Introduction to Mechanics PDF

 AN Daniel Kleppner Associate Professor of Physics Massachusetts Institute of Technology Robert J. Kolenkow Formerly Associate Professor of Physics, Massachusett s INTRODUCTION...

 AN Daniel Kleppner Associate Professor of Physics Massachusetts Institute of Technology Robert J. Kolenkow Formerly Associate Professor of Physics, Massachusett s INTRODUCTION TO Institute of Technology MECHANICS Boston, Massachusetts Burr Ridge, Illinois Dubuque, Iowa Madison, Wisconsin New York, New York San Francisco, California St. Louis, Missouri  McGraw-Hill A Division ofTheMcGraw·HiUCompanies AN Copyright © 1973 by McGraw-Hili, Inc. All rights reserved. PrInted in the United States of America. Except as permitted INTRODUCTION under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval TO system, without the prior written permission of the publisher. MECHANICS Printed and bound by Book-mart Press, Inc. 30 31 32 33 34 35 36 37 38 39 BKM BKM 098 This book was set in News Gothic by The Maple Press Company.' The editors were Jack L. Farnsworth and J. W. Maisel; the designer was Edward A. Butler; and the production supervisor was Sally Ellyson. The drawings were done by Felix Cooper. Library of Congress Cataloging in Publication Data Kleppner. Daniel. An introduction to mechanics. 1. Mechanics. I. Kolenkow. Robert. joint author. II Title. QA805.K62 531 72·11770 ISBN·13: 978·0'()7·035048-9 ISBN·IO: 0·07·035048·5 To our parents Beatrice and Otto Katherine and John CONTENTS LIST OF EXAMPLES xi PREFACE xv TO THE TEACHER xix 1 VECTORS 1.1 INTRODUCTION 2 AND 1.2 VECTORS 2 KINEMATICS Definition of a Vector, The Algebra of Vectors, 3. -A FEW 1.3 COMPONENTS OF A VECTOR 8 MATHEMATICAL 1.4 BASE VECTORS 10 PREll MINARIES 1.5 DISPLACEMENT AND THE POSITION VECTOR 11 1.6 VELOCITY AND ACCELERATION 13 Motion in One Dimension, 14; Motion in Several Dimensions, 14; A Word about Dimensions and Units, 18. 1.7 FORMAL SOLUTION OF KINEMATICAL EQ UATIONS 9 1.8 MORE ABOUT THE DERIVATIVE OF A VECTOR 23 1.9 MOTION IN PLANE POLAR COORDINATES 27 Polar Coordinates, 27; Velocity in Polar Coordinates, 27; Evaluating d;jdt, 31; Acceleration in Polar Coordinates, 36. Note 1.1 MATHEMATICAL APPROXIMATION METHODS 39 The Binomial Series, 41; Taylor's Series, 42; Differentials, 45. Some References to Calculus Texts, 47. PROBLEMS 47 2 NEWTON'S 2.1 INTRODUCTION 52 LAWS-THE 2.2 NEWTON'S LAWS 53 FOUNDATIONS Newton's First Law, 55; Newton's Second Law, 56; Newton's Third Law, 5!1. OF 2.3 STANDARDS AND UNITS 64 NEWTONIAN The Fundamental Standards, 64; Systems of Units, 67. MECHANICS 2.4 SOME APPLICATIONS OF NEWTON'S LAWS 68 2.5 THE EVERYDAY FORCES OF PHYSICS 79 Gravity, Weight, and the Gravitational Field, 80; The Electrostatic Force, 86; Contact Forces, 87; Tension-The Force of a String, 87; Tension and Atomic Forces, 91; The Normal Force, 92; Friction, 92; Viscosity, 95; The Linear Restoring Force: Hooke's Law, the Spring, and Simple Harmonic Motion, 97. Note 2.1 THE GRAVITATIONAL ATTRACTION OF A SPHERICAL SHELL 101 PROBLEMS 103 3 MOMENTUM 3.1 INTRODUCTION 112 3.2 DYNAMICS OF A SYSTEM OF PARTICLES 113 Center of Mass, 116. 3.3 CONSERVATION OF MOMENTUM 122 Center of Mass Coordinates, 127. 3.4 IMPULSE AND A RESTATEMENT OF THE MOMENTI RELATION 130 3.5 MOMENTUM AND THE FLOW OF MASS 133 viii CONTENTS 3.6 MOMENTUM TRANSPORT 139 Note 3.1 CENTER OF MASS 145 PROBLEMS 147 4 WORK 4.1 INTRODUCTION 152 AND 4.2 INTEGRATING THE EQUATION OF MOTION IN ONE ENERGY DIMENSION 153 4.3 THE WORK·ENERGY THEOREM IN ONE DIMENSION 156 4.4 INTEGRATING THE EQUATION OF MOTION IN SEVERAL DIMENSIONS 158 4.5 THE WORK·ENERGY THEOREM 160 4.6 APPLYING THE WORK·ENERGY THEOREM 162 4.7 POTENTIAL ENERGY 168 /IIustrations of Potential Energy, 170. 4.8 WHAT POTENTIAL ENERGY TI1...LS US ABOUT FORCE 173 Stability, 174. 4.9 ENERGY DIAGRAMS 176 4.10 SMALL OSCILLATIONS IN A BOUND SYSTEM 178 4.11 NONCONSERVATIVE FORCES 182 4.12 THE GENERAL LAW OF CONSERVATION OF ENERGY 184 4.13 POWER 186 4.14 CONSERVATION LAWS AND PARTICLE COLLISIONS 187 Collisions and Conservation Laws, 188; Elastic and Inelastic Collisions, 188; Collisions in One Dimension, 189; Collisions and Center of Mass Coordinates, 190. PROBLEMS 194 5 SOME 5.1 INTRODUCTION 202 MATHEMATICAL 5.2 PARTIAL DERIVATIVES 202 ASPECTS 5.3 HOW TO FIND THE FORCE IF YOU KNOW THE POTENTIAL OF FORCE ENERGY 206 AND 5.4 THE GRADIENT OPERATOR 207 ENERGY 5.5 THE PHYSICAL MEANING OF THE GRADIENT 210 Constant Energy Surfaces and Contour Lines, 211. 5.6 HOW TO FIND OUT IF A FORCE IS CONSERVATIVE 215 5.7 STOKES' THEOREM 225 PROBLEMS 228 6 ANGULAR 6.1 INTRODUCTION 232 MOMENTUM 6.2 ANGULAR MOMENTUM OF A PARTICLE 233 AND FIXED AXIS 6.3 TORQUE 238 ROTATION 6.4 ANGULAR MOMENTUM AND FIXED AXIS ROTATION 248 6.5 DYNAMICS OF PURE ROTATION ABOUT AN AXIS 253 6.6 THE PHYSICAL PENDULUM 255 The Simple Pendulum, 253; The Physical Pendulum, 257. 6.7 MOTION INVOLVING BOTH TRANSLATION AND ROTATION 260 The Work·energy Theorem, 267. 6.8 THE BOHR ATOM 270 Note 6.1 CHASLES' THEOREM 274 Note 6.2 PENDULUM MOTION 276 PROBLEMS 279  CONTENTS ix 7 RIGID BODY 7.1 INTRODUCTION 288 MOTION 7.2 THE VECTOR NATURE OF ANGULAR VELOCITY AND AND THE ANGULAR MOMENTUM 288 CONSERVATION 7.3 THE GYROSCOPE 295 OF 7.4 SOME APPLICATIONS OF GYROSCOPE MOTION 300 ANGULAR 7.5 CONSERVATION OF ANGULAR MOMENTUM 305 MOMENTUM 7.6 ANGULAR MOMENTUM OF A ROTATING RIGID BODY 308 Angular Momentum and the Tensor of Inertia, 308; Principal Axes, 313; Rotational Kinetic Energy, 313; Rotation about a Fixed Point, 315. 7.7 ADVANCED TOPICS IN THE DYNAMICS OF RIGID BODY ROTATION 316 Introduction, 316; Torque-free Precession: Why the Earth Wobbles, 317; Euler's Equations, 320. Note 7.1 FINITE AND INFINITESIMAL ROTATIONS 326 Note 7.2 MORE ABOUT GYROSCOPES 328 Case 1 Uniform Precession, 331; Case 2 Torque-free Precession, 331; Case 3 Nutation, 331. PROBLEMS 334 8 NONINERTIAL 8.1 INTRODUCTION 340 SYSTEMS 8.2 THE GALILEAN TRANSFORMATIONS 340 AND 8.3 UNIFORMLY ACCELERATING SYSTEMS 343 FICTITIOUS 8.4 THE PRINCIPLE OF EQUIVALENCE 346 FORCES 8.5 PHYSICS IN A ROTATING COORDINATE SYSTEM 355 Time Derivatives and Rotating Coordinates, 356; Acceleration Relative to Rotating Coordinates, 358; The Apparent Force in a Rotating Coordinate System, 359. Note 8.1 THE EQUIVALENCE PRINCIPLE AND THE GRAVITATIONAL RED SHIFT 369 Note 8.2 ROTATING COORDINATE TRANSFORMATION 371 PROBLEMS 372 9 CENTRAL 9.1 INTRODUCTION 378 FORCE 9.2 CENTRAL FORCE MOTION AS A ONE BODY PROBLEM 378 MOTION 9.3 GENERAL PROPERTIES OF CENTRAL FORCE MOTION 380 The Motion Is Confined to a Plane, 380; The Energy and Angular Momentum Are Constants of the Motion, 380; The Law of Equal Areas, 382. 9.4 FINDING THE MOTION IN REAL PROBLEMS 382 9.5 THE ENERGY EQUATION AND ENERGY DIAGRAMS 383 9.6 PLANETARY MOTION 390 9.7 KEPLER'S LAWS 400 Note 9.1 PROPERTIES OF THE ELLIPSE 403 PROBLEMS 406 10 THE 10.1 INTRODUCTION AND REVIEW 410 HARMONIC Standard Form of the Solution, 410; Nomenclature, 411; Energy Considerations, OSCILLATOR 412; Time Average Values, 413; Average Energy, 413. 10.2 THE DAMPED HARMONIC OSCILLATOR 414 Energy, 416; The Q of an Oscillator, 418. x CONTENTS 10.3 THE FORCED HARMONIC OSCILLATOR 421 The Undamped Forced Oscillator, 421; Resonance, 423; The Forced Damped Harmonic Oscillator, 424; Resonance in a Lightly Damped System: The Quality Factor Q, 426. 10.4 RESPONSE IN TIME VERSUS RESPONSE IN FREQUENCY 432 Note 10.1 SOLUTION OF THE EQUATION OF MOTION FOR THE UNDRIVEN DAMPED OSCILLATOR 433 The Use of Complex Variables, 433; The Damped Oscillator, 435. Note 10.2 SOLUTION OF THE EQUATION OF MOTION FOR THE FORCED OSCILLATOR 437 PROBLEMS 438 11 THE 11.1 THE NEED FOR A NEW MODE OF THOUGHT 442 SPECIAL 11.2 THE MICHELSON·MORLEY EXPERIMENT 445 THEORY 11.3 THE POSTULATES OF SPECIAL ELATIVITY 450 OF The Universal Velocity, 451; The Principle of Relativity, 451; The Postulates of RELATIVITY Special Relativity, 452. 11.4 THE GALILEAN TRANSFORMATIONS 453 11.5 THE LORENTZ TRANSFORMATIONS 455 PROBLEMS 459 1Z RELATIVISTIC 12.1 INTRODUCTION 462 KINEMATICS 12.2 SIMULTANEITY AND THE ORDER OF EVENTS 463 12.3 THE LORENTZ CONTRACTION AND TIME DILATION 466 The Lorentz Contraction, 466; Time Dilation, 468. 12.4 THE RELATIVISTIC TRANSFORMATION OF VELOCITY 472 12.5 THE DOPPLER EFFECT 475 The Doppler Shift in Sound, 475; Relativistic Doppler Effect, 477; The Doppler Effect for an Observer off the Line of Motion, 478. 12.6 THE TWIN PARADOX 480 PROBLEMS 484 13 RELATIVISTIC 13.1 MOMENTUM 490 MOMENTUM 13.2 ENERGY 493 AND 13.3 MASSLESS PARTICLES 500 ENERGY 13.4 DOES LIGHT TRAVEL AT THE VELOCITY OF LIGHT? 508 PROBLEMS 512 14 FOUR· 14.1 INTRODUCTION 516 VECTORS 14.2 VECTORS AND TRANSFORMATIONS 516 AND Rotation about the z Axis, 517; Invariants of a Transformation, 520; The Trans· RELATIVISTIC formation Properties of Physical Laws, 520; Scalar Invariants, 521. INVARIANCE 14.3 MINIKOWSKI SPACE AND FOUR·VECTORS 521 14.4 THE MOMENTUM·ENERGY FOUR·VECTOR 527 14.5 CONCLUDING REMARKS 534 PROBLEMS 536 INDEX 539  LIST OF EXAMPLES, CHAPTER 1 1.1 Law of Cosines, 5; 1.2 Work and the Dot Product, 5; 1.3 Examples of the Vector Product in Physics, 7; 1.4 Area as a Vector, 7. Vector Algebra, 9; 1.6 Construction of a Perpendicular Vector, 10. EXAMPLES 1.5 1.7 Finding v from r, 16; 1.8 Uniform Circular Motion, 17. 1.9 Finding Velocity from Acceleration, 20; 1.10 Motion in a Uniform Gravi­ tational Field, 21; 1.11 Nonuniform Acceleration-The Effect of a Radio 1 VECTORS Wave on an Ionospheric Electron, 22. AND 1.12 Circular Motion and Rotating Vectors, 25. KINEMATICS 1.13 Circular Motion and Straight line Motion in Polar Coordinates, 34; -A FEW 1.14 Velocity of a Bead on a Spoke, 35; 1.15 Off-center Circle, 35; 1.16 Ac­ MATHEMATICAL celeration of a Bead on a Spoke, 37; 1.17 Radial Motion without Accelera­ PRELl MINARIES tion, 38. 2 NEWTON'S EXAMPLES, CHAPTER 2 LAWS-THE 2.1 Astronauts in Space-Inertial Systems and Fictitious Force, 60. FOUNDATIONS 2.2 The Astronauts' Tug-of-war, 70; 2.3 Freight Train, 72; 2.4 Constraints, OF 74; 2.5 Block on String I, 75;.6 Block on String 2, 76; 2.7 The Whirling NEWTONIAN Block, 76; 2.8 The Conical Pendulum, 77. MECHANICS 2.9 Turtle in an Elevator, 84; 2.10 Block and String 3, 87; 2.11 Dangling Rope, 88; 2.12 Whirling Rope, 89; 2.13 Pulleys, 90; 2.14 Block and Wedge with Friction, 93; 2.15 The Spinning Terror, 94; 2.16 Free Motion in a Viscous Medium, 96; 2.17 Spring and Block-The Equation for Simple Harmonic Motion, 98; 2.18 The Spring Gun-An Example Illustrating I nitial Conditions, 99. 3 MOMENTUM EXAMPLES, CHAPTER 3 3.1 The Bola, 115; 3.2 Drum Maj or's Baton, 117; 3.3 Center of Mass of a Nonuniform Rod, 119; 3.4 Center of Mass of a Triangular Sheet, 120; 3.5 Center of Mass Motion, 122. 3.6 Spring Gun Recoil, 123; 3.7 Earth, Moon, and Sun-A Three Body System, 125; 3.8 The Push Me-Pull You, 128. 3.9 Rubber Ball Rebound, 131; 3.10 How to Avoid Broken Ankles, 132. 3.11 Mass Flow and Momentum, 134; 3.12 Freight Car and Hopper, 135; 3.13 Leaky Freight Car, 136; 3.14 Rocket in Free Space, 138; 3.15 Rocket in a Gravitational Field, 139. 3_16 Momentum Transport to a Surface, 141; 3.17 A Dike at the Bend of a River, 143; 3.18 Pressure of a Gas, 144. 4 WORK EXAMPLES, CHAPTER 4 AND 4.1 Mass Thrown Upward in a Uniform Gravitational Field, 154; 4.2 Solving ENERGY the Equation of Simple Harmonic Motion, 154. 4.3 Vertical Motion in an Inverse Square Field, 156. 4.4 The Conical Pendulum, 161; 4.5 Escape Velocity-The General Case, 162. 4.6 The Inverted Pendulum, 164; 4.7 Work Done by a Uniform Force, 165; 4.8 Work Done by a Central Force, 167; 4.9 A Path-dependent line Integral, 167; 4.10 Parametric Evaluation of a line Integral, 168. xii LIST OF EXAMPLES 4.11 Potential Energy of a Uniform Force Field, 170; 4.12 of an Inverse Square Force, 171; 4.13 4.14 Potential Energy Bead, Hoop, and Spring, 172. Energy and Stability-The Teeter Toy, 175. II 4.15 4.17 Molecular Vibrations, 179; 4.16 Small Oscillations, 181. Block Sliding down Inclined Plane, 183. t 4.18 Elastic Collision of Two Balls, 190; 4.19 Limitations on Laboratory i Scattering Angle, 193. t I 5 SOME EXAMPLES, CHAPTER 5 MATHEMATICAL 5.1 Partial Derivatives, 203; 5.2 Applications of the Partial Derivative, 205. ASPECTS 5.3 Gravitational Attraction by a Particle, 208; 5.4 Uniform Gravitational OF FORCE Field, 209; 5.5 Gravitational Attraction by Two Point Masses, 209. AND 5.6 Energy Contours for a Binary Star System, 212. ENERGY 5.7 The Curl of the Gravitational Force, 219; 5.8 A Nonconservative Force, 220; 5.9 A Most Unusual Force Field, 221; 5.10 Construction of the Potential Energy Function, 222; 5.11 How the Curl Got Its Name, 224. 5.12 Using Stokes' Theorem, 227. ­ i , I 6 ANGULAR EXAMPLES, CHAPTER 6 MOMENTUM 6.1 Angular Momentum of a Sliding Block, 236; 6.2 Angular Momentum AND FIXED AXIS of the Conical Pendulum, 237. ROTATION 6.3 Central Force Motion and the Law of Equal Areas, 240; 6.4 Capture Cross Section of a Planet, 241; 6.5 Torque on a Sliding Block, 244; 6.6 tr Torque on the Conical Pendulum, 245; 6.7 Torque due to Gravity, 247. I , 6.8 Moments of Inertia of Some Simple Objects, 250; 6.9 The Parallel Axis ,I Theorem, 252. 6.10 Atwood's Machine with a Massive Pulley, 254. 1 6.11 Grandfather's Clock, 256; 6.12 Kater's Pendulum, 258; 6.13 The Door· I I step, 259. 6.14. Angular Momentum of a Rolling Wheel, 262; 6.15 Disk on Ice, 264; 6.16 Drum Rolling down a Plane, 265; 6.17 Drum Rolling down a Plane: l Energy Method, 268; 6.18 The Falling Stick, 269. 7 RIGID BODY EXAMPLES, CHAPTER 7 MOTION 7.1 Rotations through Finite Angles, 289; 7.2 Rotation in the xy Plane, 291; AND THE 7.3 Vector Nature of Angular Velocity, 291; 7.4 Angular Momentum of a CONSERVATION Rotating Skew Rod, 292; 7.5 Torque on the Rotating Skew Rod, 293; 7.6 OF Torque on the Rotating Skew Rod (Geometric Method), 294. ANGULAR 7.7 Gyroscope Precession, 298; 7.8 Why a Gyroscope Precesses, 299. MOMENTUM 7.9 Precession of the Equinoxes, 300; 7.10 The Gyrocompass Effect, 301; 7.11 Gyrocompass Motion, 302; 7.12 The Stability of Rotating Objects, 304. 7.13 Rotating Dumbbell, 310; 7.14 The Tensor of Inertia for a Rotating Skew Rod, 312; 7.15 Why Flying Saucers Make Better Spacecraft than Do Flying Cigars, 314. 7.16 Stability of Rotational Motion, 322; 7.17 The Rotating Rod, 323; "".18 Euler's Equations and Torque-free Precession, 324_  LIST OF EXAMPLES xiii 8 NONINERTIAL EXAMPLES, CHAPTER 8 SYSTEMS 8.1 The Apparent Force of Gravity, 346; 8.2 Cylinder on an Accelerating AND Plank, 347; 8.3 Pendulum in an Accelerating Car, 347. FICTITIOUS 8.4 The Driving Force of the Tides, 350; 8.5 Equilibrium Height of the Tide, FORCES 352. 8.6 Surface of a Rotating Liquid, 362; 8.7 The Coriolis Force, 363; 8.8 De­ flection of a Falling Mass, 364; 8.9 Motion on the Rotating Earth, 366; 8.10 Weather Systems, 366; 8.11 The Foucault Pendulum, 369. 9 CENTRAL EXAMPLES, CHAPTER 9 FORCE 9.1 Noninteracting Particles, 384; 9.2 The Capture of Comets, 387; 9.3 MOTION Perturbed Circular Orbit, 388. 9.4 Hyperbolic Orbits, 393; 9.5 Satellite Orbit, 396; 9.6 Satellite Maneuver, 398. 9.7 The Law of Periods, 403. 10 THE EXAMPLES, CHAPTER 10 HARMONIC 10.1 Initial Conditions and the Frictionless Harmonic Oscillator, 411. OSCILLATOR 10.2 The Q of Two Simple Oscillators, 419; 10.3 Graphical Analysis of a Damped Oscillator, 420. 10.4 Forced Harmonic Oscillator Demonstration, 424; 10.5 Vibration Elimi­ nator, 428. 11 THE EXAMPLES, CHAPTER 11 SPECIAL 11.1 The Galilean Transformations, 453; 11.2 A Light Pulse as Described t;ly THEORY the Galilean Transformations, 455. OF RELATIVITY 12 RELATIVISTIC EXAMPLES, CHAPTER 12 KINEMATICS 12.1 Simultaneity, 463; 12.2 An Application of the Lorentz Transformations, 464; 12.3 The Order of Events: Timelike and Spacelike Intervals, 465. 12.4 The Orientation of a Moving Rod, 467; 12.5 Time Dilation and Meson Decay, 468; 12.6 The Role of Time Dilation in an Atomic Clock, 470_ 12.7 The Speed of Light in a Moving Medium, 474. 12.8 Doppler NaVigation, 479. 13 RELATIVISTIC EXAMPLES, CHAPTER 13 MOMENTUM 13.1 Velocity Dependence of the Electron's Mass, 492. AND 13.2 Relativistic Energy and Momentum in an Inelastic Collision, 496; 13.3 ENERGY The Equivalence of Mass and Energy, 498. 13.4 The Photoelectric Effect, 502; 13.5 Radiation Pressure of Light, 502;  f \ xiv LIST OF EXAMPLES 13.6 The Compton Effect, 503; 13.7 Pair Production, 505; 13.8 The Photon Picture of the Doppler Effect, 507. 13.9 The Rest Mass of the Photon, 510; 13.10 Light from a Pulsar, 510. 14 FOUR­ EXAMPLES, CHAPTER 14 VECTORS 14.1 Transformation Properties of the Vector Product, 518; 14.2 A Non· AND vector, 519. RELATIVISTIC 14.3 Time Dilation, 524; 14.4 Construction of a Four·vector: The Four­ INVARIANCE velocity, 525; 14.5 The Relativistic Addition of Velocities, 526. 14.6 The Doppler Effect, Once More, 530; 14.7 Relativistic Center of Mass Systems, 531; 14.8 Pair Production in Electron-electron Collisions, 533. I ! i ; , I I { i' Ii Ii. 11 ) Ii ,; i: !ij; ii' Ii. I [; II'i I I" 'ti.  There is good reason for the tradition that students of science and PREFACE engineering start college physics with the study of mechanics: mechanics is the cornerstone of pure and applied science. The concept of energy, for example, is essential for the study of the evolution of the universe, the properties of elementary particles, and the mechanisms of biochemical reactions. The concept of energy is also essential to the design of a cardiac pacemaker and to the analysis of the limits of growth of industrial society. How· ever, there are difficulties in presenting an introductory course in mechanics which is both exciting and intellectually rewarding. Mechanics is a mature science and a satisfying discussion of its principles is easily lost in a superficial treatment. At the other extreme, attempts to "enrich" the subject by emphasizing advanced topics can produce a false sophistication which empha· sizes technique rather than understanding. This text was developed from a first-year course which we taught for a number of years at the Massachusetts Institute of Technology and, earlier, at Harvard University. We have tried to present mechanics in an engaging form which offers a strong base for future work in pure and applied science. Our approach departs from tradition more in depth and style than in the choice of topics; nevertheless, it reflects a view of mechanics held by twentieth­ century physicists. Our book is written primarily for students who come to the course knowing some calculus, enough to differentiate and integrate sim­ ple functions.' It has also been used successfully in courses requiring only concurrent registration in calculus. (For a course of this nature, Chapter 1 should be treated as a resource chapter, deferring the detailed discussion of vector kinematics for a time. Other suggestions are listed in To The Teacher.) Our experi­ ence has been that the principal source of difficulty for most stu­ dents is in learning how to apply mathematics to physical problems, not with mathematical techniques as such. The elements of cal­ culus can be mastered relatively easily, but the development of problem-solving ability requires careful guidance. We have pro­ vided numerous worked examples throughout the text to help supply this guidance. Some of the examples, particularly in the early chapters, are essentially pedagogical. Many examples, how­ ever, illustrate principles and techniques by application to prob­ lems of real physical interest. The first chapter is a mathematical introduction, chiefly on vec­ tors and kinematics. The concept of rate of change of a vector, 1 The background provided in "Quick Calculus" by Daniel Kleppner and Norman Ramsey, John Wiley & Sons, New York, 1965, is adequate. xvi PREFACE probably the most difficult mathematical concept in the text, plays an important role throughout mechanics. Consequently, this topic is developed with care, both analytically and geometrically. The geometrical approach, in particular, later proves to be invalu· able for visualizing the dynamics of angular momentum. Chapter 2 discusses inertial systems, Newton's laws, and some common forces. Much.of the discussion centers on applying New· ton's laws, since analyzing even simple problems according to general principles can be a challenging task at first. Visualizing a complex system in terms of its essentials, selecting suitable inertial coordinates, and distinguishing between forces and accel· erations are all acquired skills. The numerous illustrative exam· pies in the text have been carefully chosen to help develop these skills. Momentum and energy are developed in the following two chap· ters. Chapter 3, on momentum, applies Newton's laws to extended systems. Students frequently become confused when they try to apply momentum considerations to rockets and other systems involving flow of mass. Our approach is to apply a differential method to a system defined so that no mass crosses its boundary during the chosen time interval. This ensures that no contribution to the total momentum is overlooked. The chapter concludes with a discussion of momentum flux. Chapter 4, on energy, develops the work·energy theorem and its application to conservative and nonconservative forces. The conservation laws for momentum and energy are illustrated by a discussion of collision problems. Chapter 5 deals with some mathematical aspects of conservative forces and potential energy; this material is not needed elsewhe re in the text, but it will be of interest to stUdents who want a mathe· matically complete treatment of the subject. Students usually find it difficult to grasp the properties of angular momentum and rigid body motion, partly because rotational motion lies so far from their experience that they cannot rely on intuition. As a result, introductory texts ften slight these topics, despite their importance. We have found that rotational motion can be made understandable by emphasizing physical reasoning rather than mathematical formalism, by appealing to geometric argu· ments, and by providing numerous worked examples. In Chapter 6 angular momentum is introduced, and the dynamics of fixed axis rotation is treated. Chapter 7 develops the important features of rigid body motion by applying vector arguments to systems dominated by spin angular momentum. An elementary treatment of general rigid body motion is prp.sented in the last sections of Chapter 7 to show how Euler's equations can be developed from PREFACE xvii simple physical arguments. This more advanced material is optional however; we do not usually treat it in our own course. Chapter 8, on noninertial coordinate systems, completes the development of the principles of newtonian mechanics. Up to this point in the text, inertial systems have been used exclusively in order to avoid confusion between forces and accelerations. Our discussion of noninertial systems emphasizes their value as computational tools and their implications for the foundations of mechanics. Chapters 9 and 10 treat central force motion and the harmonic oscillator, respectively. Although no new physical concepts are involved, these chapters illustrate the application of the principles of mechanics to topics of general interest and importance in phy­ sics. Much of the algebraic complexity of the harmonic oscillator is avoided by focusing the discussion on energy, and by using sim­ ple approximations. Chapters 11 through 14 present a discussion of the principles of special relativity and some of its applications. We attempt to emphasize the harmony between relativistic and classical thought, believing, for example, that it is more valuable to show how the classical conservation laws are unified in relativity than to dwell at length on the so-called "paradoxes." Our treatment is con­ cise and minimizes algebraic complexities. Chapter 14 shows how ideas of symmetry play a fundamental role in the formulation of relativity. Although we have kept the beginning students in mind, the concepts here are more subtle than in the previous chapters. Chapter 14 can be omitted if desired; but by illustrating how sym­ metry bears on the principles of mechanics, it offers an exciting mode of thought and a powerful new tool. Physics cannot be learned passively; there is absolutely no sub­ stitute for tackling challenging problems. Here is where stUdents gain the sense of satisfaction and involvement produced by a genuine understanding of the principles of physics. The collec­ tion of problems in this book was developed over many years of classroom use. A few problems are straightforward and intended for drill; most emphasize basic principles and require serious thought and effort. We have tried to choose problems which make this effort worthwhile in the spirit of Piet Hein's aphorism Problems worthy of attack prove their worth by hitting backl 1 From Grooks I, by Piet Hein, copyrighted 1966, The M.I.T. Press. xviii PREFACE It gives us pleasure to acknowledge the many contributions to I this book from our colleagues and from our students. In par· ticular, we thank Professors George B. Benedek and David E. I Pritchard for a number of examples and problems. We should also like to thank Lynne Rieck and Mary Pat Fitzgerald for their cheerful fortitude in typing the manuscript. Daniel Kleppner ! I Robert J. Kolenkow [ t \ r  The first eight chapters form a comprehensive introduction to TO classical mechanics and constitute the heart of a one-semester course. In a 12-week semester, we have generally covered the THE first 8 chapters and parts of Chapters 9 or 10. However, Chapter 5 and some of the advanced topics in Chapters 7 and 8 are usually omitted, although some students pursue them independently. EACHER Chapters 11,12,and 13 present a complete introduction to special relativity. Chapter 14, on transformation theory and four-vectors, provides deeper insight into the subject for interested students. We have used the chapters on relativity in a three-week short course and also as part of the second·term course in electricity and magnetism. The problems at the end of each chapter are generally graded in difficulty. They are also cumulative; concepts and techniques from earlier chapters are repeatedly called upon in later sections of the book. The hope is that by the end of the course the student will have developed a good intuition for tackling new problems, that he will be able to make an intelligent estimate, for instance, about whether to start from the momentum approach or from the energy approach, and that he will know how to set off on a new tack if his first approach is unsuccessful. Many students report a deep sense of satisfaction from acquiring these skills. Many of the problems require a symbolic rather than a numerical solution. This is not meant to minimize the importance of numeri­ cal work but to reinforce the habit of analyzing problems symboli­ cally. Answers are given to some problems; in others,a numerica! "answer clue" is provided to allow the student to check his sym­ bolic result. Some of the problems are challenging and require serious thought and discussion. Since too many such problems at once can result in frustration, each assignment should have a mix of easier and harder problems. Chapter 1 Although we would prefer to start a course in mechan­ ics by discussing physics rather than mathematics, there are real advantages to devoting the first few lectures to the mathematics of motion. The concepts of kinematics are straightforward for the most part, and it is helpful to have them clearly in hand before tackling the much subtler problems presented by new­ tonian dynamics in Chapter 2. A departure from tradition in this chapter is the discussion of kinematics using polar coordinates. Many students find this topic troublesome at first,requiring serious effort. However, we feel that the effort will be amply rewarded. In the first place, by being able to use polar coordinates freely, the kinematics of rotational motion are much easier to understand; xx TO THE TEACHER the mystery of radial acceleration disappears. More important, this topic gives valuable insights into the nature of a time-varying vector, insights which not only simplify the dynamics of particle motion in Chapter 2 but which are invaluable to the discussion of momentum flux in Chapter 3, angular momentum in Chapters 6 and 7, and the use of noninertial coordinates in Chapter 8. Thus, the effort put into understanding the nature of time-varying vectors in Chapter 1 pays important dividends throughout the course. If the course is intended for students who are concurrently begin­ ning their study of calculus, we recommend that parts of Chapter 1 be deferred. Chapter 2 can be started after having covered only the first six sections of Chapter 1. Starting with Example 2.5, the kinematics of rotational motion are needed; at this pOint the ideas presented in Section 1.9 should be introduced. Section 1.7, on the integration of vectors, can be postponed until the class has become familiar with integrals. Occasional examples and problems involv­ ing integration will have to be omitted until that time. Section 1.8, on the geometric interpretation of vector differentiation, is essen­ tial preparation for Chapters 6 and 7 but need not be discussed earlier. Chapter 2 The material in Chapter 2 often represents the stu­ dent's first serious attempt to apply abstract prinCiples to con­ crete situations. Newton's laws of motion are not self-evident; most people unconsciously follow aristotelian thought. We find that after an initial period of uncertainty, stUdents become accus­ tomed to unalyzing problems according to principles rather than vague intuition. A common source of difficulty at first is to con­ fuse force and acceleration. We therefore emphasize the use of inertial systems and recommend strongly that noninertial coor­ dinate systems be reserved until Chapter 8, where their correct use is discussed. In particular, the use of centrifugal force in the early chapters can lead to endless confusion between inertial and noninertial systems and, in any case, it is not adequate for the analysis of motion in rotating coordinate systems. Chapters 3 and 4 There are many different ways to derive the rocket equations. However, rocket problems are not the only ones in which there is a mass flow, so that it is important to adopt a method which is easily generalized. It is also desirable that the method be in harmony with the laws of conservation of momentum or, to put it more crudely, that there is no swindle involved. The differential approach used in Section 3.5 was developed to meet these requirements. The approach may not be elegant, but it is straightforward and quite general. TO THE TEACHER xxi In Chapter 4, we attempt to emphasize the general nature of the work-energy theorem and the difference between conserva­ tive and nonconservative forces. Although the line integral is introduced and explained, only simple line integrals need to be evaluated, and general computational techniques should not be given undue attention. Chapter 5 This chapter completes the discussion of energy and provides a useful introduction to potential theory and vector cal­ culus. However, it is relatively advanced and will appeal only to students with an appetite for mathematics. The results are not needed elsewhere in the text, and we recommend leaving this chapter for optional use, or as a special topic. Chapters 6 and 7 Most students find that angular momentum is the most difficult physical concept in elementary mechanics. The major conceptual hurdle is visualizing the vector properties of angular momentum. We th.erefore emphasize the vector nature of angular momentum repeatedly throughout these chapters. In particular, many features of rigid body motion can be understood intuitively by relying on the understanding of time-varying vectors developed in earlier chapters. It is more profitable to emphasize the qualitative features of rigid body motion than formal aspects such as the tensor of inertia. If desired, these qualitative argu· ments can be pressed quite far, as in the analysis of gyroscopic nutation in Note 7.2. The elementary discussion of Euler's equa· tions in Section 7.7 is intended as optional reading only. Although Chapters 6 and 7 require hard work, many students develop a phy· sical insight into angular momentum and rigid body motion which is seldom gained at the introductory level and which is often obscured by mathematics in advanced courses. Chapter 8 The subject of noninertial systems offers a natural springboard to such speculative and interesting topics as trans· formation theory and the principle of equivalence. From a more practical point of view, the use of noninertial systems is an impor­ tant technique for solving many physical problems. Chapters 9 and 10 In these chapters the principles developed earlier are applied to two important problems, central force motion and the harmonic oscillator. Although both topics are generally treated rather formally, we have tried to simplify the mathematical development. The discussion of central force motion relies heavily on the conservation laws and on energy diagrams. The treatment of the harmonic oscillator sidesteps much of the usual algebraic complexity by focusing on the lightly damped oscillator. Applica­ tions and examples play an important role in both chapters. xxii TO THE TEACHER Chapters 11 to 14 Special relativity offers an exciting change of pace to a course in mechanics. Our approach attempts to empha­ size the connection of relativity with classical thought. We have used the Michelson-Morley experiment to motivate the discussion. Although the prominence of this experiment in Einstein's thought has been much exaggerated, this approach has the advantage of grounding the discussion on a real experiment. We have tried to focus on the ideas of events and their trans­ formations without emphasizing computational aids such as dia­ grammatic methods. This approach allows us to deemphasize many of the so-called paradoxes. For many students, the real mystery of relativity lies not in the postulates or transformation laws but in why transformation prin­ ciples should suddenly become the fundamental concept for gen­ erating new physical laws. This touches on the deepest and most provocative aspects of Einstein's thought. Chapter 14, on four­ vectors, provides an introduction to transformation theory which unifies and summarizes the preceding development. The chapter is intended to be optional. Daniel Kleppner Robert J. Kolenkow  AN INTRODUCTION ;1 f ,.. TO MECHANICS  VEC TORS AND KINEMATICS- A FEW MATHEMATICAL PRELIMINARIES \ , I' \ J ' 2 VECTORS AND KINEMATlCS-A FEW MATHEMATICAL PRELIMINARIES 1.1 Introduction The goal of this book is to help you acquire a deep understanding of the principles of mechanics. The subject of mechanics is at the very heart of physics; its concepts are essential for under· standing the everyday physical world as well as phenomena on the atomic and cosmic scales. The concepts of mechanics, such as momentum, angular momentum, and energy, play a vital role in practically every area of physics. We shall use mathematics frequently in our discussion of physical principles, since mathematics lets us express complicated ideas quickly and transparently, and it often points the way to new insights. Furthermore, the interplay of theory and experiment in physics is based on quantitative prediction and measurement. For these reasons, we shall devote this chapter to developing some necessary mathematical tools and postpone our discussion of the principles of mechanics until Chap. 2. 1.2 Vectors The study of vectors provides a good introduction to the role of mathematics in physics. By using vector notation, physical laws can often be written in compact and simple form. (As a matter 1, of fact, modern vector notation was invented by a physicist, \ Willard Gibbs of Yale University, primarily to simplify the appear· I: ance of equations.) For example, here is how Newton's second i law (which we shall discuss in the next chapter) appears in ; nineteenth century notation: f , F", = ma", F1J = ma1J F. = ma. In vector notation, one simply writes F = mao Our principal motivation for introducing vectors is to simplify the t form of equations. However, as we shall see in the last chapter 0 of the book, vectors have a much deeper significance. Vectors are closely related to the fundamental ideas of symmetry and ! their use can lead to valuable insights into the possible forms of 1\ unknown laws. to 1 t  SEC. 1.2 VECTORS 3 I I Definition of a Vector Vectors can be approached from three points of view-geometric, w analytic, and axiomatic. Although all three points of view are use­ ''jI. ful, we shall need only the geometric and analytic approaches in "., our discussion of mechanics.. ', j " From the geometric point of view, a vector is a directed line segment. In writing, we can represent a vector by an arrow and label it with a letter capped by a symbolic arrow. In print, bold­ faced letters are traditionally used. In order 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 at left all represent the same vector. If two vectors have the same length and the same direction they are equal. The vectors Band C are equal: B= C. The length of a vector is called its magnitude. The magnitude 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 IAI, or simply A. If the length of A is V2, then IAI = A = V2. 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 A. It follows that _ A A=-, IA I and conversely A = IAIA. The Algebra of Vectors Multiplication of a Vector by a Scalar If we multiply A by a positive scalar b, the result is a new vector C = bA. The vector C is parallel to A, and its length is b times greater. Thus e = A, and II; ICI = blAI· The result of multiplying a vector by -1 is a new vector opposite in direction (antiparallel) to the original vector. Multiplication of a vector by a negative scalar evidently can change both the magnitude and the direction sense. 4 VECTORS AND KINEMATlCS-A FEW MATHEMATICAL PRELIMINARIES Addition of Two Vectors Addition of vectors has the simple geo· metrical interpretation shown by the drawing. The rule is: To add B to A, place the tail of B at the head of A. The sum is a vector from the tail of A to the head of B. Subtraction of Two Vectors Since A - B = A+(- B), in order to sUbtract B from A we can simply multiply it by -1 and then add. The sketches below show how. l. fr... , ,B I , , r f L'\l 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 right hand drawing above. It is not difficult to prove the following laws. We give a geo· metrical proof of the commutative law; try to cook up your own proofs of the others. A+B=B+A Commutative law A+(B+C) = (A+B)+C Associative law c(dA) =(cd)A (c+ d)A = cA+dA Distributive law c(A+B) = cA+cB Proof of the Commutative law of vector addition t Although there is no great mystery to addition, subtraction, 1 and multiplication of a vector by a scalar, the result of "multiply· i.. ing" one vector by another is somewhat less apparent. Does ' I. multiplication yield a vector, a scalar, or some other quantity? The choice.is up to us, and we shall define two types of products f which are useful in our applications to physics.  SEC. 1.2 VECTORS 5 Scalar Product ("Dot" Product) The first type of product is called the scalar product, since it represents a way of combining two vectors to form a scalar. The scalar product of A and B is denoted by A. B and is often called the dot product. A· B is defined by A· B == IAIIBI cos e. Here f) is the angle between A and B when they are drawn tail to tail. Since IBI cos f) is the projection of B along the direction of A, A. B = IAI X (projection of B on A). Similarly, A· B = IBI X (projection of A on B). If A· B = 0, then IAI = 0 or IBI = 0, or A is perpendicular to B (that is, cos f) = 0). Scalar multiplication is unusual in that the dot product of two nonzero vectors can be O. Note that A· A = IAI2. By way of demonstrating the usefulness of the dot product, here is an almost trivial proof of the law of cosines. Example 1.1 Law of Cosines C=A+B C C (A + B). (A + B) = ICI2=IAI2+IBI2 + ZIAIIBI cos f) This result is generally expressed in terms of the angle : C2 = A 2 + B 2 - 2AB cos. (We have used cos f) = cos (rr - fjJ) = -cos.) Example 1.2 Work and the Dot Product The dot product finds its most important application in the discussion of F work and energy in Chap. 4. As you may already know, the work W done 8 by a force F on an object is the displacement d of the object times the d component of F along the direction of d. If the force is applied at an angle f) to the displacement, W = (F cos f) d. Granting for the time being that force and displacement are vectors. W = F· d. 6 VECTORS AND KINEMATlCS-A FEW MATHEMATICAL PRELIMINARIES Vector Product ("Cross" Product) The second type of product we need is the vector product. In this case, two vectors A and B are combined to form a third vector C. The symbol for vector product is a cross: C= AX B. An alternative name is the cross product. The vector product is more complicated than the scalar product because we have to specify both the magnitude and direction of AX B. The magnitude is defined as follows: if C= AX B, then ICI = I AIIBI sin 8, where 8 is the angle between A and B when they are drawn tail to tail. (To eliminate ambiguity, 8 is always taken as the angle smaller than '11". ) Note that the vector product is zero when 8 = 0 or'll", even if I AI and I B I are not zero. When we draw A and B tail to tail, they determine a plane. We define the direction of C to be perpendicular to the plane of A C and B. A, B, and C form what is called a right hand triple. Imag· ine a right hand coordinate system with A and B in the xy plane as / ---------7 / shown in the sketch. A lies on the x axis and B lies toward the Y / / I y axis. If A, B, and C form a right hand triple, then C lies on the / z axis. We shall always use right hand coordinate systems such as I / the one shown at left. Here is another way to determine the I direction of the cross product. Think of a right hand screw with x __________1 the axis perpendicular to A and B. Rotate it in the direction which (A is into paper) swings A into B. C lies in the direction the screw advances. (Warning: Be sure not to use a left hand screw. Fortunately, they are rare. Hot water faucets are among the chief offenders; your honest everyday wood screw is right handed.) A result of our definition of the cross product is that BX A= -AX B. Here we have a case in which the order of multiplication is impor­ tant. The vector product is not commutative. (In fact, since reversing the order reverses the sign, it is anticommutative.) We see that AXA=O for any vector A.  SEC. 1.2 VECTORS 7 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 g, the magnetic field B, and the velocity of the particle v. The force varies as the sine of the angle between v and B, and is perpendicular to the plane formed by v and B, in the direction indicated. A simpler way to give all these rules is F = gv X B. Another application is the definition of torque. We shall develop this F idea later. For now we simply mention in passing that the torque is (J defined by = r X F, u 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 the familiar idea that torque is a measure of the ability of an applied force to produce a twist. Note that a large force directed parallel to r produces no twist; it merely pulls. Only F sin e, the component of force perpen· dicular to r, produces a torque. The torque increases as the lever arm gets larger. As you will see in Chap. 6, it is extremely useful to associate a direction with torque. The natural direction is along the axis of rotation which the torque tends to produce. All these ideas are summarized in a nutshell by the simple equation = r X F. Top view Example 1.4 Area as a Vector We can use the cross product to describe an area. Usually one thinks of area in terms of magnitude only. However, many applications in ----------7 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 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. (In the latter / case the flow through the loop is zero.) Here is how the vector product c accomplishes this: Consider the area of a quadrilateral formed by two vectors, e and O. The area of the parallelogram A. is given by A base X height CD sin e lex 01· If we think of A. as a vector, we have c A = ex O. 8 VECTORS AND KINEMATlCS-A FEW MATHEMATICAL PRELIMINARIES We have already shown that the magnitude of A is the area of the parallelogram, and the vector product defines the convention for assigning a direction to the area. The direction is defined to be perpendicular to the plane of the area; that is, the direction is parallel to a normal to the surface. The sign of the direction is to some extent arbitrary; we could just as well have defined the area by A = D X C. However, once the sign is chosen, it is unique. 1.3 Components of a Vector The fact that we have discussed vectors without introducing a particular coordinate system shows why vectors are so useful; y vector operations are defined without reference to coordinate systems. However, eventually we have to translate our results from the abstract to the concrete, and at this point we have to choose a coordinate system in which to work. For simplicity, let us restrict ourselves to a two·dimensional system, the familiar xy plane. The diagram shows a vector A in the xy plane. The projections of A along the two coordinate axes are called the components of A. The components of A along the x and y axes are, respectively, Ax and All' The magnitude of ------ -x A is IAI = (Ax2 + AII2)1, and the direction of A is such that it makes an angle () = arctan (AliiAx) with the x axis. Since the components of a vector define it, we can specify a vector entirely by its components. Thus A = (Ax,AlI) y or, more generally, in three dimensions, A = (Ax,Al/,A.). A Prove for yourself that IAI = (Ax2 + A1I2 + A.2)l. The vector A has a meaning independent of any coordinate system. However, the components of A depend on the coordinate system being used. x To illustrate this, here is a vector A drawn in two different coordi· nate systems. In the first case, x' A = (A,O) (x,y system), while in the second A A = (O,-A) (x';y' system). Unless noted otherwise, we shall restrict ourselves to a single coordinate system, so that if A B, t y' = ; ,  SEC. 1.3 COMPONENTS OF A VECTOR 9 then A. = B. The single vector equation A = B symbolically represents three scalar equations. All vector operations can be written as equations for com­ ponents. For instance, multiplication by a scalar gives cA = (cA""cAlI). The law for vector addition is By writing A and B as the sums of vectors along each of the coordinate axes, you can verify that A B = k.B", + AliBli + A.B. We shall defer evaluating the cross product until the next section. Example 1.5 Vector Algebra Let A= (3,5,-7) B= (2,7,1). Find A + B. A-B. IAI. IBI. A· B. and the cosine of the angle between A and B. A + B= (3 + 2, 5 + 7. -7 + I) =(5 12, -6). A - B= (3 - 2, 5 - 7. -7-I) = (1,-2, -8) IAI = (32 + 52 + 72}l = V83 = 9. 11 IB I = (22 + 72 + 12)\ = V54 =7.35 A·B=3X2+5X7-7X1 = 34 A·B 34 cos (A,B)= = 0.507 IAI IBI (9.11)(7.35) = 10 VECTORS AND KINEMATlCS-A FEW MATHEMATICAL PRELIMINARIES Example 1.6 Construction of a Perpendicular Vector Find a unit vector in the xy plane which is perpendicular to A = (3,5,1). We denote the vector by B = (Bx,By,B.). Since B is in the xy plane, B. = O. For B to be perpendicular to A, we have A · B = O. A.B=3Bx+5By =0 Hence By = -tBx. However, B is a unit vector, which means that ·1 Bz:2 + By2 = 1. Bx2 + /"5B%2 = 1, or Bx - Combining these gives = vH = ±0.857 and By = -tBx = +0.514. The ambiguity in sign of Bx and By indicates that B can point along a.. line perpendicular to A in either of two directions. 1.4 Base Vectors Base vectors are a set of orthogonal (perpendicular) unit vectors, one for each dimension. For example, if we are dealing with the familiar cartesian coordinate system of three dimensions, the base k vectors lie along the x, y, and z axes. The x unit vector is denoted by i, the y unit vector by j, and the z unit vector by The base vectors have the following properties, as you can 1 tv. If this condi· tion is not satisfied, the mass does not follow a circular path but starts to fall; r is no longer zero. The tangential acceleration is given by Eq. (2). Since f = 0 we have a8 = RO W cos (J m The mass does not move with constant speed; it accelerates tangentially. On the downswing the tangential speed increases, on the upswing it decreases. The next example involves rotational motion, translational motion, and constraints. Example 2.7 The Whirling Block A horizontal frictionless table has a small hole in its center. Block.1 on ,,------ the table is connected to block B hanging beneath by a string of negligible / \ \ -- -- mass which passes through the hole. " Initially, B is held stationary and A. rotates at constant radius ro with If B is released at t 0, what is its accel·..... _------ steady angular velocity woo = eration immediately afterward? The forcl! diagrams for A and B after the moment of release are shown in the sketches. j q,  SEC. 2.4 SOME APPLICATIONS OF NEWTON'S LAWS 77 The vertical forces acting on A are in balance and we need not consider them. The only horizontal force acting on A is the string force T. The forces on B are the string force T and the weight WB. It is natural to use polar coordinates r, e for.'t, and a single linear coordinate z for B, as shown in the force diagrams. As usual, the unit vector r is radially outward. The equations of motion are T -T = MA 0, our solution predicts cos IX -> 00, which is nonsense since cos ex S 1. Something has gone wrong. Here is the trouble. Our solution predicts cos ex > 1 for w < Vgjl. When w = Vgll, cos ex = 1 and sin ex = 0; the bob simply hangs vertically. In going from Eq. (2) to Eq. (3) we divided both sides of Eq. (2) by sin IX and, in this case L------- w we divided by 0, which is not permissible. However, we see that we have ; ; overlooked a second possible solution, namely. sin ex = 0, T = lr, which is true for all values of w. The solution corresponds to the pendulum hanging straight down. Here is a plot of the complete solution. Physically, for w S V(ijz the only acceptable solution is ex = 0, cos ex = 1. For w > V(ijz there are two acceptable solutions: cos a 1. cos IX = 1 \ 2. cos IX = JL. \ cos a = g/(/w2) Iw2 \/ \ " cos a = 1 Solution 1 corresponds to the bob rotating rapidly but hanging verti· " cally. Solution 2 corresponds to the bob flying around at an angle with the vertical. For w > V(ijz, solution 1 is unstable-if the system is in that state and is slightly perturbed, it will jump outward. Can you see w why this is so? SEC. 2.5 THE EVERYDAY FORCES OF PHYSICS 79 The moral of this example is that you have to be sure that the mathe­ matics makes good physical sense. 2.5 The Everyday Forces of Physics When a physicist sets out to design an accelerator, he uses the laws of mechanics and his knowledge of electric and magnetic forces to determine the paths that the particles will follow. Pre­ dicting motion from known forces is an impor:tant part of physics and underlies most of its applications. Equally important, how­ ever, is the converse process of deducing the physical interaction by observing the motion; this is how new laws are discovered. A classic example is Newton's deduction of the law of gravitation from Kepler's laws of planetary motion. The current attempt to understand the interactions between elementary particles from high energy scattering experiments provides a more contemporary illustration. Unscrambling experimental observations to find the force can be difficult. In a facetious mood, Eddington once said that force is the mathematical expression we put into the left hand side of Newton's second law to obtain results that agree with observed moti.ons. Fortunately, force has a more concrete physical reality. Much of our effort in the following chapters will be to learn how systems behave under applied forces. If every pair of particles in the universe had its own special interaction, the task would be impossible. Fortunately, nature is kinder than this. As far as we know, there are only four fundamentally different types of interactions in the universe: gravity, electromagnetic interactions, the so-called weak interaction, and the strong interaction. Gravity and the electromagnetic interactions can act over a long range because they decrease only as the inverse square of the distance. However, the gravitational force always attracts, whereas electrical forces can either attract or repel. I n large systems, electrical attraction and repulsion cancel to a high degree, and gravity alone is left. For this reason, gravitational forces dominate the cosmic scale of our universe. In contrast, the world immediately around us is dominated by the electrical forces, since they are far stronger than gravity on the atomic scale. Electrical forces are responsible for the structure of atoms, molecules, and more complex forms of matter, as well as the existence of light. The weak and strong interactions have such short ranges that they are important only at nuclear distances, typically 10-16 m.  80 NEWTON'S LAWS-THE FOUNDATIONS OF NEWTONIAN MECHANICS They are negligible even at atomic distances, 10-10 m. As its name implies, the strong interaction is very strong, much stronger than the electromagnetic force at nuclear distances. It is the "glue" that binds the atomic nucleus, but aside from this it has little effect in the everyday world. The weak interaction plays a less dramatic role; it mediates in the creation and destruction of neutrinos-particles of no mass and no charge which are essential to our understanding of matter but which can be detected only by the most arduous experiments. Our object in the remainder of the chapter is to become familiar with the forces which are important in everyday mechanics. Two of these, the forces of gravity and electricity, are fundamental and cannot be explained in simpler terms. The other forces we shall discuss, friction, the contact force, and the viscous force, can be understood as the macroscopic manifestation of interatomic forces. Gravity, Weight, and the Gravitational Field Gravity is the most familiar of the fundamental forces. It has close historical ties to the development of mechanics; Newton discovered the law of universal gravitation in 1666, the same year that he formulated his laws of motion. By calculating the motion of two gravitating particles, he was able to derive Kepler's empiri· cal laws of planetary motion, (By accomplishing all this by age 26, Newton established a tradition which still maintains-that great advances are often made by young physicists.) According to Newton's law of gravitation, two particles attract each other with a force directed along their line of centers. The magnitude of the force is proportional to the product of the masses and decreases as the inverse square of the distance between the particles. In verbal form the law is bulky and hard to use. However. we can reduce it to a simple mathematical expression. I" "I Consider two particles, a and b, with masses Ma and Mb, respec· Ma --------.--. Mb a Fa Fb b tively, separated by distance r. Let Fb be the force exerted on particle b by particle a. Our verbal description of the magnitude of the force is summarized by GMaMb IFbl = r 2 G is a constant of proportionality called the graVitational constant. Its value is found by measuring the force between masses in a Ii  SEC. 2.5 THE EVERYDAY FORCES OF PHYSICS 81 known geometry. The first measurements were performed by Henry Cavendish in 1771 using a torsion balance. The modern value of G is 6.67 X 10-11 N·m2/kg2. (G is the least accurately known of the fundamental constants. Perhaps you can devise a new way to measure it more precisely.) Experimentally, G is the same for all materials-aluminum, lead, neutrons, or what have you. For this reason, the law is called the universal law of gravitation. The gravitational force between two particles is central (along r ob the line of centers) and attractive. The simplest way to describe ---- ---.... -- rab b these properties is to use vectors. By convention, we introduce - a a vector rab from the particle exerting the force, particle a in this case, to the particle experiencing the force, particle b. Note that Irabl = 1'. Using the unit vector rill> = rab/1', we have GM"Mb. Fb = - 2 rab· l' The negative sign indicates that the force is attractive. The force on a due to b is since rba = -rab. The forces are equal and opposite, and New· ton's third law is automatically satisfied. The gravitational force has a unique and mysterious property. Consider the equation of motion of particle b under the gravita­ tional attraction of particle a. or The acceleration of a particle under gravity is independent of its mass! There is a subtle point connected with our cancelation of Mb, however. The "mass" (gravitational mass) in the law of gravi­ tation, which measures the strength of gravitational interaction, is operationally distinct from the "mass" (inertial mass) which char­ acterizes inertia in Newton's second law. Why gravitational mass is proportional to inertial mass for all matter is one of the great mysteries of physics. However, the proportionality has been 82 NEWTON'S LAWS-THE FOUNDATIONS OF NEWTONIAN MECHANICS experimentally verified to very high accuracy, approximately 1 part in 1011; we shall have more to say about this in Chap. 8. The Gravitational Force of a Sphere The law of gravitation applies only to particles. How can we find the gravitational force on a particle due to an extended body like the earth? Fortunately, the gravitational force obeys the law of superposition: the force due to a collection of particles is the vector sum of the forces exerted by the particles individually. This allows us to mentally divide @R '-- __ the bOdy into a cOliection Of smalielements which can be treated as particles. Using integral calculus, we can sum the forces from _ ----, _4 __ ----I. all the particles. This method is applied in Note 2.1 to calculate the force between a particle of mass m and a uniform thin spher· ical shell of mass M and radius R. The result is Mm_ F = -G - r r> R r2 F = 0 r < R,...... --....... where r is the distance from the center of the shell to the particle. / "- / If the particle lies outside the shell, the force is the same as if all \ F I M. I - the mass of the shell were concentrated at its center. \ / \ The reason the gravitational force vanishes inside the spherical " /....... _-/ shell can be seen by a simple argument due to Newton. Consider