,which 2 is directed outwards, and the CORIOLIS force FC D 2mu! D 2m! R which is directed towards the center.

3 2 4 7.3. aC D 2!v sin ' D 1:9 10 m/s 10 g (g D acceleration of gravity). The CORIOLIS force is thus negligible compared to the inertial forces which are due for example to the imperfect ﬂatness and smoothness of the rails. p 7.4. T D 2 R=g,i.e.theSCHULER period!

7.5. s D 1:5 109 m 1nm

7.6. During seven complete swings, that is after 48 s, the image of the pendulum wire has shifted by seven wire diameters, or by 2.8 mm on a circle of radius A D 1 m. From this, at the latitude of 5 1 ı Göttingen, we ﬁnd an angular velocity of !G D 5:83 10 s . Dividing by the factor sin 51:5 D 5 1 5 1 0:78 then yields !E D 7:5 10 s (the precise value is 7:3 10 s ).

8.1. E D 1:25 105 N/mm2

8.2. From L and H, with a certain amount of trigonometry, we ﬁnd for the radius of curvature r D 2:56 m. s is measured to be 0.26 m. Then we can compute the torque which deforms the rod: M D 1kg g 0:26 m D 2:6 N m (note that the torque which is due to the second kg weight serves only to keep the rod from being accelerated). The geometrical moment follows from Eq. (8.23): J D 64 1012 m4. Then, ﬁnally, we obtain E D 10:3 104 N=mm2, in good agreement with the value given in Table 8.1. Solutions to the Exercises 515

8.3. The light pointer (broadened by diffraction) is displaced by nearly ds.Fromthis,weﬁnd˛ D 7 104. With the geometrical moment from Eq. (8.26), we obtain G D 8:0 104 N=mm2,in agreement with the value given in Table 8.1.

8.4. s D 11:5m

9.1. F2 D 34:5N

9.2. A2 D 5A1

3 9.3. F D ..4=3/d =8/g%water (g D acceleration of gravity)

9.4. F D 12 700 N

9.5. F D 104 N

9.6. m D 7:35 g (weight 0.072 N)

9.7. The heat of evaporation (vaporization) for one molecule is 7:6 1020 W s. In each cm2 of water surface, there are 1:03 1015 molecules, as can be found from the number density of the water molecules. Then from Table 9.1, we ﬁnd the surface work for one molecule to be 7:2 1021 Ws. Around 10 % of the heat of vaporization is thus required to bring the molecules to the surface where they can evaporate.

9.8. D 4:8 s (in order to keep the effort required for counting to a minimum, it was begun after 6 s following the end of the pouring).

3 3 9.9. The constants of BOYLE’slaware,at0°C:0.8m bar/kg; and at 200 °C: 1.4 m bar/kg; po 2 pc D 1:6 10 Pa (D 1:6mbar)

9.10. .p2 p1/A D gAh.%air %hydrogen/; this is the same as for the buoyant force.

9.11. The pressure in the methane is greater than in the surrounding air, by p D 0:57 Pa (this corresponds to the gravity pressure (Sect. 9.4) of a column of water 0.057 mm “high”!).

2 2.% % /= . 2 2 / % . 2 % 2 % /=. 2 2 / 10.1. D 2gR1R2 2 1 9 R1u2 R2u1 I D R1u2 1 R2u1 2 R1u2 R2u1

10.2. From energy conservation, Eq. (10.12), and neglecting the viscosity of the water, we ﬁnd p p p r D .1= 2gh/ cm3=s, as long as R r, so that we can consider the water in the cylinder to be practically at rest.

II. Vibrations and Waves

11.1. These beats are produced by the harmonics 22 and 31.

11.2. a) d1 D 0:326 m; b) d2 D 0:163 m (the reduction of the resonance frequency at a ﬁxed length is used for example in constructing organ pipes; see E. Skudrzyk, “The Foundations of Acoustics” Springer (Heidelberg, New York) 1971). 516 Solutions to the Exercises

11.3. D 2:4cm; D 14:17 kHz

11.4. cl D 3840 m/s

1 2 3 11.5. ı D D 50 s; e D 0:42 Hz; D 4:8 10 ; K D 1:05; H D 6:4 10 Hz

11.6. For the symmetrical normal mode oscillation, we obtain the frequency s D 0:900 Hz, and for die antisymmetrical normal mode, as D 0:997 Hz. For the beat frequency, B D 0:100 Hz. The difference D s as D 0:097 Hz is found, in agreement with the beat frequency determined experimentally. p p p p = = = = = = . 0/= = 11.7. pa) 0 D 2D m p2 , 1 D D m 2 , 2 D 3D m 2 ;b) 0 D D C D m 2 , 0 1 D D=m=2, 2 D .D C 2D /=m=2. Note that in both cases, one of the eigenfrequencies is always larger and the other is always smaller than the frequency of a single pendulum as deﬁned in the problem. The difference between the two eigenfrequencies becomes smaller as the coupling becomes weaker.

12.1. 0 D 531 Hz

12.2. Maxima appear along the axis of symmetry and at a spacing of ˙my=a from it, where m D 1; 2; 3;::::

12.3. On the diagonals of the square, since in these directions, the openings are at the greatest distances from each other.

12.4. D D a=b

12.5. cl D 5000 m/s

12.6. d D 108 m

12.7. u D 60 km/h

ı 12.8. sin ˛max D 1=n D 0:78;˛max D 51

12.9. m D 1: D 9:9ı (grazing angle), ˇ D 80:1ı; m D 2: D 20ı, ˇ D 70ı; m D 3: D 31ı, ˇ D 59ı; m D 4: D 43:4ı, ˇ D 46:4ı.

12.10. c D c.1 C .dn=d/=n/

12.11. D 4:25 m to 0.425 m

III. Thermodynamics

13.1. The clock advances too slowly by 104, thus losing about one minute per week.

14.1. Applying Eq. (14.19), we ﬁnd p D 166 105 Pa (D 166 bar).

5 14.2. Addition of the partial pressures yields ptot D 9:4 10 Pa. Solutions to the Exercises 517

14.3. With V1 D initial volume, V2 D ﬁnal volume and the temperatures T1 D 291 K (room temperature) and T2 D 773 K, we obtain from Eq. (14.48): V1=V2 D 15:1. The volume must therefore be reduced by a factor of about 1/15.

4 3 15.1. a) Vm D 1:217 10 m /mol D 3:5Vm;solid, i.e. the density of the gas is 28 % of the density in the solid phase. b) From the VA N -DER-WAALS equation, with this value of Vm, we calculate a value of RT which is only around 1 % smaller than the true value. An exact solution of the VA N - 4 3 DER-WAALS equation yields Vm D 1:23 10 m /mol, that is a value which is about 1 % larger. The ideal-gas equation thus holds rather precisely for nitrogen, even at a packing density at which the intermolecular distance is only 50 % larger than in the solid state.

18.1. Make use of Eq. (18.15) and take the standard conditions, T1 D 273 K and p1 D 1013 hPa. Then you obtain the molar entropies as shown in Fig. 18.8.

19.1. Wel D 0:143 kJ Index

A Auto control, 283 Boundary energy, 188 Acceleration, 20, 35 Auto-controlling Boundary layer, 211, 212 definition, 21 of a tuning fork, 249 in the RUBENS’ flame tube, 267 measurement, 23 AVOGADRO constant, 370 thickness, 213 of a sinusoidal oscillation, 50 Avoidance flow, 226, 238 Bow wave, 307 Acceleration of gravity = Apparent Axis BOYLE’s law, 194 gravity, 25 free, 108 BRAGG angle, 321, 335 Acceleration of gravity = Free-fall Axis of incidence, 302 Breakers, 192 acceleration, 35 Axis of rotation, 91 Breathing sphere, 351 Acoustic replica method, 323 BROWN,R.,175 Acoustic wave resistance, 350 B Brownian motion, 175, 445, 479 actio D reactio, 32, 59, 76, 141 Balance, 43, 104 in gases, 193, 204 Adiabatic change of state, 400, 489 Ballistic curve, 65 Bulk modulus, 156 Adiabatic curve, 401 Ballistic pendulum, 81 Buoyancy, 86, 184 Adiabatic expansion, 487 Bar exercises, 108 in gases, 205 Adiabatic exponent, 390, 402 Bar, unit of pressure, 178 in liquids, 183 measurement, 405 Barometric equation, 445 Air friction, 83 Barometric pressure formula, 203, C Air pressure, 199 204, 446, 449 Calorimeter, 389 Air resistance, 26, 83 Base quantities, 29 Capillary depression, 190 Aircraft Base units, 29 Capillary elevation, 190 lift, 241 Basilar membrane, 363 Carbon dioxide propulsion, 86 Bats, 325 isotherms, 410, 411 thrust, 244 Beam balance, 43, 104 phase diagram, 422 Airfoil, 238 Beam of sound, 303 CARNOT cycle, 497 Amount of substance, 29, 369 Beat frequency, 253 CARNOT,SADI, 496 Anemometer, 236 Beats, 14, 253, 284, 342 Cartesian diver, 185 Angle measurement, 9 BEHN’s tube, 206 CAVENDISH,H.,60 Angle of incidence, 302 Bending vibrations, 270 Cavitation, 190, 349 Angle, unit of, 10 BERNOULLI’s equation, 223, 327 Celestial mechanics, 62 Angular acceleration, 97 BEUERMANN,G.,254 CELSIUS temperature scale, 373, Angular momentum, 97, 104, 115, Bicycle, 84 392 117 Bicycle riding, 119 CELSIUS,A.,371 as vector, 104 Bicycle, riding ‘no hands’, 119 Center of gravity, 77, 93 Angular momentum conservation, Bimetallic strip, 371 Centrifugal force, 134, 177, 206 105 Blood circulation, 216 Centripetal force = radial force, 44 Angular velocity, 22, 27, 96, 97, 277 Blood reservoir, 216 Chain ‘syphon’, 200 as a vector, 96 Boiling, 394, 425 Changes of state Apparent gravity, 25 Boiling temperature, 421 by irreversible processes, 478 Apparent viscosity, 217 BOLTZMANN distribution, 449 of ideal gases, 398 ARCHIMEDES’ principle, 184 BOLTZMANN,LUDWIG, 483 reversible, 487 Area law, 55 BOLTZMANN’s constant, 393, 483 Chemical potential, 384 Atmosphere (atm), 178 determination by J. PERRIN, 445 Chimney, 206 Atmosphere = “ocean of air”, 198, Boom, supersonic, 354 CHLADNI sound patterns, 272 445 Boomerang, 121 Circular frequency, 50 Atmosphere of the earth, 198 Bound energy, 506 Circular motion, 44 Index 519

Circular orbits, 26 DEWAR vessel, 376 Elastic aftereffect, 165 Circulation, 229 Diabolo top, 116 Elastic aftereffects, 165 CLAUSIUS,RUDOLF, 496 DIESEL engine, 501 Elastic coefficient, 94 CLAUSIUS-CLAPEYRON equation, Difference frequency, 253, 358 Elastic deformation, 153, 165, 264 509 Difference oscillation, 255, 280 Elastic oscillator, 52 Closed system, 478, 487 Diffraction, 299 Elementary oscillator, 259 Cloud chamber, 425 by a slit, 311, 331 Elementary waves, 301, 308, 312 Clouds, sink velocity, 216 FRAUNHOFER, 311 Elevator feeling, 43 Cold working, 167 FRESNEL, 311 Elliptical orbit, 56 Collision Diffraction grating, 319 Elongation, 153, 154 elastic, 79 Diffusion, 176, 413, 432, 453, 469, Energy, 71 inelastic, 80 471 bound, 506 non-central, 82 Diffusion boundary as surface conservation of, 71, 74 Combination oscillations, 280 for gases, 201 free, 505 Compressibility, 156, 177, 193 of liquids, 413 applications, 506 isothermal, 398 Diffusion constant, 455, 456 internal, 379, 384, 396 of water, 181 Dilation, cubic, 156 kinetic, 73, 98, 429 Compression, 153 Dipole, 228 of rotation, 97, 98, 100 Conical waves, 307, 354 Directional characteristic, 331 potential, 72 Conservation of angular Directional hearing:, 359 Energy conservation, 385 momentum, 115 Directional receiver, 356 Energy release, stimulated, 278 Consonants, 357 Discus, 117, 121 Enthalpy, 387, 416, 489 Convection, 460 Dispersion, 301, 341, 342, 347 Enthalpy of vaporization, 378, 387 Cooling bottle, 378 Distance, 4 Entropy, 479, 481, 489 Coriolis force, 138, 149, 435 Distance measurement and probability, 483 Countercurrent heat exchanger, 420 direct, 4 calculations, 483 Countercurrent process, 419 Doorbell, 249 in a molecular picture, 482 Coupled pendulums, 283 DOPPLER effect, 293, 325 Entropy changes of ideal gases, 486 Critical temperature, 386, 414 Drag, 212 Entropy elasticity, 507 Curl, 229 Droplets, 191, 192 Entropy increase, 481, 482 Current impulse, 75 Dry ice, 423 by melting, 484 Cutting, 171 DULONG-PETIT on evaporation, 485 Cyclic process, CARNOT, 497 rule, 439 on heating, 484 Dynamic pressure = Stagnation Entropy production, 481 D pressure, 223 Envelope, 410 DALTON’s law, 394 Dynamics, 31 Equation of motion Damping ratio, 273 for rotations, 97–99, 117 Dancing balls, 75 E Equation of state dB(A), 361 Ear, 362 caloric, 395 de LAVA L nozzle, 492 detection threshold, 360 for ideal gases, 391, 392 Dead water, 340 frequency range, 356 for real gases, 414 DEBYE temperature, 389 model of the cochlea, 363 thermal, 384, 391 Decibel (dB), 351 spectral sensitivity distribution, VA N D ER WAALS’, 414, 415 Decompression, 396 361 Equilibrium Deep drilling, 165 Earth indifferent, 47 Deformation as a top, 123 stable, 47 elastic, 153, 165 as an accelerated frame of Equipartition principle, 437, 438 of solid bodies, 31, 96 reference, 146, 148 Escape velocity, 64 of solids, 153 atmosphere, 198 Expansion, 479, 480 plastic, 167 Earth’s rotation Expansion coefficient, 370 Deformation ellipsoid, 158 detection by HAGEN, 150 Degree Celsius (°C), 371, 392 verification by FOUCAULT, 149 F Degrees of freedom, 123, 437 Eddy-current damping, 274 Feedback, 247, 250, 283 Delay in boiling, 425 Efficiency, 85 FICK’s law Density, 38 of a heat engine, 496, 501 first, 455, 469 of air, 193 of a heat pump, 503, 504 second, 459 Density = Mass density, 38 of a refrigeration device, 503 Field-line patterns Density fluctuations, statistical, 414, of muscles, 510 in electrodynamics, 228 448 Elastic, 74 Figure axis, 112, 113 520 Index

Fire pump, 405 Free fall, 23, 38 Gyrocompass, 140, 150 First Law, 385, 496, 505 Frequency, 13, 19, 27, 28 Fixed temperature points, 372 amplitude-dependent, 74 H Flame, sensitive, 218 Frequency meter, 280 HAGEN-POISEUILLE law, 216 Floatation, 184 Frequency spectrum, 358 Hair, diameter of, 6 Flow FRESNEL diffraction, 311 Hardening, 155, 167 laminar, 214 FRESNEL zones, 315, 331 Harmonic oscillator, 50, 247 turbulent, 217 Friction Harmonics, 261 Flow apparatus, 220 external, 32, 169 Heat, 375 Flow around a disk, 225 internal, 211, 466, 470 as random motion, 429 Flow field, 220, 226 sliding, 169 conversion into work, 495 irrotational, 227 Frictional resistance, 212, 214, 216, latent, 377 Flow limit, 155 236 Heat capacity, 376 Flow resistance, 234 Frictional work, 219 molar, 376, 389 Flow resistance around objects, 235 Frictionless fluid motion, 220 of gases, 391, 438 Flow through a nozzle, 490, 491 Fundamental equation of some solids, 389 Flow velocity in a nozzle, 491 of kinetic theory of gases, 196 temperature dependence, 439 Flue pipe, 268, 269, 323 of motion, 37 specific, 376, 388 Fluid, 211 Fundamental vibration, 261 Heat conduction, 460, 481 Focal length, 304 genuine, 460 Focal point, 303, 311, 326 G in gases, 465, 470 Fog formation, 411, 412, 425, 487 Gas non-stationary, 464 Force flow out of a nozzle, 491 stationary, 463 as vector, 32 ideal, 194 Heat conductivity conservative, 73 in an accelerated frame, 206 temperature dependence, 463 hydrodynamic, 224 real, 194 Heat current, 463 measurement of, 35 Gas compressors, 406 Heat engine, 495 unit of, 35, 37 Gas constant, universal, 393 efficiency, 496, 501 Force law Gas law, ideal, 391 technical, 500 linear, 46, 52 Gas liquefaction, 420 Heat exchanger nonlinear, 47 Gas molecules, velocity, 198, 430, countercurrent, 419, 420, 461 Forced oscillations, 273 435 Heat of condensation, 377 amplitude, 275 Gas syphon, 201 Heat of crystallization, 378 phase difference between Gas thermometer, 372 Heat of melting, 378 excitation and resonator, Gases Heat of transition, 379 275 heat conduction, 465, 470 Heat of vaporization, 377, 378, 450, Forces ideal, 195, 392, 395 485 aerodynamic, 266 internal friction, 466, 470 of water, 386 hydrodynamic, 349 model experiment, 196, 204, 432 Heat pump, 501, 502 Formant range, 259, 357 real, 409 efficiency, 503, 504 FOUCAULT’s pendulum experiment, transport processes, 465 Heat transport, 375, 460 149 Gauge block, 5 through radiation, 375, 468 FOURIER analysis, 255, 259 GAY-LUSSAC’s throttle experiment, Heating pad, 379 FOURIER integral, 258 395 Hectopascal (hPa), 178 Frame of reference, 17, 33 Geometrical moment of inertia, 162, HELE-SHAW flow, 215 accelerated, 129, 141, 206 163 Helium, superfluid, 191 with pure path acceleration, 129, GIBBS’ phase rule, 423 HELMHOLTZ resonator, 273 130 Glass teardrops, 182 HELMHOLTZ’s equation, 506 with pure radial acceleration, Gliding flight, 242 Herpolhode cone, 116 130 Gravitation, 34, 35 Hertz D 1s1, 13, 247 Frame of reference with radial Gravitational constant, 60, 62 Hold point, 379 acceleration, 133 Gravity pendulum, 53 Hollow vortex, 231 FRAUNHOFER diffraction, 311 in a rotating frame of reference, HOOKE’s law, 155, 322 Free axes, 108 134, 138 Hour glass, 15 of humans and animals, 111 Gravity pressure, 182, 183 H-S diagram = MOLLIER diagram, Free balloon, 205 in the air, 203 489 Free energy, 505 in water, 202 HUYGHENS’ principle, 308 applications, 506 Group velocity, 341, 342 HUYGHENS,CHR., 26 GUERICKE,OTTO VON, 199 Hydraulic press, 181 Index 521

Hydrodynamic attraction, 225 temperature scale, 373, 504 tensile strength, 186 Hydrofoil, 238 KEMPELEN,W. V., 358 LISSAJOUS orbits, 57 Hysteresis, 166 KEPLER,JOH., 62 Logarithmic decrement, 273 Hysteresis curve, 166 KEPLER’s elliptical orbits, 58, 63 Longitudinal vibrations, 260, 263, Hysteresis loop, 166 KEPLER’s laws, 62 270 Kilogram, 34 of linear solid bodies, 263 I Kilopond, 38 Longitudinal waves, 321 Ideal gas, 194, 195, 395 Kilowatt (kW), 71 in air, 323 changes of state, 398 Kilowatt hour, 68 Loudness, 360 equation of state, 392 Kinematics, 17 Loudspeaker, 354 molar heat capacity, 398 Kinetic theory of gases, 196 Lubrication, 171 Image formation, 303 Kite, 242 Lunar motion, 27 Image point, 303, 311 KNUDSEN effect, 471 Impact = impulse, 75 KUNDT’s dust figures, 266, 341, M Impulse, 75 405 MACH’s angle, 309, 354 Index of refraction, 302, 330 Magdeburg hemispheres, 199 Indices, 296, 318 L Magnitude of a quantity, 20 Individual unit of mass, 370 Larynx, 357 MAGNUS effect, 240 Inertia, 34 Latent heat, 377 Manometer Inertial forces, 129 of melting, 484 liquid, 184 Initial vortex, 238 Latent heat of vaporization, 378, Mass, 34 Instantaneous axis, 94 444 molar, 370, 376, 393 Interference, 295 Lattice constant, 13, 319, 334 Mass density, 38 Interferometer, 334 Lattice plane, 329, 334 Materials, solid, 508 Internal combustion engine, 500 Law of areas, 63, 108 MAX WIEN’s experiment, 285 Internal energy, 379, 384, 396 Law of gravity, 60 Maxima, 261 Internal friction Law of reflection, 302 MAXWELL-BOLTZMANN velocity in liquids, 211 Law of refraction, 302 distribution, 450 independence of pressure, 466 Law of thermodynamics MAXWELL’s velocity distribution, internal friction, 470 first, 385, 496, 505 436 Internal pressure, 416, 418 second, 495, 505, 509 MAXWELL’s wheel, 43 International temperature scale, 372 Leaf spring, 15, 279 Mean free path, 435, 436, 467, 468 Intonation, 357 Length, 4 Mean squared deviation, 448 Inversion temperature, 417 Length measurement Melting temperature, 422 Irradiance, 348, 360 by interference, 7 pressure dependence, 508 Irreversibility, 479, 482 indirect, 7 Memory effect, 165 Irreversible process, 477, 481 microscopic, 6 Metacenter, 185 Irrotational vortex field, 228, 231 Lens for water waves, 303 Meter, 5–7 Isochoric change of state, 400 Lens wheel, 263 Metronome, 12 Isotherm, 398 Lift, 237, 240 MICHELSON interferometer, 335 Isothermal change of state, 398 of a wing, 116, 239 Microphones, 355 Isotherms Light mill, 433 Microscope, 6 of CO2, 410, 411 LINDE, liquefaction of air, 419 Midpoint of oscillation, 103 Line spectrum, 256 Mobility, 455 J Liquefaction Model gas, 195, 197, 479 Jet engines, 244 of air, 419 Model gas atmosphere, 446 Jet propulsion, 86 of gases, 418, 420 Model of liquid structure, 414 JOULE, 68 Liquid Modulus of compression, 349, 405 JOULE-THOMSON effect, 417, 418, in an accelerated frame, 206 Modulus of elasticity, 155 420 supercooled, 424 Molar heat capacity, 376, 389 JOULE-THOMSON throttle tensile strength of, 425 in a molecular picture, 437 experiment, 416 Liquid helium, 419 of ideal gases, 398 Jumping techniques, 70 Liquid manometer, 184 Molar mass, 370, 376, 393 Liquid surface, 412 determination, 393 K as diffusion boundary, 413 Molar volume, 370, 393 KELTING,H.,359 in an accelerated frame of of ideal gases, 393 Kelvin reference, 177 Mole, 370 as unit, 373 Liquids Molecular beam, 435 temperature, 392 internal friction, 211 Molecular mass 522 Index

comparison after R. BUNSEN, Optics, geometric, 302 Phase differences in sound waves, 431 Orbital angular momentum, 104 357 Molecular velocity Ordinal number, 296 Phase rule of GIBBS, 423 calculation, 197 Oscillation Phase shift, 51 measurement, 435 damped, 273 Phase transition Molecular weight, 376 elliptically polarized, 56 hindrance, 424, 425 MOLLIER diagram, 488, 489 gravity pendulum, 53 liquid-gas, 425 MOLLWO,E.,287, 344, 346, 359 linearly polarized, 52 of CO2, 413 Moment of inertia, 97, 98 Oscillation frequency solid-liquid, 412, 424 computation, 99 of an elastic pendulum, 52 Phase velocity, 291, 341 measurement of, 100 of torsional oscillations, 97 Phon, 361 of a human body, 102, 108 Oscillation period Phonometry, 360 of cylinders, 101 of a gravity pendulum, 54 Photophoresis, 434 Momentary axis, 94 Oscillations, 247 Physical quantities, expressions, 38 Momentum, 76, 470 amplification of, 282 Pipe vibrations, 268 Momentum conservation, 76, 77, 81 damped, 258 Pirouette, 111 Motion excitation of, 247 Pitch around a central point, 54 forced, 147, 273 of the voice, 357 Muscular work, 510 non-sinusoidal, 251, 262 PITOT tube, 224 Musical instruments, 262, 353 of strings, 262 PLANCK’s radiation law, 373 Muzzle report or bang, 354 sinusoidal, 49, 252 Plasticity, 167 Muzzle velocity, 81 spectral representation of, 256 Pneumatic motors, 406, 504 superposition of, 252 POCKELS,AGNES, 191 N Osmosis, 440 POHL’s pendulum, 273 Natural gas, 206 Osmotic pressure, 442, 444 Point mass, 47 Neper (Np), 351 OTTO engine, 501 POISSON’s law, 401 Neutral fiber, 161 POISSON’s number, 155 NEWTON ISAAC, 59, 76, 87 POISSON’s relation, 155 Newton meter, 68 P Polhode cone, 116 Newton, unit of force, 37 Pan mill, 124 Polytropic curve, 402 NEWTON’s equation Parabolic trajectory, 64 Polytropic exponent, 402 applications, 41 Parametric excitation, 261, 298 measurement, 403 NEWTON’s fundamental equation Parametric excitation of Potential flow, 226, 228 of motion, 34 oscillations, 108 Power, 71 Nodal lines, 271 Partial pressures, 394 of a human, 71 Nodes, 261, 297 PASCAL,BLAISE, 183 of rotational motions, 97 Normal modes Pascal, unit of pressure, 178 PRANDTL tube, 224 in liquids and gases, 265 Path acceleration, 21, 22, 54, 130 Precession, 117 maxima, 261, 264 constant, 25 of the earth, 123 nodes of, 261, 264 Path difference, 312 Precession cone, 118, 121 of 2- and 3-dimensional bodies, Pendulum Press, hydraulic, 181 271 coupled, 283 Pressure, 154, 178 of linear solid bodies, 263 mathematical, 54, 103 in liquids, 178 of stiff linear objects, 269 period of, 54 in the lungs of a person, 394 of vibration, 260 physical, 103 osmotic, 442 Normal stress, 159 simple, 54 production of, 195 Normalized stress, 154 Pendulum clock static, in fluids, 223 Nose, blocked, 219 automatic control, 247 units of, 178 Note, 357 Pendulum length, reduced, 102, 103 Pressure distribution Number density, 369, 454 Pendulum-Top, 121 in a gravitational field, 183 Nutation, 114, 115, 121 Period, 13 in air, 203 cone of, 116, 118 of torsional oscillation, 101, 103 in gas lines, 205 Phase, 50, 292 in standing waves, 324 O Phaseangle=Phase,50 in water, 202 Object point, 303 Phase changes Pressure probe, 223 Observation point, 315 of real gases, 409 Pressure receiver, 355 Octave, 357 Phase diagram Principal ray, 303 OHM,GEORG SIMON, 357 of CO2, 422 Principal stress, 159 Optical lattice, 319 of water, 422 Index 523

Prism Resonator, 276 Sinking, 83 for sound waves, 330 Restoring force, 52 Sliding friction, 170 spectral resolving power, 347 Reversible processes, 475, 480 coefficient of, 170 Probability, 482, 483 Reversion pendulum, 103 Slit, 299 Productivity, 226 REYNOLDS number, 217, 219 Solid bodies, 153 Projectile velocity, 18 Ripple tank, 293 deformation, 31, 96, 153 Propeller, 244 Rocket equation, 86 Solids, 153, 508 Proper angular momentum, 104 Rockets, 86 Somersault, 111 Pulsed excitation, 247, 257 Rolling friction, 172 Sonoluminescence, 190 p-V diagram, 382 Rosette orbit Sound detection threshold, 361 of carbon dioxide, 410 of a pendulum, 139, 148 Sound field Rotation pendulum energy of, 328, 347 Q POHL’s pendulum, 273 pressure amplitude, 348, 350 Quantities referred to the amount of Rotation, sign or sense, 104 velocity amplitude, 349 substance (molar), 376 Rotations of liquids, 228 Sound radiometer, 326 Quantity Rubber Sound receivers, 355 derived, 29 elasticity, 370, 508 Sound shadows, 328 of of substance, 369 mechanical hysteresis, 166 Sound source, 351 physical, 28 RUBENS’ flame tube, 266 primary, 353 Quasi-static processes, 476 Rule of DULONG-PETIT, 439 secondary, 353 Rupture strength Sound velocity, 322 R of solids, 167 in gases, 349 Radar, 325 supersonic, 354, 488, 492 Radial acceleration, 21, 26, 44, 130, S Sound wave resistance, 347 133 Sailing Sound waves Radial force, 44 close-hauled, 242 diffraction of, 331 Radian, 10, 95 Satellite, artificial, 135 in air, 323 Radiation power, 326, 348 Saturation pressure, 377, 386, 422 standing, 323 Radiation pressure of sound, 326 Scales, 34 Sound-pressure amplitude, 348 Radiometer force, 433 Scattering, 330 Source, 226 Radius vector, 55, 62 Schlieren Speaking machine, 358 Rapidity, 348 in air, 331 Specific, 39 RAYLEIGH disk, 349 SCHULER period, 145 Specific enthalpy of vaporization, Real gases, 194, 409 Screw caliper, 6 378 isotherms, 410 Screw micrometer, 5 Specific heat, 376, 388 phase changes of, 409 Screws, loosening of, 171 Specific heat of vaporization, 444 Recording, distortion-free, 281 Seasickness, 43 Specific volume, 39 Rectification, 420 Second, 11 Spectral apparatus, 279 Reference frame, 44 Second Law, 505, 509 Spectral representation, 256 Reflection coefficient, 350 of thermodynamics, 495 Spin, 104 Reflection of sound waves by warm Secondary radiation, 325 Spiral illusion, 4 air, 329 Seismograph, 281 Spring balance, 36 Reflection of water waves, 302 Separation fracture, 168 Spring constant, 46, 52, 94 Refraction of sound waves, 330 Separation of gases, 420 Spring pendulum, 52, 53 Refraction of water waves, 302 Shadow, 299, 328 Stabilizing tank, 286 Refrigeration device, 501, 502 colored, 3 Stagnation point, 223 Refrigeration technology, basic rule, Shaft Stagnation pressure, 223 503 supple, 109 Standard conditions, 393 Regenerator, 498 Shallow-water lens, 303 Standing waves, 297, 323 Relaxation oscillations, 288 Shape memory alloys, 167 State variables Relaxation time, 165, 288 Shear modulus, 155, 157 critical, 409 Resolving power, 347 Shear strain, 157 for water, 485 of the ear, 357 Shear stress, 157, 159 simple, 369 Resonance, 276 Shockwave, 307 thermal, 384 importance for the detection of SI = Système International Statistical fluctuations, 447 sinusoidal oscillations, d’Unités, 29 Steam 279 Simple pendulum, 53 superheated, 411 Resonance chamber, 352 Sine curve, 13, 251 Steam engine, 500 Resonance curves, 276 Sink, 226 524 Index

Steamboats, 236 of molecules, 430 TROUTON’s rule, 485 STEINER’s law, 100, 102, 143 Thermal vibrations, 271 Tsunami, 340 Steradian, 10 Thermocouple, 372 Tuning fork, 282, 352 Stereogrammetry, 8 Thermodiffusion, 473 auto-control, 249 Sticking friction, 169 Thermometer, 371, 372 damping, 285 Stimulated energy transfer, 278 Thermos bottle, 376 improvement of emission, 353 STIRLING engine, 498, 501 Throttle experiment of Tunnel effect, 306 STOKES’law,216 GAY-LUSSAC, 396 Turbulence, 217, 414 Strain, 157 Throttled release, 396 Twisting, 95, 96, 163 Strain ellipsoid, 158 Tides, 147 Stream formation, 233 Tightrope artist, 125 U Streamlines, 216, 220, 225, 237, Time, 10, 16 Ultrasound, 268 239 Time integral, 75 Unit of work, 68 in a model experiment, 224, 225, Time measurement, 10 Unit system 239 true, 10 SI, 29 in a water wave, 336 Time measurements Units, 5 Stretching limit, 155 indirect, 15 Universal gas constant, 393 String Toggle oscillations, 288 emission of, 352 Tone, 357 V vibrations of, 262, 352 Top, 105, 112 VA N D ER WAALS’ equation of Stroboscope, 14 angular momentum axis, 113 state, 414, 415 Sublimation, 423 force-free, 115 Vapor Sublimation temperature, 422 instantaneous axis of rotation, saturation pressure, 377 Sum frequency, 358 113 Vapor pressure, 421, 450 Supercooling, 424 momentary axis of rotation, 113 Vapor-pressure curve Superfluidity of helium, 191 precessional oscillations, 125 of ice, 424 Supersonic jet, 492 symmetry axis, 113 of supercooled water, 424 Supersonic velocities, 354, 488 with two degrees of freedom, Vector, 20 Surface 123 Vector product, 92 formation of, 413, 425 Torque, 97 Velocity, 28 Surface defects, 168 definition of, 92 definition, 18, 29 Surface energy, 168, 187, 188 of an electric motor, 93 of a bullet, 137 Surface tension, 188 production of, 94 of molecules, 198, 430 abnormal, 192 Torsion, 163, 164 vector addition, 20 Surface waves on water, 336 Torsion balance, 61 Velocity distribution Syphon, 200 Torsion coefficient, 94, 97, 163 in laminar flow, 215 for gases, 201 Torsion shaft, 94, 95 MAXWELL-BOLTZMANN, 436, for liquids, 200 Torsional modulus = shear modulus, 450 157 of gas molecules, 435 T Torsional oscillation, 97, 100, 264, Velocity of light, 88 Technical work, 381, 387 271, 273 Velocity of sound, 405 Temperature, 370 Torsional pendulum, 95 Velocity receiver, 355 critical, 386, 414 Torsional waves, 322 Vernier caliper, 5 on the molecular scale, 429 Total pressure = dynamic head Vibrating-reed frequency meter, thermodynamic definition, 504 in fluids, 223 279, 363 Temperature conductivity, 464 Total reflection, 304, 306 Vibrations, 292 Temperature scale, international, Trade winds, 150 excitation of, 247 372 Transport processes in gases, 465 in pipes, 268 Tennis ball, ‘cut’, 241 Transverse acceleration, 21 of a doorbell, 249 Tensile strength Transverse contraction, 155 of a gravity pendulum, 248 of liquids, 186, 426 Transverse force of strings, 352 of solids, 168 applications, 241 self-controlling, 247 of water, 186, 426 dynamic, 237, 240 undamped, 247 Tensile stress, 154, 426 Transverse vibrations, 259, 260, 270 Violin Thermal conductivity, 463, 470 Transverse waves, 322 as sound source, 353 Thermal expansion coefficient, 370 Travelling waves, 291 frequency spectrum, 354 Thermal motions TREVELYAN’s rocker, 288 Viscosity, 212, 213, 470 in gases, 195 Triple point, 411, 421, 423 dynamic, 218 in solids, 175 kinematic, 218 Index 525

of gases, 466 Wave tank, 336 in rotations, 97 Voltage impulse, 75 Waves of acceleration, 70, 396, 490 Volume current, 214, 216 diffracted, 299 of lifting, 68, 69 Vo l u m e , m o l a r, 393 on liquid surfaces, 336 of rotation, 92 Vortex field, irrotatonal, 228, 231 standing, 297 Work of displacement, 381 Vortex ring, 234 travelling, 291 Work of expansion, 381, 383 Vortex tube, 474 Weight, 31, 35 Work, technical, 383 Vortices and separation surfaces, of the moon, 59 232 WESTPHAL, W.H., 434 Y Vorticity, 231 Wet steam, 411 YOUNG,THOMAS, 293, 295, 317, Vow e l s , 357 Wetting, 185, 189 334 Wobble oscillations, 286 W YOUNG’s modulus = modulus of amplitude dependence of the elasticity, 155 Water-wave experiments, 294 frequency, 286 Watt = N m/s, 71 Work, 67 Watt second, 68 against a spring, 69 Z Wave field on a water surface, 294 definition, 68 Zone plate, 317 Wave maxima, 297 Zones, FRESNEL, 315