Introduction to Mechanics (Kleppner & Kolenkow) PDF

An Introduction to Mechanics For 40 years, Kleppner and Kolenkow’s classic text has introduced stu- dents to the principles of mechanics. Now brought up-to-date, this re- vised and improved Second Edition is ideal for classical mechanics courses for first- and second-year undergraduates with foun...

An Introduction to Mechanics For 40 years, Kleppner and Kolenkow’s classic text has introduced stu- dents to the principles of mechanics. Now brought up-to-date, this re- vised and improved Second Edition is ideal for classical mechanics courses for first- and second-year undergraduates with foundation skills in mathematics. The book retains all the features of the first edition, including numer- ous worked examples, challenging problems, and extensive illustrations, and has been restructured to improve the flow of ideas. It now features New examples taken from recent developments, such as laser slowing of atoms, exoplanets, and black holes A “Hints, Clues, and Answers” section for the end-of-chapter prob- lems to support student learning A solutions manual for instructors at www.cambridge.org/kandk d a n i e l k l e p p n e r is Lester Wolfe Professor of Physics, Emeritus, at Massachusetts Institute of Technology. For his contributions to teaching he has been awarded the Oersted Medal by the American Association of Physics Teachers and the Lilienfeld Prize of the American Physical Society. He has also received the Wolf Prize in Physics and the National Medal of Science. r o b e r t k o l e n k o w was Associate Professor of Physics at Mas- sachusetts Institute of Technology. Renowned for his skills as a teacher, Kolenkow was awarded the Everett Moore Baker Award for Outstanding Teaching.  Daniel Kleppner Robert Kolenkow AN INTRODUCTION TO MECHANICS SECOND EDITION University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is a part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521198110  c D. Kleppner and R. Kolenkow 2014 This edition is not for sale in India. This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First edition previously published by McGraw-Hill Education 1973 First published by Cambridge University Press 2010 Reprinted 2012 Second edition published by Cambridge University Press 2014 Printed in the United States by Sheridan Inc. A catalogue record for this publication is available from the British Library ISBN 978-0-521-19811-0 Hardback Additional resources for this publication at www.cambridge.org/kandk Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. CONTENTS PREFACE TO THE TEACHER page xi xv LIST OF EXAMPLES xvii 1 VECTORS AND KINEMATICS 1 1.1 Introduction 2 1.2 Vectors 2 1.3 The Algebra of Vectors 3 1.4 Multiplying Vectors 4 1.5 Components of a Vector 8 1.6 Base Vectors 11 1.7 The Position Vector r and Displacement 12 1.8 Velocity and Acceleration 14 1.9 Formal Solution of Kinematical Equations 19 1.10 More about the Time Derivative of a Vector 22 1.11 Motion in Plane Polar Coordinates 26 Note 1.1 Approximation Methods 36 Note 1.2 The Taylor Series 37 Note 1.3 Series Expansions of Some Common Functions 38 Note 1.4 Differentials 39 Note 1.5 Significant Figures and Experimental Uncertainty 40 Problems 41 vi CONTENTS 2 NEWTON’S LAWS 47 2.1 Introduction 48 2.2 Newtonian Mechanics and Modern Physics 48 2.3 Newton’s Laws 49 2.4 Newton’s First Law and Inertial Systems 51 2.5 Newton’s Second Law 51 2.6 Newton’s Third Law 54 2.7 Base Units and Physical Standards 59 2.8 The Algebra of Dimensions 63 2.9 Applying Newton’s Laws 64 2.10 Dynamics Using Polar Coordinates 72 Problems 77 3 FORCES AND EQUATIONS OF MOTION 81 3.1 Introduction 82 3.2 The Fundamental Forces of Physics 82 3.3 Gravity 83 3.4 Some Phenomenological Forces 89 3.5 A Digression on Differential Equations 95 3.6 Viscosity 98 3.7 Hooke’s Law and Simple Harmonic Motion 102 Note 3.1 The Gravitational Force of a Spherical Shell 107 Problems 110 4 MOMENTUM 115 4.1 Introduction 116 4.2 Dynamics of a System of Particles 116 4.3 Center of Mass 119 4.4 Center of Mass Coordinates 124 4.5 Conservation of Momentum 130 4.6 Impulse and a Restatement of the Momentum Relation 131 4.7 Momentum and the Flow of Mass 136 4.8 Rocket Motion 138 4.9 Momentum Flow and Force 143 4.10 Momentum Flux 145 Note 4.1 Center of Mass of Two- and Three-dimensional Objects 151 Problems 155 5 ENERGY 161 5.1 Introduction 162 5.2 Integrating Equations of Motion in One Dimension 162 5.3 Work and Energy 166 5.4 The Conservation of Mechanical Energy 179 5.5 Potential Energy 182 5.6 What Potential Energy Tells Us about Force 185 CONTENTS vii 5.7 Energy Diagrams 185 5.8 Non-conservative Forces 187 5.9 Energy Conservation and the Ideal Gas Law 189 5.10 Conservation Laws 192 5.11 World Energy Usage 194 Note 5.1 Correction to the Period of a Pendulum 199 Note 5.2 Force, Potential Energy, and the Vector Operator ∇ 200 Problems 205 6 TOPICS IN DYNAMICS 211 6.1 Introduction 212 6.2 Small Oscillations in a Bound System 212 6.3 Stability 217 6.4 Normal Modes 219 6.5 Collisions and Conservation Laws 225 Problems 233 7 ANGULAR MOMENTUM AND FIXED AXIS ROTATION 239 7.1 Introduction 240 7.2 Angular Momentum of a Particle 241 7.3 Fixed Axis Rotation 245 7.4 Torque 250 7.5 Torque and Angular Momentum 252 7.6 Dynamics of Fixed Axis Rotation 260 7.7 Pendulum Motion and Fixed Axis Rotation 262 7.8 Motion Involving Translation and Rotation 267 7.9 The Work–Energy Theorem and Rotational Motion 273 7.10 The Bohr Atom 277 Note 7.1 Chasles’ Theorem 280 Note 7.2 A Summary of the Dynamics of Fixed Axis Rotation 282 Problems 282 8 RIGID BODY MOTION 291 8.1 Introduction 292 8.2 The Vector Nature of Angular Velocity and Angular Momentum 292 8.3 The Gyroscope 300 8.4 Examples of Rigid Body Motion 304 8.5 Conservation of Angular Momentum 310 8.6 Rigid Body Rotation and the Tensor of Inertia 312 8.7 Advanced Topics in Rigid Body Dynamics 320 Note 8.1 Finite and Infinitesimal Rotations 329 Note 8.2 More about Gyroscopes 331 Problems 337 viii CONTENTS 9 NON-INERTIAL SYSTEMS AND FICTITIOUS FORCES 341 9.1 Introduction 342 9.2 Galilean Transformation 342 9.3 Uniformly Accelerating Systems 344 9.4 The Principle of Equivalence 347 9.5 Physics in a Rotating Coordinate System 356 Note 9.1 The Equivalence Principle and the Gravitational Red Shift 368 Problems 370 10 CENTRAL FORCE MOTION 373 10.1 Introduction 374 10.2 Central Force Motion as a One-body Problem 374 10.3 Universal Features of Central Force Motion 376 10.4 The Energy Equation and Energy Diagrams 379 10.5 Planetary Motion 386 10.6 Some Concluding Comments on Planetary Motion 402 Note 10.1 Integrating the Orbit Integral 403 Note 10.2 Properties of the Ellipse 405 Problems 407 11 THE HARMONIC OSCILLATOR 411 11.1 Introduction 412 11.2 Simple Harmonic Motion: Review 412 11.3 The Damped Harmonic Oscillator 414 11.4 The Driven Harmonic Oscillator 421 11.5 Transient Behavior 425 11.6 Response in Time and Response in Frequency 427 Note 11.1 Complex Numbers 430 Note 11.2 Solving the Equation of Motion for the Damped Oscillator 431 Note 11.3 Solving the Equation of Motion for the Driven Harmonic Oscillator 434 Problems 435 12 THE SPECIAL THEORY OF RELATIVITY 439 12.1 Introduction 440 12.2 The Possibility of Flaws in Newtonian Physics 440 12.3 The Michelson–Morley Experiment 442 12.4 The Special Theory of Relativity 445 12.5 Transformations 447 12.6 Simultaneity and the Order of Events 450 12.7 The Lorentz Transformation 451 12.8 Relativistic Kinematics 454 12.9 The Relativistic Addition of Velocities 463 12.10 The Doppler Effect 466 CONTENTS ix 12.11 The Twin Paradox 470 Problems 472 13 RELATIVISTIC DYNAMICS 477 13.1 Introduction 478 13.2 Relativistic Momentum 478 13.3 Relativistic Energy 481 13.4 How Relativistic Energy and Momentum are Related 487 13.5 The Photon: A Massless Particle 488 13.6 How Einstein Derived E = mc2 498 Problems 499 14 SPACETIME PHYSICS 503 14.1 Introduction 504 14.2 Vector Transformations 504 14.3 World Lines in Spacetime 506 14.4 An Invariant in Spacetime 508 14.5 Four-Vectors 509 14.6 The Energy–Momentum Four-Vector 512 14.7 Epilogue: General Relativity 513 Problems 515 HINTS, CLUES, AND ANSWERS TO SELECTED PROBLEMS 519 APPENDIX A MISCELLANEOUS PHYSICAL AND ASTRONOMICAL DATA 527 APPENDIX B GREEK ALPHABET 529 APPENDIX C SI PREFIXES 531 INDEX 533 PREFACE An Introduction to Mechanics grew out of a one-semester course at the Massachusetts Institute of Technology—Physics 8.012—intended for students who seek to understand physics more deeply than the usual freshman level. In the four decades since this text was written physics has moved forward on many fronts but mechanics continues to be a bedrock for concepts such as inertia, momentum, and energy; fluency in the physicist’s approach to problem-solving—an underlying theme of this book—remains priceless. The positive comments we have received over the years from students, some of whom are now well advanced in their careers, as well as from faculty at M.I.T. and elsewhere, reassures us that the approach of the text is fundamentally sound. We have received many suggestions from colleagues and we have taken this opportunity to incorporate their ideas and to update some of the discussions. We assume that our readers know enough elementary calculus to dif- ferentiate and integrate simple polynomials and trigonometric functions. We do not assume any familiarity with differential equations. Our expe- rience is that the principal challenge for most students is not with un- derstanding mathematical concepts but in learning how to apply them to physical problems. This comes with practice and there is no substitute for solving challenging problems. Consequently problem-solving takes high priority. We have provided numerous worked examples to help pro- vide guidance. Where possible we try to tie the examples to interesting physical phenomena but we are unapologetic about totally pedagogical problems. A block sliding down a plane is sometimes mocked as the quintessentially dull physics problem but if one allows the plane to ac- celerate, the system takes on a new complexion. xii PREFACE The problems in the first edition have challenged, instructed, and occa- sionally frustrated generations of physicists. Some former students have volunteered that working these problems gave them the confidence to pursue careers in science. Consequently, most of the problems in the first edition have been retained and a number of new problems have been added. We continue to respect the wisdom of Piet Hein’s aphoristic ditty1 Problems worthy of attack, Prove their worth by hitting back. In addition to this inspirational thought, we offer students a few prac- tical suggestions: The problems are meant to be worked with pencil and paper. They generally require symbolic solutions: numerical values, if needed, come last. Only by looking at a symbolic solution can one de- cide if an answer is reasonable. Diagrams are helpful. Hints and answers are given for some of the problems. We have not included solutions in the book because checking one’s approach before making the maximum effort is often irresistible. Working in groups can be instructional for all parties. A separate solutions manual with restricted distribution is how- ever available from Cambridge University Press. Two revolutionary advances in physics that postdate the first edition deserve mention. The first is the discovery, more accurately the rediscov- ery, of chaos in the 1970’s and the subsequent emergence of chaos the- ory as a vital branch of dynamics. Because we could not discuss chaos meaningfully within a manageable length, we have not attempted to deal with it. On the other hand, it would have been intellectually dishonest to present evidence for the astounding accuracy of Kepler’s laws without mentioning that the solar system is chaotic, though with a time-scale too long to be observable, and so we have duly noted the existence of chaos. The second revolutionary advance is the electronic computer. Compu- tational physics is now a well-established discipline and some level of computational fluency is among the physicist’s standard tools. Never- theless, we have elected not to include computational problems because they are not essential for understanding the concepts of the book, and because they have a seductive way of consuming time. Here is a summary of the second edition: The first chapter is a math- ematical introduction to vectors and kinematics. Vector notation is stan- dard not only in the text but throughout physics and so we take some care to explain it. Translational motion is naturally described using fa- miliar Cartesian coordinates. Rotational motion is equally important but its natural coordinates are not nearly as familiar. Consequently, we put special emphasis on kinematics using polar coordinates. Chapter 2 in- troduces Newton’s laws starting with the decidedly non-trivial concept of inertial systems. This chapter has been converted into two, the first (Chapter 2) discussing principles and the second (Chapter 3) devoted to applying these to various physical systems. Chapter 4 introduces the concepts of momentum, momentum flux, and the conservation of 1 From Grooks 1 by Piet Hein, copyrighted 1966, The M.I.T. Press. PREFACE xiii momentum. Chapter 5 introduces the concepts of kinetic energy, po- tential energy, and the conservation of energy, including heat and other forms. Chapter 6 applies the preceding ideas to phenomena of general in- terest in mechanics: small oscillations, stability, coupled oscillators and normal modes, and collisions. In Chapter 7 the ideas are extended to ro- tational motion. Fixed axis rotation is treated in this chapter, followed by the more general situation of rigid body motion in Chapter 8. Chapter 9 returns to the subject of inertial systems, in particular how to understand observations made in non-inertial systems. Chapters 10 and 11 present two topics that are of general interest in physics: central force motion and the damped and forced harmonic oscillator, respectively. Chapters 12–14 provide an introduction to non-Newtonian physics: the special theory of relativity. When we created Physics 8.012 the M.I.T. semester was longer than it is today and there is usually not enough class time to cover all the ma- terial. Chapters 1–9 constitute the intellectual core of the course. Some combination of Chapters 9–14 is generally presented, depending on the instructor’s interest. We wish to acknowledge contributions to the book made over the years by colleagues at M.I.T. These include R. Aggarwal, G. B. Benedek, A. Burgasser, S. Burles, D. Chakrabarty, L. Dreher, T. J. Greytak, H. T. Imai, H. J. Kendall (deceased), W. Ketterle, S. Mochrie, D. E. Pritchard, P. Rebusco, S. W. Stahler, J. W. Whitaker, F. A. Wilczek, and M. Zwierlein. We particularly thank P. Dourmashkin for his help. Daniel Kleppner Robert J. Kolenkow TO THE This edition of An Introduction to Mechanics, like the first edition, is intended for a one-semester course. Like the first edition, there are 14 TEACHER chapters, though much of the material has been rewritten and two chap- ters are new. The discussion of Newton’s laws, which sets the tone for the course, is now presented in two chapters. Also, the discussion of energy and energy conservation has been augmented and divided into two chap- ters. Chapter 5 on vector calculus from the first edition has been omitted because the material was not essential and its presence seemed to gen- erate some math anxiety. A portion of the material is in an appendix to Chapter 5. The discussion of energy has been extended. The idea of heat has been introduced by relating the ideal gas law to the concept of momentum flux. This simultaneously incorporates heat into the principle of energy conservation, and illustrates the fundamental distinction between heat and kinetic energy. At the practical end, some statistics are presented on international energy consumption, a topic that might stimulate thinking about the role of physics in society, The only other substantive change has been a recasting of the dis- cussion of relativity with more emphasis on the spacetime description. Throughout the book we have attempted to make the math more user friendly by solving problems from a physical point of view before pre- senting a mathematical solution. In addition, a number of new examples have been provided. The course is roughly paced to a chapter a week. The first nine chap- ters are vital for a strong foundation in mechanics: the remainder covers material that can be picked up in the future. The first chapter introduces xvi TO THE TEACHER the language of vectors and provides a background in kinematics that is used throughout the text. Students are likely to return to Chapter 1, using it as a resource for later chapters. On a few occasions we have been able to illustrate concepts by ex- amples based on relatively recent advances in physics, for instance exo- planets, laser-slowing of atoms, the solar powered space kite, and stars orbiting around the cosmic black hole at the center of our galaxy. The question of student preparation for Physics 8.012 at M.I.T. comes up regularly. We have found that the most reliable predictor of per- formance is a quiz on elementary calculus. At the other extreme, oc- casionally a student takes Physics 8.012 having already completed an AP physics course. Taking a third introductory physics course might be viewed as cruel and unusual, but to our knowledge, these students all felt that the experience was worthwhile. LIST OF Chapter 1 VECTORS AND KINEMATICS 1.1 The Law of Cosines 5; 1.2 Work and the Dot Product 5; 1.3 Ex- EXAMPLES amples of the Vector Product in Physics 7; 1.4 Area as a Vector 8; 1.5 Vector Algebra 10; 1.6 Constructing a Vector Perpendicular to a Given Vector 10; 1.7 Finding Velocity from Position 17; 1.8 Uniform Circular Motion 18; 1.9 Finding Velocity from Acceleration 19; 1.10 Motion in a Uniform Gravitational Field 21; 1.11 The Effect of Radio Waves on an Ionospheric Electron 21 1.12 Circular Motion and Rotat- ing Vectors 24; 1.13 Geometric Derivation of dr̂/dt and dθ̂/dt 30; 1.14 Circular Motion in Polar Coordinates 31; 1.15 Straight Line Motion in Polar Coordinates 32; 1.16 Velocity of a Bead on a Spoke 33; 1.17 Motion on an Off-center Circle 33; 1.18 Acceleration of a Bead on a Spoke 34; 1.19 Radial Motion without Acceleration 35 Chapter 2 NEWTON’S LAWS 2.1 Inertial and Non-inertial Systems 55; 2.2 Converting Units 63; 2.3 Astronauts’ Tug-of-War 67; 2.4 Multiple Masses: a Freight Train 69; 2.5 Examples of Constrained Motion 70; 2.6 Masses and Pulley 71; 2.7 Block and String 1 73; 2.8 Block and String 2 73; 2.9 The Whirling Block 74; 2.10 The Conical Pendulum 75 Chapter 3 FORCES AND EQUATIONS OF MOTION 3.1 Turtle in an Elevator 87; 3.2 Block and String 89; 3.3 Dangling Rope 90; 3.4 Block and Wedge with Friction 93; 3.5 The Spinning xviii LIST OF EXAMPLES Terror 94; 3.6 Whirling Rope 95; 3.7 Pulleys 97; 3.8 Terminal Veloc- ity 99; 3.9 Falling Raindrop 101; 3.10 Pendulum Motion 104; 3.11 Spring Gun and Initial Conditions 106 Chapter 4 MOMENTUM 4.1 The Bola 118; 4.2 Drum Major’s Baton 120; 4.3 Center of Mass of a Non-uniform Rod 122; 4.4 Center of Mass of a Triangular Plate 123; 4.5 Center of Mass Motion 124; 4.6 Exoplanets 125; 4.7 The Push Me–Pull You 128; 4.8 Spring Gun Recoil 130; 4.9 Measuring the Speed of a Bullet 132; 4.10 Rubber Ball Rebound 133; 4.11 How to Avoid Broken Ankles 135 4.12 Mass Flow and Momentum 136; 4.13 Freight Car and Hopper 138; 4.14 Leaky Freight Car 138; 4.15 Center of Mass and the Rocket Equation 139; 4.16 Rocket in Free Space 140; 4.17 Rocket in a Constant Gravitational Field 141; 4.18 Saturn V 142; 4.19 Slowing Atoms with Laser Light 144; 4.20 Reflection from an Irregular Object 147; 4.21 Solar Sail Spacecraft 148; 4.22 Pressure of a Gas 149; 4.23 Dike at the Bend of a River 150 Chapter 5 ENERGY 5.1 Mass Thrown Upward Under Constant Gravity 163; 5.2 Solving the Equation for Simple Harmonic Motion 164; 5.3 Vertical Motion in an Inverse Square Field 166; 5.4 The Conical Pendulum 171; 5.5 Escape Velocity—the General Case 171; 5.6 Empire State Building Run-Up 173; 5.7 The Inverted Pendulum 174; 5.8 Work by a Uniform Force 175; 5.9 Work by a Central Force 176; 5.10 A Path-dependent Line Integral 177; 5.11 Parametric Evaluation of a Line Integral 179 5.12 Energy Solution to a Dynamical Problem 180; 5.13 Potential Energy of a Uniform Force Field 182; 5.14 Potential Energy of a Central Force 183; 5.15 Potential Energy of the Three-Dimensional Spring Force 183; 5.16 Bead, Hoop, and Spring 184; 5.17 Block Sliding Down an Inclined Plane 188; 5.18 Heat Capacity of a Gas 191; 5.19 Conservation Laws and the Neutrino 193; 5.20 Energy and Water Flow from Hoover Dam 195 Chapter 6 TOPICS IN DYNAMICS 6.1 Molecular Vibrations 213; 6.2 Lennard-Jones Potential 214; 6.3 Small Oscillations of a Teeter Toy 216; 6.4 Stability of the Teeter Toy 218; 6.5 Energy Transfer Between Coupled Oscillators 221; 6.6 Nor- mal Modes of a Diatomic Molecule 222; 6.7 Linear Vibrations of Car- bon Dioxide 224; 6.8 Elastic Collision of Two Balls 228; 6.9 Limita- tions on Laboratory Scattering Angle 231 LIST OF EXAMPLES xix Chapter 7 ANGULAR MOMENTUM AND FIXED AXIS ROTATION 7.1 Angular Momentum of a Sliding Block 1 243; 7.2 Angular Mo- mentum of the Conical Pendulum 244; 7.3 Moments of Inertia of Some Simple Objects 247; 7.4 Torque due to Gravity 251; 7.5 Torque and Force in Equilibrium 252; 7.6 Central Force Motion and the Law of Equal Areas 253; 7.7 Capture Cross-section of a Planet 254; 7.8 An- gular Momentum of a Sliding Block 2 257; 7.9 Dynamics of the Coni- cal Pendulum 258; 7.10 Atwood’s Machine with a Massive Pulley 261; 7.11 Kater’s Pendulum 264; 7.12 Crossing Gate 265; 7.13 Angular Momentum of a Rolling Wheel 269; 7.14 Disk on Ice 271; 7.15 Drum Rolling down a Plane 272; 7.16 Drum Rolling down a Plane: Energy Method 275; 7.17 The Falling Stick 276 Chapter 8 RIGID BODY MOTION 8.1 Rotations through Finite Angles 292; 8.2 Rotation in the x−y Plane 295; 8.3 The Vector Nature of Angular Velocity 295; 8.4 Angular Mo- mentum of Masses on a Rotating Skew Rod 296; 8.5 Torque on the Ro- tating Skew Rod 298; 8.6 Torque on the Rotating Skew Rod (Geometric Method) 299; 8.7 Gyroscope Precession 302; 8.8 Why a Gyroscope Precesses 303; 8.9 Precession of the Equinoxes 304; 8.10 The Gyro- compass 305; 8.11 Gyrocompass Motion 307; 8.12 The Stability of Spinning Objects 309; 8.13 Rotating Dumbbell 314; 8.14 The Tensor of Inertia for a Rotating Skew Rod 316; 8.15 Why A Flying Saucer Is Better Than A Flying Cigar 318; 8.16 Dynamical Stability of Rigid Body Motion 325; 8.17 The Rotating Rod 327; 8.18 Euler’s Equations and Torque-free Precession 327 Chapter 9 NON-INERTIAL SYSTEMS AND FICTITIOUS FORCES 9.1 The Apparent Force of Gravity 345; 9.2 Cylinder on an Accelerating Plank 346; 9.3 Pendulum in an Accelerating Car 347; 9.4 The Driving Force of the Tides 349; 9.5 Equilibrium Height of the Tides 351; 9.6 Surface of a Rotating Liquid 360; 9.7 A Sliding Bead and the Coriolis Force 361; 9.8 Deflection of a Falling Mass 361; 9.9 Motion on the Rotating Earth 363; 9.10 Weather Systems 364; 9.11 The Foucault Pendulum 366 Chapter 10 CENTRAL FORCE MOTION 10.1 Central Force Description of Free-particle Motion 380; 10.2 How the Solar System Captures Comets 382; 10.3 Perturbed Circular Orbit 384; 10.4 Rutherford (Coulomb) Scattering 389; 10.5 Geostationary Orbit 394; 10.6 Satellite Orbit Transfer 1 395; 10.7 Satellite Orbit xx LIST OF EXAMPLES Transfer 2 397; 10.8 Trojan Asteroids and Lagrange Points 398; 10.9 Cosmic Keplerian Orbits and the Mass of a Black Hole 400 Chapter 11 THE HARMONIC OSCILLATOR 11.1 Incorporating Initial Conditions 413; 11.2 Physical Limitations to Damped Motion 417; 11.3 The Q of Two Simple Oscillators 419; 11.4 Graphical Analysis of a Damped Oscillator 420; 11.5 Driven Harmonic Oscillator Demonstration 423; 11.6 Harmonic Analyzer 426; 11.7 Vi- bration Attenuator 427 Chapter 12 THE SPECIAL THEORY OF RELATIVITY 12.1 Applying the Galilean Transformation 448; 12.2 Describing a Light Pulse by the Galilean Transformation 449; 12.3 Simultaneity 451; 12.4 The Role of Time Dilation in an Atomic Clock 456; 12.5 Time Di- lation, Length Contraction, and Muon Decay 460; 12.6 An Application of the Lorentz Transformation 461; 12.7 The Order of Events: Time- like and Spacelike Intervals 462; 12.8 The Speed of Light in a Moving Medium 465; 12.9 Doppler Navigation 468 Chapter 13 RELATIVISTIC DYNAMICS 13.1 Speed Dependence of the Electron’s Mass 480; 13.2 Relativistic Energy and Momentum in an Inelastic Collision 483; 13.3 The Equiva- lence of Mass and Energy 485; 13.4 The Photoelectric Effect 490; 13.5 The Pressure of Light 491; 13.6 The Compton Effect 492; 13.7 Pair Production 495; 13.8 The Photon Picture of the Doppler Effect 496; 13.9 The Photon Picture of the Gravitational Red Shift 497 Chapter 14 SPACETIME PHYSICS 14.1 Relativistic Addition of Velocities 511 1 VECTORS AND KINEMATICS 1.1 Introduction 2 1.2 Vectors 2 1.2.1 Definition of a Vector 2 1.3 The Algebra of Vectors 3 1.3.1 Multiplying a Vector by a Scalar 3 1.3.2 Adding Vectors 3 1.3.3 Subtracting Vectors 3 1.3.4 Algebraic Properties of Vectors 4 1.4 Multiplying Vectors 4 1.4.1 Scalar Product (“Dot Product”) 4 1.4.2 Vector Product (“Cross Product”) 6 1.5 Components of a Vector 8 1.6 Base Vectors 11 1.7 The Position Vector r and Displacement 12 1.8 Velocity and Acceleration 14 1.8.1 Motion in One Dimension 14 1.8.2 Motion in Several Dimensions 15 1.9 Formal Solution of Kinematical Equations 19 1.10 More about the Time Derivative of a Vector 22 1.10.1 Rotating Vectors 23 1.11 Motion in Plane Polar Coordinates 26 1.11.1 Polar Coordinates 27 1.11.2 dr̂/dt and dθ̂/dt in Polar Coordinates 29 1.11.3 Velocity in Polar Coordinates 31 1.11.4 Acceleration in Polar Coordinates 34 Note 1.1 Approximation Methods 36 Note 1.2 The Taylor Series 37 Note 1.3 Series Expansions of Some Common Functions 38 Note 1.4 Differentials 39 Note 1.5 Significant Figures and Experimental Uncertainty 40 Problems 41 2 VECTORS AND KINEMATICS 1.1 Introduction Mechanics is at the heart of physics; its concepts are essential for under- standing the world around us and phenomena on scales from atomic to cosmic. Concepts such as momentum, angular momentum, and energy play roles in practically every area of physics. The goal of this book is to help you acquire a deep understanding of the principles of mechanics. The reason we start by discussing vectors and kinematics rather than plunging into dynamics is that we want to use these tools freely in dis- cussing physical principles. Rather than interrupt the flow of discussion later, we are taking time now to ensure they are on hand when required. 1.2 Vectors The topic of vectors provides a natural introduction to the role of math- ematics in physics. By using vector notation, physical laws can often be written in compact and simple form. Modern vector notation was invented by a physicist, Willard Gibbs of Yale University, primarily to simplify the appearance of equations. For example, here is how New- ton’s second law appears in nineteenth century notation: F x = ma x Fy = may Fz = maz. In vector notation, one simply writes F = ma, where the bold face symbols F and a stand for vectors. Our principal motivation for introducing vectors is to simplify the form of equations. However, as we shall see in Chapter 14, vectors have a much deeper significance. Vectors are closely related to the fundamen- tal ideas of symmetry and their use can lead to valuable insights into the possible forms of unknown laws. 1.2.1 Definition of a Vector Mathematicians think of a vector as a set of numbers accompanied by rules for how they change when the coordinate system is changed. For our purposes, a down to earth geometric definition will do: we can think of a vector as a directed line segment. We can represent a vector graphi- cally by an arrow, showing both its scale length and its direction. Vectors are sometimes labeled by letters capped by an arrow, for instance A,  but we shall use the convention that a bold face letter, such as A, stands for a vector. To describe a vector we must specify both its length and its direction. Unless indicated otherwise, we shall assume that parallel translation does not change a vector. Thus the arrows in the sketch all represent the same vector. 1.3 THE ALGEBRA OF VECTORS 3 If two vectors have the same length and the same direction they are C equal. The vectors B and C are equal: B B = C. The magnitude or size of a vector is indicated by vertical bars or, if no confusion will occur, by using italics. For example,√ the magnitude of√A is written |A|, or simply A. If the length of A is 2, then |A| = A = 2. Vectors can have physical dimensions, for example distance, velocity, acceleration, force, and momentum. If the length of a vector is one unit, we call it a unit vector. A unit vector is labeled by a caret; the vector of unit length parallel to A is Â. It follows that A Â = A and conversely A = AÂ. The physical dimension of a vector is carried by its magnitude. Unit vectors are dimensionless. 1.3 The Algebra of Vectors We will need to add, subtract, and multiply two vectors, and carry out some related operations. We will not attempt to divide two vectors since the need never arises, but to compensate for this omission, we will define two types of vector multiplication, both of which turn out to be quite useful. Here is a summary of the basic algebra of vectors. 1.3.1 Multiplying a Vector by a Scalar C = bA If we multiply A by a simple scalar, that is, by a simple number b, the result is a new vector C = bA. If b > 0 the vector C is parallel to A, and A its magnitude is b times greater. Thus Ĉ = Â, and C = bA. −A If b < 0, then C = bA is opposite in direction (antiparallel) to A, and its magnitude is C = |b| A. 1.3.2 Adding Vectors Addition of two vectors has the simple geometrical interpretation shown A+B B by the drawing. The rule is: to add B to A, place the tail of B at the head of A by parallel translation of B. The sum is a vector from the tail of A to the head of B. A 1.3.3 Subtracting Vectors Because A − B = A + (−B), to subtract B from A we can simply multi- ply B by –1 and then add. The sketch shows how. 4 VECTORS AND KINEMATICS B −B A −B A B A A + (−B) = A − B A−B An equivalent way to construct A − B is to place the head of B at the head of A. Then A − B extends from the tail of A to the tail of B, as shown in the drawing. 1.3.4 Algebraic Properties of Vectors It is not difficult to prove the following: Commutative law A + B = B + A. Associative law A + (B + C) = (A + B) + C c(dA) = (cd)A. Distributive law c(A + B) = cA + cB (c + d)A = cA + dA. A A A+B B +A B B B= B B B+A A+ A A The sketch shows a geometrical proof of the commutative law A + B = B + A; try to cook up your own proofs of the others. 1.4 Multiplying Vectors Multiplying one vector by another could produce a vector, a scalar, or some other quantity. The choice is up to us. It turns out that two types of vector multiplication are useful in physics. 1.4.1 Scalar Product (“Dot Product”) The first type of multiplication is called the scalar product because the result of the multiplication is a scalar. The scalar product is an operation 1.4 MULTIPLYING VECTORS 5 that combines vectors to form a scalar. The scalar product of A and B is B written as A · B, therefore often called the dot product. A · B (referred to as “A dot B”) is defined by θ A A · B ≡ AB cos θ. Here θ is the angle between A and B when they are drawn tail to tail. B Because B cos θ is the projection of B along the direction of A, it follows θ A that Projection of A · B = A times the projection of B on A B on A = B times the projection of A on B. Note that A · A = |A|2 = A2. Also, A · B = B · A; the order does not change the value. We say that the dot product is commutative. If either A or B is zero, their dot product is zero. However, because cos π/2 = 0 the dot product of two non-zero vectors is nevertheless zero if the vectors happen to be perpendicular. A great deal of elementary trigonometry follows from the properties of vectors. Here is an almost trivial proof of the law of cosines using the dot product. Example 1.1 The Law of Cosines The law of cosines relates the lengths of three sides of a triangle to the B θ cosine of one of its angles. Following the notation of the drawing, the law of cosines is C φ C 2 = A2 + B2 − 2AB cos φ. A The law can be proved by a variety of trigonometric or geometric con- structions, but none is so simple and elegant as the vector proof, which merely involves squaring the sum of two vectors. C=A+B C · C = (A + B) · (A + B) = A · A + B · B + 2(A · B) C = A2 + B2 + 2AB cos θ. 2 Recognizing that cos φ = − cos θ completes the proof. Example 1.2 Work and the Dot Product The dot product has an important physical application in describing the work done by a force. As you may already know, the work W done F on an object by a force F is defined to be the product of the length θ d of the displacement d and the component of F along the direction of displacement. If the force is applied at an angle θ with respect to the displacement, as shown in the sketch, 6 VECTORS AND KINEMATICS then W = (F cos θ)d. Assuming that force and displacement can both be written as vectors, then W = F · d. 1.4.2 Vector Product (“Cross Product”) The second type of product useful in physics is the vector product, in which two vectors A and B are combined to form a third vector C. The symbol for vector product is a cross, so it is often called the cross product: C = A × B. The vector product is more complicated than the scalar product be- cause we have to specify both the magnitude and direction of the vec- tor A × B (called “A cross B”). The magnitude is defined as follows: if C=A×B B then θ A C = AB sin θ where θ is the angle between A and B when they are drawn tail to tail. To eliminate ambiguity, θ is always taken as the angle smaller than π. Even if neither vector is zero, their vector product is zero if θ = 0 or π, the situation where the vectors are parallel or antiparallel. It follows that z A×A=0 for any vector A. C Two vectors A and B drawn tail to tail determine a plane. Any plane can be drawn through A. Simply rotate it until it also contains B. y We define the direction of C to be perpendicular to the plane of A and A B B. The three vectors A, B, and C form what is called a right-hand triple. Imagine a right-hand coordinate system with A and B in the x−y plane x as shown in the sketch. A lies on the x axis and B lies toward the y axis. When A, B, and C form a right-hand triple, then C lies along the positive z axis. We shall always use right-hand coordinate systems such as the one shown. Here is another way to determine the direction of the cross product. Think of a right-hand screw with the axis perpendicular to A and B. 1.4 MULTIPLYING VECTORS 7 (A is into paper) If we rotate it in the direction that swings A into B, then C lies in the A direction the screw advances. (Warning: be sure not to use a left-hand screw. Fortunately, they are rare, with hot water faucets among the chief offenders. Your honest everyday wood screw is right-handed.) B A result of our definition of the cross product is that C=A×B B × A = −A × B. Here we have a case in which the order of multiplication is important. The vector product is not commutative. Since reversing the order re- verses the sign, it is anticommutative. Example 1.3 Examples of the Vector Product in Physics The vector product has a multitude of applications in physics. For instance, if you have learned about the interaction of a charged particle with a magnetic field, you know that the force is proportional to the charge q, the magnetic field B, and the velocity of the particle F v. The force varies as the sine of the angle between v and B, and B is perpendicular to the plane formed by v and B, in the direction indicated. q v All these rules are combined in the one equation F = qv × B. τ=r×F F Another application is the definition of torque, which we shall develop θ in Chapter 7. For now we simply mention in passing that the torque r vector τ is defined by τ = r × F, F sin θ F where r is a vector from the axis about which the torque is evaluated to the point of application of the force F. This definition is consistent with θ the familiar idea that torque is a measure of the ability of an applied r force to produce a twist. Note that a large force directed parallel to r produces no twist; it merely pulls. Only F sin θ, the component of force perpendicular to r, produces a torque. Top view Imagine that we are pushing open a garden gate, where the axis of rota- tion is a vertical line through the hinges. When we push the gate open, we instinctively apply force in such a way as to make F closely perpen- dicular to r , to maximize the torque. Because the torque increases as the lever arm gets larger, we push at the edge of the gate, as far from the hinge line as possible. As you will see in Chapter 7, the natural direction of τ is along the axis of the rotation that the torque tends to produce. All these ideas are summarized in a nutshell by the simple equation τ = r × F. 8 VECTORS AND KINEMATICS Example 1.4 Area as a Vector D We can use the cross product to describe an area. Usually one thinks D sin θ of area in terms of magnitude only. However, many applications in θ physics require that we also specify the orientation of the area. For example, if we wish to calculate the rate at which water in a stream C flows through a wire loop of given area, it obviously makes a difference whether the plane of the loop is perpendicular or parallel to the flow. A (If parallel, the flow through the loop is zero.) Here is how the vector D product accomplishes this: nˆ Consider the area of a quadrilateral formed by two vectors C and D. C The area A of the parallelogram is given by A = base × height = CD sin θ = |C × D|. The magnitude of the cross product gives us the area of the parallel- ogram, but how can we assign a direction to the area? In the plane of the parallelogram we can draw an infinite number of vectors pointing every which-way, so none of these vectors stands out uniquely. The only unique preferred direction is the normal to the plane, specified by a unit vector n̂. We therefore take the vector A describing the area as parallel to n̂. The magnitude and direction of A are then given com- pactly by the cross product A = C × D. A minor ambiguity remains, because n̂ can point out from either side of the area. We could just as well have defined the area by A = D × C = −C × D, as long as we are consistent once the choice is made. 1.5 Components of a Vector The fact that we have discussed vectors without introducing a particular coordinate system shows why vectors are so useful; vector operations are defined independently of any particular coordinate system. However, eventually we have to translate our results from the abstract to the con- crete, and at this point we have to choose a coordinate system in which to work. The combination of algebra and geometry, called analytic geometry, is a powerful tool that we shall use in many calculations. Analytic geometry has a consistent procedure for describing geometrical objects by a set of numbers, greatly easing the task of performing quantitative calculations. With its aid, students still in school can routinely solve problems that would have taxed the ancient Greek geometer Euclid. Analytic geometry was developed as a complete subject in the first half of the seventeenth 1.5 COMPONENTS OF A VECTOR 9 century by the French mathematician René Descartes, and independently by his contemporary Pierre Fermat. y For simplicity, let us first restrict ourselves to a two-dimensional sys- tem, the familiar x−y plane. The diagram shows a vector A in the x−y plane. Ay A The projections of A along the x and y coordinate axes are called the components of A, A x and Ay , respectively. The magnitude of A is A =  θ A x 2 + Ay 2 , and the direction of A makes an angle θ = arctan (Ay /A x ) with the x axis. Since its components define a vector, we can specify a vector entirely x by its components. Thus Ax A = (A x , Ay ) or, more generally, in three dimensions, A = (A x , Ay , Az ).  Prove for yourself that A = A x 2 + Ay 2 + Az 2. If two vectors are equal A = B, then in the same coordinate system their corresponding components are equal. A x = Bx Ay = By Az = Bz. The single vector equation A = B symbolically represents three scalar equations. y The vector A has a meaning independent of any coordinate system. However, the components of A depend on the coordinate system being used. To illustrate this, here is a vector A drawn in two different coordi- A nate systems. In the first case, A = (A, 0) (x, y system), x while in the second x′ A = (0, −A) (x , y system). A All vector operations can be written as equations for components. For instance, multiplication by a scalar is written cA = (cA x , cAy , cAz ). y′ The law for vector addition is A + B = (A x + Bx , Ay + By , Az + Bz ). By writing A and B as the sums of vectors along each of the coordinate axes, you can verify that A · B = A x Bx + Ay By + Az Bz. We shall defer evaluating the cross product until the next section. 10 VECTORS AND KINEMATICS Example 1.5 Vector Algebra Let A = (3, 5, −7) B = (2, 7, 1). Find A + B, A − B, A, B, A · B, and the cosine of the angle between A and B. A + B = (3 + 2, 5 + 7, −7 + 1) = (5, 12, −6) A − B = (3 − 2, 5 − 7, −7 − 1) = (1, −2, −8)  A = (32 + 52 + 72 ) √ = 83 ≈ 9.11  B = (22 + 72 + 12 ) √ = 54 ≈ 7.35 A·B=3×2+5×7−7×1 = 34 A·B 34 cos(A, B) = ≈ ≈ 0.508. AB (9.11)(7.35) Example 1.6 Constructing a Vector Perpendicular to a Given Vector The problem is to find a unit vector lying in the x−y plane that is perpendicular to the vector A = (3, 5, 1). A vector B in the x−y plane has components (Bx , By ). For B to be perpendicular to A, we must have A · B = 0: A · B = 3Bx + 5By = 0. Hence By = − 35 Bx. For B to be a unit vector, Bx 2 + By 2 = 1. Combining these gives 9 2 Bx 2 + Bx = 1, 25 or  25 Bx = 34 ≈ ±0.858 By = − 35 Bx ≈ ∓0.515. 1.6 BASE VECTORS 11 There are two solutions, given by the upper and lower signs. Each vec- tor is the negative of the other, so they are equal in magnitude but point in opposite directions. 1.6 Base Vectors Base vectors are a set of orthogonal (mutually perpendicular) unit vec- tors, one for each dimension. For example, if we are dealing with the z familiar Cartesian coordinate system of three dimensions, the base vec- tors lie along the x, y, and z axes. We shall designate the x unit vector k by î, the y unit vector by ĵ, and the z unit vector by k̂. (Sometimes the symbols x̂, ŷ, and ẑ are used.) jˆ y The base vectors have the following properties, as you can readily iˆ verify: î · î = ĵ · ĵ = k̂ · k̂ = 1 x î · ĵ = ĵ · k̂ = k̂ · î = 0 î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ. As shown in the drawing, we can write any three-dimensional vector in terms of its components and the base vectors: z A = A x î + Ay ĵ + Az k̂ Az k̂ To find the component of a vector in any direction, take the dot product with a unit vector in that direction. For instance, the z component of A vector A is Ay jˆ Az = A · k̂. y The base vectors are particularly useful in deriving the general rule for Axiˆ evaluating the cross product of two vectors in terms of their components: x A × B = (A x î + Ay ĵ + Az k̂) × (Bx î + By ĵ + Bz k̂). Consider the first term: A x î × B = A x Bx (î × î) + A x By (î × ĵ) + A x Bz (î × k̂). (The associative law holds here.) Because î × î = 0, î × ĵ = k̂, and î × k̂ = −ĵ, we find A x î × B = A x (By k̂ − Bz ĵ). The same argument applied to the y and z components gives Ay ĵ × B = Ay (Bz î − Bx k̂) Az k̂ × B = Az (Bx ĵ − By î). 12 VECTORS AND KINEMATICS A quick way to derive these relations is to work out the first and then to obtain the others by cyclically permuting x, y, z, and î, ĵ, k̂ (that is, x → y, y → z, z → x, and î → ĵ, ĵ → k̂, k̂ → î). A compact mnemonic for expressing this result is to write the base vectors and the components of A and B as three rows of a determinant, like this:    î ĵ k̂  A × B = A x Ay Az    Bx By Bz  = (Ay Bz − Az By ) î − (A x Bz − Az Bx ) ĵ + (A x By − Ay Bx ) k̂. For instance, if A = î + 3 ĵ − k̂ and B = 4 î + ĵ + 3 k̂, then    î ĵ k̂  A × B = 1 3 −1   4 1 3 = 10 î − 7 ĵ − 11 k̂. 1.7 The Position Vector r and Displacement So far we have discussed only abstract vectors. However, the reason for introducing vectors is that many physical quantities are conveniently de- scribed by vectors, among them velocity, force, momentum, and gravi- tational and electric fields. In this chapter we shall use vectors to discuss kinematics, which is the description of motion without regard for the causes of the motion. Dynamics. which we shall take up in Chapter 2, looks at the causes of motion. z Kinematics is largely geometric and perfectly suited to characteriza- 5 tion by vectors. Our first application of vectors will be to the description 4 of position and motion in familiar three-dimensional space. 3 To locate the position of a point in space, we start by setting up a co- 2 ordinate system. For convenience we choose a three-dimensional Carte- 1 sian system with axes x, y, and z, as shown. In order to measure position, the axes must be marked in some convenient unit of length—meters, for y 0 1 2 3 4 instance. The position of the point of interest is given by listing the val- 1 2 ues of its three coordinates, x1 , y1 , z1 , which we can write compactly as 3 a position vector r(x1 , y1 , z1 ) or more generally as r(x, y, z). This nota- x 4 tion can be confusing because we normally label the axes of a Cartesian coordinate system by x, y, z. However, r(x, y, z) is really shorthand for r(x-axis, y-axis, z-axis). The components of r are the coordinates of the point referred to the particular coordinate axes. The three numbers (x, y, z) do not represent the components of a vec- tor according to our previous discussion because they specify only the position of a single point, not a magnitude and direction. Unlike other physical vectors such as force and velocity, r is tied to a particular coor- dinate system. 1.7 THE POSITION VECTOR R AND DISPLACEMENT 13 The position of an arbitrary point P at (x, y, z) is written as r = (x, y, z) = xî + yĵ + zk̂. z If we move from the point x1 , y1 , z1 to some new position, x2 , y2 , z2 , then the displacement defines a true vector S with coordinates S x = x2 − x1 , S y = y2 − y1 , S z = z2 − z1. (x2, y2, z2) S S is a vector from the initial position to the final position—it defines the displacement of a point of interest. Note, however, that S contains no information about the initial and final positions separately—only about (x1, y1, z1) y the relative position of each. Thus, S z = z2 − z1 depends on the differ- ence between the final and initial values of the z coordinates; it does not specify z2 or z1 separately. Thus S is a true vector: the values of the co- x ordinates of its initial and final points depend on the coordinate system but S does not, as the sketches indicate. (x2′, y2′, z2′) y′ (x2, y2, z2,) y S x′ (x1′, y1′, z1′) (x1, y1, z1) x z′ z One way in which our displacement vector differs from vectors in pure mathematics is that in mathematics, vectors are usually pure quantities, with components described by simple numbers, whereas the magnitude S has the physical dimension of length associated with it. We will use the convention that the physical dimension of a vector is attached to its z′ magnitude, so that the associated unit vector is dimensionless. Thus, a P displacement of 8 m (8 meters) in the x direction is S = (8 m, 0, 0). S = z r′ 8 m, and Ŝ = S/S = î. r The sketch shows position vectors r and r indicating the position of R y′ the same point in space but drawn in different coordinate systems. If R x′ is the vector from the origin of the unprimed coordinate system to the y origin of the primed coordinate system, we have r = R + r , or alterna- x tively, r = r − R. 14 VECTORS AND KINEMATICS We use these results to show that displacement S, a true vector, is independent of coordinate system. As the sketch indicates, S = r2 − r1 = (R + r2 ) − (R + r1 ) = r2 − r1. 1.8 Velocity and Acceleration 1.8.1 Motion in One Dimension Before employing vectors to describe velocity and acceleration in three z′ dimensions, it may be helpful to review one-dimensional motion: motion along a straight line. z r ′2 Let x be the value of the coordinate of a particle moving on a line, with S x measured in some convenient unit such as meters. We assume that we r2 r ′1 have a continuous record of position versus time. y′ r1 The average velocity v of the point between two times t1 and t2 is R x′ defined by y x(t2 ) − x(t1 ) v=. t 2 − t1 x (We shall generally use a bar to indicate the time average of a quantity.) The instantaneous velocity v is the limit of the average velocity as the time interval approaches zero: x(t + Δt) − x(t) v = lim. Δt→0 Δt The limit we introduced in defining v is exactly the definition of a deriva- tive in calculus. In the latter half of the seventeenth century Isaac Newton invented calculus to give him the tools he needed to analyze change and motion, particularly planetary motion, one of his greatest achievements in physics. We therefore write dx v= , dt using notation due to Gottfried Leibniz, who independently invented calculus. Newton would have written v = ẋ where the dot stands for d/dt. Following a convention frequently used in physics, we shall use Newton’s notation only for derivatives with respect to time. The derivative of a function f (x) can also be written f  (x) ≡ d f (x)/dx. In a similar fashion, the instantaneous acceleration a is v(t + Δt) − v(t) a = lim Δt→0 Δt dv = = v̇. dt 1.8 VELOCITY AND ACCELERATION 15 Using v = dx/dt, d2 x a= = ẍ. dt2 Here d2 x/dt2 is called the second derivative of x with respect to t. The concept of speed is sometimes useful. Speed s is simply the mag- nitude of the velocity: s = |v|. In one dimension, speed and velocity are synonymous. 1.8.2 Motion in Several Dimensions Our task now is to extend the ideas of velocity and acceleration to several dimensions using vector notation. Consider a particle moving in the x−y plane. As time goes on, the particle traces out a path. We assume that we know the particle’s coordinates at every value of time. The instantaneous position of the particle at some time t1 is r(t1 ) = (x(t1 ), y(t1 )) or r(t1 ) = (x1 , y1 ) Position at time t2 (x2, y2) r2 r2 r2 – r1 (x1, y1) Position r1 r1 at time t1 where x1 is the value of x at t = t1 , and so forth. At time t2 the position is similarly r(t2 ) = (x2 , y2 ). The displacement of the particle between times t1 and t2 is r(t2 ) − r(t1 ) = (x2 − x1 , y2 − y1 ). ) We can generalize our example by considering the position at some Δt Δr t+ time t and also at some later time t + Δt. We put no restrictions on the r( size of Δt—it can be as large or as small as we please. The displacement of the particle during the interval Δt is r(t) Δr = r(t + Δt) − r(t). 16 VECTORS AND KINEMATICS y This vector equation is equivalent to the two scalar equations y(t + Δt) Δx = x(t + Δt) − x(t) Δy = y(t + Δt) − y(t). Δy Δr The velocity v of the particle as it moves along the path is y(t) Δr v = lim Δx Δt→0 Δt x dr x(t) x(t + Δt) = , dt which is equivalent to the two scalar equations Δx dx v x = lim = Δt→0 Δt dt Δy dy vy = lim =. Δt→0 Δt dt Extension of the argument to three dimensions is trivial. The third com- ponent of velocity is z(t + Δt) − z(t) dz vz = lim =. Δt→0 Δt dt Our definition of velocity as a vector is a straightforward generaliza- tion of the familiar concept of motion in a straight line. Vector notation allows us to describe motion in three dimensions with a single equation, a great economy compared with the three equations we would need oth- erwise. The equation v = dr/dt expresses concisely the results we have just found. An alternative approach to calculating the velocity is to start with the definition r = x î + y ĵ + z k̂, and then differentiate: dr d(x î + y ĵ + z k̂) =. dt dt To evaluate this expression, we use a key property of vectors—they can change with time in magnitude or in direction or in both. But base vectors are unit vectors and therefore have constant magnitude, so they cannot change in magnitude. The Cartesian base vectors also have the special property that they are fixed in direction, and therefore cannot change direction. Hence we can treat the Cartesian base vectors as constants when we differentiate: dr dx dy dz = î + ĵ + k̂ dt dt dt dt as before. Similarly, acceleration a is defined by dv dv x dvy dvz a= = î + ĵ + k̂ dt dt dt dt d2 r = 2. dt 1.8 VELOCITY AND ACCELERATION 17 Δt″′ > Δt″ > Δt ′ We could continue to form new vectors by taking higher derivatives of v(t ) r, but in the study of dynamics it turns out that r, v, and a are of chief interest. ′) Let the particle undergo a displacement Δr in time Δt. In the limit Δt ″ ″) Δt Δt → 0, Δr becomes tangent to the trajectory, as the sketch indicates. + Δt ′) + r(t r(t + r(t r(t ) The relation dr Δr ≈ Δt dt = vΔt becomes exact in the limit Δt → 0, and shows that v is parallel to Δr; the instantaneous velocity v of a particle is everywhere tangent to the trajectory. Example 1.7 Finding Velocity from Position Suppose that the position of a particle is given by r = A(eαt î + e−αt ĵ), where A and α are constants. Find the velocity, and sketch the trajec- tory. dr v= dt = A(αeαt î − αe−αt ĵ) or v x = Aαeαt vy = −Aαe−αt. The magnitude of v is  v= v x 2 + vy 2 √ = Aα e2αt + e−2αt. To sketch the trajectory it is often helpful to look at limiting cases. At t = 0, we have r(0) = A (î + ĵ) v(0) = αA (î − ĵ). Note that v(0) is perpendicular to r(0). y A v(0) r(0) r(t ) v(t ) Trajectory ĵ A v(t >> 0) x î 18 VECTORS AND KINEMATICS As t → ∞, eαt → ∞ and e−αt → 0. In this limit r → Aeαt î, which is a vector along the x axis, and v → αAeαt î; in this unrealistic exam- ple, the point rushes along the x axis and the speed increases without limit. y Example 1.8 Uniform Circular Motion Circular motion plays an important role in physics. Here we look at the simplest and most important case—uniform circular motion, which is circular motion at constant speed. x = r cos ωt r y = r sin ωt Consider a particle moving in the x−y plane according to r = ĵ ωt r(cos ωt î + sin ωt ĵ), where r and ω are constants. Find the trajectory, x the velocity, and the acceleration. î y  |r|= r2 cos2 ωt + r2 sin2 ωt. r Using the familiar identity sin2 θ + cos2 θ = 1, ωt | r | = r = constant. x The trajectory is a circle. The particle moves counterclockwise around the circle, starting from (r, 0) at t = 0. It traverses the circle in a time T such that ωT = 2π. ω is called the angular speed (or less precisely the angular velocity) of the motion and is measured in radians per second. T, the time required y to execute one complete cycle, is called the period. v dr v= dt r = rω(− sin ωt î + cos ωt ĵ). ωt x We can show that v is tangent to the trajectory by calculating v · r: v · r = r2 ω(− sin ωt cos ωt + cos ωt sin ωt) = 0. Because v is perpendicular to r, the motion is tangent to the circle, as we expect. It is easy to show that the speed |v| = r ω is constant. y dv a= dt r a = r ω2 (− cos ωt î − sin ωt ĵ) ωt = −ω2 r. x The acceleration is directed radially inward and is known as the centripetal acceleration. We shall have more to say about it later in this chapter when we look at how motion is described in polar coordinates. 1.9 FORMAL SOLUTION OF KINEMATICAL EQUATIONS 19 1.9 Formal Solution of Kinematical Equations Dynamics, which we shall take up in Chapter 2, enables us to find the acceleration of a body if we know the interactions. Once we have the acceleration, finding the velocity and position is a simple matter of inte- gration. Here is the formal integration procedure. If the acceleration is a known function of time, the velocity can be found from the defining equation dv(t) = a(t) dt by integration with respect to time. Writing this equation in more detail, we have dv x dvy dvz î + ĵ + k̂ = a x î + ay ĵ + az k̂. dt dt dt We can separate the corresponding components on each side into sepa- rate equatio