Fundamentals of Physics the open yale courses seriesis designed to bring the depth and breadth of a Yale education to a wide variety of readers. Based on Yale’s Open Yale Courses program (http://oyc.yale.edu), these books bring out- standing lectures by Yale faculty to the curious reader, whether student or adult. Covering a wide variety of topics across disciplines in the social sci- ences, physical sciences, and humanities, Open Yale Courses books offer accessible introductions at affordable prices. The production of Open Yale Courses for the Internet was made possible by a grant from the William and Flora Hewlett Foundation. RECENT TITLES Paul H. Fry, Theory of Literature Christine Hayes, Introduction to the Bible Shelly Kagan, Death Dale B. Martin, New Testament History and Literature Giuseppe Mazzotta, Reading Dante R. Shankar, Fundamentals of Physics Ian Shapiro, The Moral Foundations of Politics Steven B. Smith, Political Philosophy Fundamentals of Physics Mechanics, Relativity, and Thermodynamics r. shankar New Haven and London Published with assistance from the foundation established in memory of Amasa Stone Mather of the Class of 1907, Yale College. Copyright c 2014 by Yale University. All rights reserved. This book may not be reproduced, in whole or in part, including illustrations, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press), without written permission from the publishers. Yale University Press books may be purchased in quantity for educational, business, or promotional use. For information, please e-mail sales.press@ yale.edu (U.S. office) or [email protected] (U.K. office). SetinMiniontypebyNewgenNorthAmerica. Printed in the United States of America. ISBN: 978-0-300-19220-9 Library of Congress Control Number: 2013947491 A catalogue record for this book is available from the British Library. This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). 10 9 8 7 6 5 4 3 2 1 To my students for their friendship and inspiration Deep and original, but also humble and generous, the physicist Josiah Willard Gibbs spent much of his life at Yale University. His father was a professor of sacred languages at Yale, and Gibbs received his bachelor’s and doctorate degrees from the university before teaching there until his death in 1903. The sculptor Lee Lawrie created the memorial bronze tablet pictured above, which was installed in Yale’s Sloane Physics Laboratory in 1912. It now resides in the entrance to the J. W. Gibbs Laboratories, Yale University. Contents Preface xiii 1. The Structure of Mechanics 1 1.1 Introduction and some useful tips 1 1.2 Kinematics and dynamics 2 1.3 Average and instantaneous quantities 4 1.4 Motion at constant acceleration 6 1.5 Sample problem 10 2 = 2 + − 1.6 Deriving v v0 2a(x x0) using calculus 13 2. Motion in Higher Dimensions 15 2.1 Review 15 2.2 Vectors in d = 216 2.3 Unit vectors 19 2.4 Choiceofaxesandbasisvectors 22 2.5 Derivatives of the position vector r 26 2.6 Application to circular motion 29 2.7 Projectile motion 32 3. Newton’s Laws I 36 3.1 Introduction to Newton’s laws of motion 36 3.2 Newton’s second law 38 3.3 Two halves of the second law 41 3.4 Newton’s third law 45 3.5 Weight and weightlessness 49 4. Newton’s Laws II 51 4.1 A solved example 51 4.2 Never the whole story 54 4.3 Motion in d = 255 viii Contents 4.4 Friction: static and kinetic 56 4.5 Inclined plane 57 4.6 Coupled masses 61 4.7 Circular motion, loop-the-loop 64 5.LawofConservationofEnergy 70 5.1 Introduction to energy 70 5.2 The work-energy theorem and power 71 5.3 Conservation of energy: K2 + U2 = K1 + U1 75 5.4 Friction and the work-energy theorem 78 6. Conservation of Energy in d = 282 6.1 Calculus review 82 6.2 Work done in d = 284 6.3 Work done in d = 2 and the dot product 88 6.4 Conservative and non-conservative forces 92 6.5 Conservative forces 95 6.6 Application to gravitational potential energy 98 7. The Kepler Problem 101 7.1 Kepler’s laws 101 7.2 The law of universal gravity 104 7.3 Details of the orbits 108 7.4 Law of conservation of energy far from the earth 112 7.5 Choosing the constant in U 114 8. Multi-particle Dynamics 118 8.1 The two-body problem 118 8.2 The center of mass 119 8.3 Law of conservation of momentum 128 8.4 Rocket science 134 8.5 Elastic and inelastic collisions 136 8.6 Scattering in higher dimensions 140 9. Rotational Dynamics I 143 9.1 Introduction to rigid bodies 143 9.2 Angle of rotation, the radian 145 Contents ix 9.3 Rotation at constant angular acceleration 147 9.4 Rotational inertia, momentum, and energy 148 9.5 Torque and the work-energy theorem 154 9.6 Calculating the moment of inertia 156 10. Rotational Dynamics II 159 10.1 The parallel axis theorem 159 10.2 Kinetic energy for a general N-body system 163 10.3 Simultaneous translations and rotations 165 10.4 Conservation of energy 167 τ = dL 10.5 Rotational dynamics using dt 168 10.6 Advanced rotations 169 10.7 Conservation of angular momentum 171 10.8 Angular momentum of the figure skater 172 11. Rotational Dynamics III 175 11.1 Static equilibrium 175 11.2 The seesaw 176 11.3 A hanging sign 178 11.4 The leaning ladder 180 11.5 Rigid-body dynamics in 3d 182 11.6 The gyroscope 191 12. Special Relativity I: The Lorentz Transformation 194 12.1 Galilean and Newtonian relativity 195 12.2 Proof of Galilean relativity 196 12.3 Enter Einstein 200 12.4 The postulates 203 12.5 The Lorentz transformation 204 13. Special Relativity II: Some Consequences 209 13.1 Summary of the Lorentz transformation 209 13.2 The velocity transformation law 212 13.3 Relativity of simultaneity 214 13.4 Time dilation 216 13.4.1 Twin paradox 219 13.4.2 Length contraction 220 x Contents 13.5 More paradoxes 222 13.5.1 Too big to fall 222 13.5.2 Muons in flight 226 14. Special Relativity III: Past, Present, and Future 227 14.1 Past, present, and future in relativity 227 14.2 Geometry of spacetime 232 14.3 Rapidity 235 14.4 Four-vectors 238 14.5 Proper time 239 15. Four-momentum 241 15.1 Relativistic scattering 249 15.1.1 Compton effect 249 15.1.2 Pair production 251 15.1.3 Photon absorption 252 16. Mathematical Methods 255 16.1 Taylor series of a function 255 16.2 Examples and issues with the Taylor series 261 16.3 Taylor series of some popular functions 263 16.4 Trigonometric and exponential functions 265 16.5 Properties of complex numbers 267 16.6 Polar form of complex numbers 272 17. Simple Harmonic Motion 275 17.1 More examples of oscillations 280 17.2 Superposition of solutions 283 17.3 Conditions on solutions to the harmonic oscillator 288 17.4 Exponential functions as generic solutions 290 17.5 Damped oscillations: a classification 291 17.5.1 Over-damped oscillations 291 17.5.2 Under-damped oscillations 292 17.5.3 Critically damped oscillations 294 17.6 Driven oscillator 294 Contents xi 18. Waves I 303 18.1 The wave equation 306 18.2 Solutions of the wave equation 310 18.3 Frequency and period 313 19. Waves II 316 19.1 Wave energy and power transmitted 316 19.2 Doppler effect 320 19.3 Superposition of waves 323 19.4 Interference: the double-slit experiment 326 19.5 Standing waves and musical instruments 330 20. Fluids 335 20.1 Introduction to fluid dynamics and statics 335 20.1.1 Density and pressure 335 20.1.2 Pressure as a function of depth 336 20.2 The hydraulic press 341 20.3 Archimedes’ principle 343 20.4 Bernoulli’s equation 346 20.4.1 Continuity equation 346 20.5 Applications of Bernoulli’s equation 349 21. Heat 352 21.1 Equilibrium and the zeroth law: temperature 352 21.2 Calibrating temperature 354 21.3 Absolute zero and the Kelvin scale 360 21.4 Heat and specific heat 361 21.5 Phase change 365 21.6 Radiation, convection, and conduction 368 21.7 Heat as molecular kinetic energy 371 22. Thermodynamics I 375 22.1 Recap 375 22.2 Boltzmann’s constant and Avogadro’s number 376 22.3 Microscopic definition of absolute temperature 379 22.4 Statistical properties of matter and radiation 382 22.5 Thermodynamic processes 384 xii Contents 22.6 Quasi-static processes 386 22.7 The first law of thermodynamics 387 22.8 Specific heats: cv and cp 391 23. Thermodynamics II 394 23.1 Cycles and state variables 394 23.2 Adiabatic processes 396 23.3 The second law of thermodynamics 399 23.4 The Carnot engine 403 23.4.1 Defining T using Carnot engines 409 24. Entropy and Irreversibility 411 24.1 Entropy 411 24.2 The second law: law of increasing entropy 418 24.3 Statistical mechanics and entropy 423 24.4 Entropy of an ideal gas: full microscopic analysis 430 24.5 Maximum entropy principle illustrated 434 24.6 The Gibbs formalism 437 24.7 The third law of thermodynamics 441 Index 443 Preface Given that the size of textbooks has nearly tripled during my own career, without a corresponding increase in the cranial dimensions of my stu- dents, I have always found it necessary, like my colleagues elsewhere, to cull the essentials into a manageable size.