DOCSLIB.ORG

Relational-Mechanics.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Relational-Mechanics.Pdf Relational Mechanics Andre K.T. Assis Apeiron Montreal Published by Apeiron 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com © Andre K.T. Assis Dr. Andre Koch Torres Assis Institute of Physics State University of Campinas 13083-970 Campinas - SP, Brazil E-mail: [email protected] Homepage: www.ifi.unicamp.br/~assis First Published 1999 Canadian Cataloguing in Publication Data Assis, Andre Koch Torres, 1962- Relational mechanics Translation of Mecanica Relacional Includes bibliographical references and index. ISBN 0-9683689-2-1 1. Mechanics. 2. Gravitation. 3. Inertia (Mechanics). I. Title. QC125.2.A88 1999 531 C99-900382-8 Front cover: Newton’s bucket experiment. The water rises in the second case because of its rotation. But: rotation relative to what? This is the central theme of this book. Back cover: Galileo’s free fall experiment. Why do two bodies of different weight fall in vacuum with the same acceleration toward the surface of the earth? Relational Mechanics offers a simple explanation for this remarkable fact. In memory of Isaac Newton who paved the way for past, present and future generations 2 Contents I Old World 13 1 Newtonian Mechanics 15 1.1 Introduction . 15 1.2 Newtonian Mechanics . 15 1.3 Energy . 26 2 Applications of Newtonian Mechanics 29 2.1 Uniform Rectilinear Motion . 30 2.2 Constant Force . 30 2.2.1 Free Fall . 30 2.2.2 Charge Moving Inside an Ideal Capacitor . 33 2.2.3 Accelerated Train . 36 2.3 Oscillatory Motions . 37 2.3.1 Spring . 37 2.3.2 Simple Pendulum . 38 2.3.3 Electrically Charged Pendulum . 42 2.4 Uniform Circular Motion . 46 2.4.1 Circular Orbit of a Planet . 47 2.4.2 Two Globes . 49 2.4.3 Newton’s Bucket Experiment . 51 3 Non-inertial Frames of Reference 59 3.1 Constant Force . 59 3.1.1 Free Fall . 59 3.1.2 Accelerated Train . 61 3.2 Uniform Circular Motion . 62 3.2.1 Circular Orbit of a Planet . 62 3.2.2 Two Globes . 65 3.2.3 Newton’s Bucket Experiment . 66 3.3 Rotation of the Earth . 67 3 4 CONTENTS 3.3.1 Kinematical Rotation of the Earth . 67 3.3.2 The Figure of the Earth . 71 3.3.3 Foucault’s Pendulum . 75 3.3.4 Comparison of the Kinematical and Dynamical Rotations 81 3.4 General fictitious Force . 82 4 Gravitational Paradox 85 4.1 Newton and the Infinite Universe . 85 4.2 The Force Paradox . 86 4.3 The Paradox based on Potential . 88 4.4 Solutions of the Paradox . 90 4.5 Absorption of Gravity . 94 5 Leibniz and Berkeley 97 5.1 Leibniz and Relative Motion . 97 5.2 Berkeley and Relative Motion . 101 6 Mach and Newton’s Mechanics 107 6.1 Inertial Frame of Reference . 107 6.2 Absolute Time . 109 6.3 The Two Rotations of the Earth . 110 6.4 Inertial Mass . 111 6.5 Mach’s Formulation of Mechanics . 114 6.6 Relational Mechanics . 115 6.7 Mach and the Bucket Experiment . 116 6.8 Mach’s Principle . 119 6.9 What Mach did not Show . 122 7 Einstein’s Theories of Relativity 125 7.1 Introduction . 125 7.2 Einstein’s Special Theory of Relativity . 125 7.2.1 Asymmetry in Electromagnetic Induction . 127 7.2.2 Postulate of Relativity . 131 7.2.3 Twin Paradox . 132 7.2.4 Constancy of the Velocity of Light . 133 7.2.5 Velocity in Lorentz’s Force . 140 7.2.6 Michelson-Morley Experiment . 144 7.3 Einstein’s General Theory of Relativity . 146 7.3.1 Relational Quantities . 146 7.3.2 Invariance in the Form of the Equations . 148 7.3.3 Implementation of Mach’s Ideas . 148 7.3.4 Newton’s Bucket Experiment . 152 CONTENTS 5 7.4 General Comments . 157 II New World 161 8 Relational Mechanics 163 8.1 Basic Concepts and Postulates . 163 8.2 Electromagnetic and Gravitational Forces . 166 8.3 Spherical Shell Interacting with a Particle . 170 8.4 Implementation of Mach’s Principle . 173 8.5 General Consequences . 180 8.6 Cosmology . 188 8.7 Ptolemaic and Copernican World Views . 190 8.8 Implementation of Einstein’s Ideas . 192 9 Applications of Relational Mechanics 197 9.1 Uniform Rectilinear Motion . 197 9.2 Constant Force . 198 9.2.1 Free Fall . 199 9.2.2 Charge Moving Inside an Ideal Capacitor . 205 9.2.3 Accelerated Train . 206 9.3 Harmonic Oscillator . 208 9.3.1 Spring . 208 9.3.2 Simple Pendulum . 209 9.4 Uniform Circular Motion . 210 9.4.1 Circular Orbit of a Planet . 210 9.4.2 Two Globes . 212 9.4.3 Newton’s Bucket Experiment . 214 9.5 Rotation of the Earth . 218 9.5.1 The Figure of the Earth . 218 9.5.2 Foucault’s Pendulum . 219 10 Beyond Newton 227 10.1 Precession of the Perihelion of the Planets . 227 10.2 Anisotropy of Inertial Mass . 231 10.3 High Velocity Particles . 233 10.4 Experimental Tests of Relational Mechanics . 236 11 History of Relational Mechanics 243 11.1 Gravitation . 243 11.2 Electromagnetism . 244 11.3 Weber’s Law Applied to Gravitation . 249 6 CONTENTS 11.4 Relational Mechanics . 250 12 Conclusion 259 Acknowledgments To the undergraduate and graduate students who followed our courses on Rela- tional Mechanics, Mach’s Principle, Weber’s Electrodynamics and Cosmology, for the many constructive remarks they presented. To our undergraduate and Ph.D. students who are developing researches in these areas. To our friends who are giving us support to continue this work. In special we wish to thank those to whom we gave a first version of this book: C. Roy Keys, Thomas E. Phipps Jr., A. Ghosh, D. Roscoe, A. Martin, R. de A. Martins, M. A. Bueno, W. M. Vieira, M. A. de Faria Rosa, A. Zylbersztajn, D. S. L. Soares, J. I. Cisneros, H. C. Velho, D. Gardelli, J. A. Hernandez and J. E. Lamesa. The suggestions and ideas which we received contributed greatly to the improvement of the book. To the following Publishers and individuals for granting permission to quote from their publications: University of California Press (I. Newton, Mathemat- ical Principles of Natural Philosophy, 1934, Cajori edition); Open Court Pub- lishing Company, a division of Carus Publishing Company, Peru, IL (E. Mach, The Science of Mechanics, 1960, translated by T. J. McCormack); Everyman’s Library, David Campbell Publications Ltd. (G. Berkeley, Philosophical Works - including De Motu and A Treatise Concerning the Principles of Human Knowl- edge, 1992); Dover Publications (A. Einstein, H. A. Lorentz, H. Weyl and H. Minkowski, The Principle of Relativity, 1952); Cambridge University Press (J. B. Barbour, Absolute or Relative Motion?, 1989); Albert Einstein Archives, The Hebrew University of Jerusalem, Israel (A. Einstein, The Meaning of Rel- ativity, 1980); Dr. Peter Gray Lucas, in name of Dr. H. G. Alexander (The Leibniz-Clarke Correspondence, edited by H. G. Alexander, 1984); Birkhauser Boston and Dr. J. B. Barbour for his translations of E. Schr¨odinger, H. Reiss- ner, A. Einstein, B. and I. Friedlaender (Mach’s Principle - From Newton’s Bucket to Quantum Gravity, edited by J. B. Barbour and H. Pfister, 1995). To the Center for Electromagnetics Research, Northeastern University (Bos- ton, USA), which received us for one year in which we had the first idea to write this book and discussed its contents with some friends. To the Institutes of Physics and Mathematics and to the Center of Logics, Epistemology and History of Sciences of the State University of Campinas - UNICAMP (Brazil), 7 8 CONTENTS that gave the necessary support to undertake this work. Above all, to my parents, my wife and children for helping me grow as a human being. Preface This book presents Relational Mechanics, a new mechanics which implements the ideas of Leibniz, Berkeley, Mach and many others. Relational mechan- ics is based only on relative quantities, such as the distance between material bodies, their relative radial velocity and relative radial acceleration. In this new mechanics the absolute concepts of space, time and motion do not appear. The same can be said of inertia, inertial mass and inertial frames of reference. When we compare relational mechanics with Newtonian mechanics, we will gain a new and clear understanding of these old concepts. Relational mechanics is a quantitative implementation of Mach’s ideas utilizing a Weber’s force law for gravitation. Many people have contributed to its development, including Erwin Schr¨odinger. This is the first time such a book has been written, bringing together all the features and characteristics of this new world view. This allows it to be seen in its proper light, and a comparison with old worldviews is easily accomplished. Considerable emphasis is placed on Galileo’s free fall experiment and on Newton’s bucket experiment. These are some of the simplest experiments ever performed in physics. Despite this fact, no other experiment has had such far-reaching consequences for the foundations of classical mechanics. An ex- planation of these two experiments without utilizing the concepts of absolute space and inertia is one of the major accomplishments of relational mechanics. In order to show all the power of relational mechanics and put it in perspec- tive, we first present Newtonian mechanics and Einstein’s theories of relativity. We address the criticisms of Newton’s theory made by Leibniz, Berkeley and Mach. Then we present relational mechanics and show how it solves all these problems quantitatively with a clarity and simplicity unsurpassed by any other model. We also discuss the history of relational mechanics in detail, emphasiz- ing the achievements and limitations of all major works along these lines.
Load more

Recommended publications
  • Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
    Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions.
    [Show full text]
  • Fundamental Governing Equations of Motion in Consistent Continuum Mechanics
    Fundamental governing equations of motion in consistent continuum mechanics Ali R. Hadjesfandiari, Gary F. Dargush Department of Mechanical and Aerospace Engineering University at Buffalo, The State University of New York, Buffalo, NY 14260 USA [email protected], [email protected] October 1, 2018 Abstract We investigate the consistency of the fundamental governing equations of motion in continuum mechanics. In the first step, we examine the governing equations for a system of particles, which can be considered as the discrete analog of the continuum. Based on Newton’s third law of action and reaction, there are two vectorial governing equations of motion for a system of particles, the force and moment equations. As is well known, these equations provide the governing equations of motion for infinitesimal elements of matter at each point, consisting of three force equations for translation, and three moment equations for rotation. We also examine the character of other first and second moment equations, which result in non-physical governing equations violating Newton’s third law of action and reaction. Finally, we derive the consistent governing equations of motion in continuum mechanics within the framework of couple stress theory. For completeness, the original couple stress theory and its evolution toward consistent couple stress theory are presented in true tensorial forms. Keywords: Governing equations of motion, Higher moment equations, Couple stress theory, Third order tensors, Newton’s third law of action and reaction 1 1. Introduction The governing equations of motion in continuum mechanics are based on the governing equations for systems of particles, in which the effect of internal forces are cancelled based on Newton’s third law of action and reaction.
    [Show full text]
  • Application of Newton's Second
    Chapter 8 Applications of Newton’s Second Law 8.1 Force Laws ............................................................................................................... 1 8.1.1 Hooke’s Law ..................................................................................................... 1 8.2.2 Principle of Equivalence: ................................................................................ 5 8.2.3 Gravitational Force near the Surface of the Earth ....................................... 5 8.2.4 Electric Charge and Coulomb’s Law ............................................................. 6 Example 8.1 Coulomb’s Law and the Universal Law of Gravitation .................. 7 8.3 Contact Forces ......................................................................................................... 7 8.3.1 Free-body Force Diagram ................................................................................... 9 Example 8.2 Tug-of-War .......................................................................................... 9 8.3.3 Normal Component of the Contact Force and Weight ............................... 11 8.3.4 Static and Kinetic Friction ............................................................................ 14 8.3.5 Modeling ......................................................................................................... 16 8.4 Tension in a Rope .................................................................................................. 16 8.4.1 Static Tension in a Rope ...............................................................................
    [Show full text]
  • Stress, Cauchy's Equation and the Navier-Stokes Equations
    Chapter 3 Stress, Cauchy’s equation and the Navier-Stokes equations 3.1 The concept of traction/stress • Consider the volume of fluid shown in the left half of Fig. 3.1. The volume of fluid is subjected to distributed external forces (e.g. shear stresses, pressures etc.). Let ∆F be the resultant force acting on a small surface element ∆S with outer unit normal n, then the traction vector t is defined as: ∆F t = lim (3.1) ∆S→0 ∆S ∆F n ∆F ∆ S ∆ S n Figure 3.1: Sketch illustrating traction and stress. • The right half of Fig. 3.1 illustrates the concept of an (internal) stress t which represents the traction exerted by one half of the fluid volume onto the other half across a ficticious cut (along a plane with outer unit normal n) through the volume. 3.2 The stress tensor • The stress vector t depends on the spatial position in the body and on the orientation of the plane (characterised by its outer unit normal n) along which the volume of fluid is cut: ti = τij nj , (3.2) where τij = τji is the symmetric stress tensor. • On an infinitesimal block of fluid whose faces are parallel to the axes, the component τij of the stress tensor represents the traction component in the positive i-direction on the face xj = const. whose outer normal points in the positive j-direction (see Fig. 3.2). 6 MATH35001 Viscous Fluid Flow: Stress, Cauchy’s equation and the Navier-Stokes equations 7 x3 x3 τ33 τ22 τ τ11 12 τ21 τ τ 13 23 τ τ 32τ 31 τ 31 32 τ τ τ 23 13 τ21 τ τ τ 11 12 22 33 x1 x2 x1 x2 Figure 3.2: Sketch illustrating the components of the stress tensor.
    [Show full text]
  • Model Comparison of DBD-PA-Induced Body Force in Quiescent Air and Separated Flow Over NACA0015
    Model comparison of DBD-PA-induced body force in quiescent air and separated flow over NACA0015 Di Chen, Kengo Asada, Satoshi Sekimoto, Kozo Fujii Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan Hiroyuki Nishida Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan Numerical simulations of plasma flows induced by dielectric barrier discharge plasma actuators (DBD-PA) are conducted with two different body-force models: Suzen-Huang (S- H) model and drift-diffusion (D-D) model. The induced flow generated in quiescent air over a flat plate in continuous actuation and the PA-based flow control effect with burst actuation in separated flow over NACA0015 are studied. In the comparative study, the body-force field and the induced velocity field are firstly investigated in the quiescent field to see the spatial difference and the temporal difference in a single discharge cycle. The D-D body force is computed with flush-mounted and bulge configuration of the exposed electrode, which is operated at the peak-to-peak AC voltage of 7kV and 10kV. The D-D models generate momentarily higher body force in the positive-going phase of the AC power, but activate smaller flow region than the S-H model with Dc = 0.0117, which is given by the experiment beforehand at 7kV.[21] The local induced velocity of the D-D bulge case at 7kV measured in the downstream flow has the best agreement with the experimental result.[36] The maximum wall-parallel induced velocity in the S-H case with Dc = 0.0117 is consistent with that in the experiment, however, the local induced velocity is relatively high with different flow structure.
    [Show full text]
  • Dynamic Meteorology - Introduction
    Dynamic Meteorology - Introduction Atmospheric dynamics – the study of atmospheric motions that are associated with weather and climate We will consider the atmosphere to be a continuous fluid medium, or continuum. Each “point” in the atmosphere will be made up of a large number of molecules, with certain properties. These properties are assumed to be continuous functions of position and time. The basic laws of thermodynamics and fluid mechanics can be expressed in terms of partial differential equations, with space and time as independent variables and the atmospheric properties as dependent variables. Physical Dimensions and Units Dimensional homogeneity – all terms in the equations that describe the atmosphere must have the same physical dimensions (units) The four base units we will use are: From these we will also use the following derived units: Fundamental Forces Newton’s Second Law: F=ma In atmospheric science it is typical to consider the force per unit mass acting on the atmosphere: Force = a mass In order to understand atmospheric motion (accelerations) we need to know what forces act on the atmosphere. € What are the fundamental forces of interest in atmospheric science? Body (or volume) force – a force that acts on the center of mass of a fluid parcel Surface force – a force that acts across the boundary separating a fluid parcel from its surroundings. The magnitude of surface forces are independent of the mass of the parcel. What are examples of body and surface forces? Pressure Gradient Force Example: Real-time weather map The force exerted on the left face of this air parcel due to pressure is: pA = pδyδz The force exerted on the right face of this air parcel due to pressure is: € % ∂p ( −(p + δp)δyδz = −' p + δx*δ yδz & ∂x ) The net force exerted by pressure on this air parcel is the sum of these forces and is equal to: € ∂p − δxδyδz ∂x By dividing by the mass of the air parcel (ρδxδyδz) we get the force per unit mass due to changes in pressure (i.e.
    [Show full text]
  • Introduction to Atmospheric Dynamics Chapter 1
    Introduction to Atmospheric Dynamics Chapter 1 Paul A. Ullrich [email protected] How to Read These Slides Defnition: A defnition is an explanation or outline for relevant jargon or terms. Concept: An idea that draws a connection between subjects or provides an answer for a question. Question: What is something that motivates delving into this topic? Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Part 1: Forces in the Atmosphere Radius of the Earth 6371.22 km Atmosphere Depth 100 km Troposphere Depth 10 km Mountain Height 8.8 km Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: How do we understand the dynamics of the atmosphere? Answer: The principles of atmospheric dynamics are drawn from basic physical principles. Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Question: What are the basic physical principles that govern the atmosphere? Newton’s Second Law: The change in momentum of an object is equal to the sum of forces acting on that object. d(mv) = F dt i allXi Conservation of Momentum: With no external forces momentum must be conserved. Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Basic Principles of Physics dx Defnition: Velocity is the change of u = position with respect to time dt du Defnition: Acceleration is the change a = of velocity with respect to time dt Hence, for an object of constant mass: d(mu) du 1 = m = ma Newton’s a = F dt dt Second Law m i allXi Paul Ullrich The Equations of Atmospheric Dynamics March 2014 Basic Principles of Physics l How do these forces induce acceleration? l We assume the existence of an idealized “parcel” of fuid.
    [Show full text]
  • Lecture 18: Planar Kinetics of a Rigid Body
    ME 230 Kinematics and Dynamics Wei-Chih Wang Department of Mechanical Engineering University of Washington Planar kinetics of a rigid body: Force and acceleration Chapter 17 Chapter objectives • Introduce the methods used to determine the mass moment of inertia of a body • To develop the planar kinetic equations of motion for a symmetric rigid body • To discuss applications of these equations to bodies undergoing translation, rotation about fixed axis, and general plane motion W. Wang 2 Lecture 18 • Planar kinetics of a rigid body: Force and acceleration Equations of Motion: Rotation about a Fixed Axis Equations of Motion: General Plane Motion - 17.4-17.5 W. Wang 3 Material covered • Planar kinetics of a rigid body : Force and acceleration Equations of motion 1) Rotation about a fixed axis 2) General plane motion …Next lecture…Start Chapter 18 W. Wang 4 Today’s Objectives Students should be able to: 1. Analyze the planar kinetics of a rigid body undergoing rotational motion 2. Analyze the planar kinetics of a rigid body undergoing general plane motion W. Wang 5 Applications (17.4) The crank on the oil-pump rig undergoes rotation about a fixed axis, caused by the driving torque M from a motor. As the crank turns, a dynamic reaction is produced at the pin. This reaction is a function of angular velocity, angular acceleration, and the orientation of the crank. If the motor exerts a constant torque M on Pin at the center of rotation. the crank, does the crank turn at a constant angular velocity? Is this desirable for such a machine? W.
    [Show full text]
  • Glossary of Terms In
    Glossary of Terms in Powder and Bulk Technology Prepared by Lyn Bates ISBN 978-0-946637-12-6 The British Materials Handling Board Foreward. Bulk solids play a vital role in human society, permeating almost all industrial activities and dominating many. Bulk technology embraces many disciplines, yet does not fall within the domain of a specific professional activity such as mechanical or chemical engineering. It has emerged comparatively recently as a coherent subject with tools for quantifying flow related properties and the behaviour of solids in handling and process plant. The lack of recognition of the subject as an established format with monumental industrial implications has impeded education in the subject. Minuscule coverage is offered within most university syllabuses. This situation is reinforced by the acceptance of empirical maturity in some industries and the paucity of quality textbooks available to address its enormous scope and range of application. Industrial performance therefore suffers. The British Materials Handling Board perceived the need for a Glossary of Terms in Particle Technology as an introductory tool for non-specialists, newcomers and students in this subject. Co-incidentally, a draft of a Glossary of Terms in Particulate Solids was in compilation. This concept originated as a project of the Working Part for the Mechanics of Particulate Solids, in support of a web site initiative of the European Federation of Chemical Engineers. The Working Party decided to confine the glossary on the EFCE web site to terms relating to bulk storage, flow of loose solids and relevant powder testing. Lyn Bates*, the UK industrial representative to the WPMPS leading this Glossary task force, decided to extend this work to cover broader aspects of particle and bulk technology and the BMHB arranged to publish this document as a contribution to the dissemination of information in this important field of industrial activity.
    [Show full text]
  • Electrostatic Forces and Stored Energy for Deformable Dielectric Materials
    Electrostatic Forces and Stored Energy for Deformable Dielectric Materials An isothermal energy balance is formulated for a system consisting of deformable dielec- tric bodies, electrodes, and the surrounding space. The formulation in this paper is Robert M. McMeeking obtained in the electrostatic limit but with the possibility of arbitrarily large deformations Fellow ASME of polarizable material. The energy balance recognizes that charges may be driven onto Department of Mechanical and Environmental or off of the electrodes, a process accompanied by external electrical work; mechanical Engineering and Materials Department, loads may be applied to the bodies, thereby doing work through displacements; energy is University of California, stored in the material by such features as elasticity of the lattice, piezoelectricity, and Santa Barbara, CA 93106 dielectric and electrostatic interactions; and nonlinear reversible material behavior such e-mail: [email protected] as electrostriction may occur. Thus the external work is balanced by (1) internal energy consisting of stress doing work on strain increments, (2) the energy associated with Chad M. Landis permeating free space with an electric field, and (3) by the electric field doing work on Mem. ASME increments of electric displacement or, equivalently, polarization. For a conservative Department of Mechanical Engineering and system, the internal work is stored reversibly in the body and in the underlying and Materials Science, surrounding space. The resulting work statement for a conservative system is considered Rice University, in the special cases of isotropic deformable dielectrics and piezoelectric materials. We Houston, TX 77251 identify the electrostatic stress, which provides measurable information quantifying the e-mail: [email protected] electrostatic effects within the system, and find that it is intimately tied to the constitutive formulation for the material and the associated stored energy and cannot be independent of them.
    [Show full text]
  • Forces & Newton's Laws
    PHYSICS 1 Forces & Newton’s Laws Advanced Placement Presenter 2014-2015 Forces & Newton’s Laws What I Absolutely Have to Know to Survive the AP* Exam Force is any push or pull. It is a vector. Newton’s Second Law is the workhorse of the AP Physics 1 exam. It allows you to write down mathematical relationships that are true. Thus, for a single body, if you pick any direction and sum up all the positive and negative forces that act on the body along that line, the sum will equal the product of the body’s mass and its acceleration along that line. A Free Body Diagram allows you to identify all of the forces acting on a single body. Neglect one force or add a fictitious force on your FBD and you are in trouble. • Newton’s 1st Law: in an inertial frame of reference, an object in a state of constant velocity (including zero velocity) will continue in that state unless impinged upon by a net external force. If ΣF=0, then a=0 and the object is at rest or moving at a constant velocity in a straight line. The converse is true also, if an object is in a state of constant velocity (including zero velocity) then a=0 and ΣF=0. • Newton’s 2nd Law: A net force acting on a mass causes that mass to accelerate in the direction of the net force. The acceleration (vector) is directly proportional to the net force (vector) acting on the mass and ΣF inversely proportional to the mass of the object being accelerated.
    [Show full text]
  • Numerical Solution of the Incompressible Navier–Stokes Equations with Coriolis Forces Based on the Discretization of the Total Time Derivative
    Journal of Computational Physics 148, 467–496 (1999) Article ID jcph.1998.6126, available online at http://www.idealibrary.com on Numerical Solution of the Incompressible Navier–Stokes Equations with Coriolis Forces Based on the Discretization of the Total Time Derivative Ramon Codina Escola Tecnica` Superior d’Enginyers de Camins, Canals i Ports, Universitat Politecnica` de Catalunya, Jordi Girona 1–3, Edifici C1, 08034 Barcelona, Spain E-mail: [email protected] Received September 8, 1998 In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on a finite difference discretization in time and a finite element discretization in space. When the viscosity is very small, numerical oscillations may appear due both to the high Reynolds number and to the presence of the Coriolis forces. To overcome these oscillations, a special discretization in time is proposed. The idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system. After this is done, a further high-order approxima- tion is introduced, leading to a problem posed in the rotating frame of reference and in spatial coordinates. In contrast with the straightforward discretization of the original equations, some additional terms appear that enhance the stability of the numerical scheme. In the absence of Coriolis forces, the method is a generalization of the char- acteristic Galerkin technique for convection-dominated flows. c 1999 Academic Press Key Words: incompressible flows; Coriolis forces; method of characteristics; in- ertial bases; stabilized finite element methods.
    [Show full text]