Taylor classical mechanics chapter 10 solutions

Trigonometric Identities

sin(9 ± 0) = sin 0 cos 4) ± cos 0 sin 4, cos(0 f 0) = cos a cos 0 sin 0 sin 0

cos 0 cos (/' = l[cos(0 + 0) + cos(8 - 0)] sine sin 0 = 4.[coso - coo + 4,)]

sin e cos 0 = [sin(e + 0) + sin(O - 0)]

cos26. 4[1+ cos 20] sin2 8 = [1 - cos 20] 2

0 + 0 0 — 0 +0 0 0 — 0 cos 0 + cos 0 = 2 cos cos cos 0 cos 0 = 2 sin sin

2 2 2 2

4) 0 sin 8 ± sin 0= 2 sin 0 ± cos 0 + 2 2

cost 0 + sin2 0 = 1 sect 8 - tan2 8 = 1

ei8 = cos 0 + i sin 0 [Euler's relation]

cos 0 = + e-`0 ) sin 0 = - e-i°)

Hyperbolic Functions

cosh z = (ez + e') = cos(iz) sinh z = (ez - e-z) = sin(iz)

tanh z = sinh z sechz = 1

cosh z cosh z

cosh2 z — sinh2 z = 1 sech2z + tanh2 z = 1

Series Expansions

f (z) = f (a) + f'(a)(z - a) + Z, f"(a)(z - a)2 + 3,f'(a)(z - a)3 + • • [Taylor's series] ez = 1 + z + + +... ln(1 + z) = z - -1z2 + Az3 - • • [ Izi < 1]

cos z = 1 - I,z2+z— • • • sin z = z - Iz3 + -L! z5 — • 3! 5 COSh Z = 1 + -1;22 + +1,24 + • • • sinh z = z + fl,z 3 + + • • •

tan z = z + iz3 + Az 5 +•••[ lz I < ir/2] tanh z = z - + 1Z5- z5 - • • • [ I z I < 7/2]

(1 + = 1 + nz + 2!

n(n - 1) z

2 + • [ IZI < 1] [binomial series]

f dx J x2 -

= arccosh x

J x x2 -1 dx = arccos(1/x) dx

+ x2)3/2 ± x2)1/2

Some Derivatives

— tan z= sect z - d

tanh z= sech2 z dz dz

d . — sink z = cosh z —

d cosh z = sinh z

dz dz

Some Integrals

I dx arctan x

f dx = arctanh x J + x2 J - x2

dx f dx = arcsin x = arcsinh x J 1-x2 J x2

f tan x dx = — In cos x f tanh x dx = 1n cosh x dx ( f x dx

J x + x2 = +x x) J 1 + x2 = In(1 + x 2)

f x dx = + x 2 J x2

dx = arcsin(li) — (1 — x) J

f ln(x)dx = xin(x) — x

[ 1 dx

Jo -11--x 24 — mx 2 = K (m), complete elliptic integral of first kind

Classical mechanics

John r. Taylor

2005

University Science Books www. uscibooks. corn

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

Taylor, John R. (John Robert), 1939– Classical mechanics / John R. Taylor.

p. cm. Includes bibliographical references ISBN 1-891389-22-X (acid-free paper) 1. Mechanics. I. Title. QC125.2.T39 2004 531—dc22

2004054971

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Contents

Preface xi

Ess

CHAPTER 1 Newton's Laws of Motion 3

1.1 Classical Mechanics 3

1.2 Space and Time 4

1.3 Mass and Force 9

1.4 Newton's First and Second Laws; Inertial Frames 13

1.5 The Third Law and Conservation of Momentum 17

1.6 Newton's Second Law in Cartesian Coordinates 23

1.7 Two-Dimensional Polar Coordinates 26

Principal Definitions and Equations of Chapter 1 33

Problems for Chapter 1 34

CHAPTER 2 Projectiles and Charged Particles 43

2.1 Air Resistance 43

2.2 Linear Air Resistance 46

2.3 Trajectory and Range in a Linear Medium 54

2.4 Quadratic Air Resistance 57

2.5 Motion of a Charge in a Uniform Magnetic Field 65

2.6 Complex Exponentials 68

2.7 Solution for the Charge in a B Field 70

Principal Definitions and Equations of Chapter 2 71

Problems for Chapter 2 72

vi Contents

CHAPTER 3 Momentum and Angular Momentum 83

3.1 Conservation of Momentum 83

3.2 Rockets 85

3.3 The Center of Mass 87

3.4 Angular Momentum for a Single Particle 90

3.5 Angular Momentum for Several Particles 93

Principal Definitions and Equations of Chapter 3 98

Problems for Chapter 3 99

CHAPTER 4 Energy 105

4.1 Kinetic Energy and Work 105

4.2 Potential Energy and Conservative Forces 109

4.3 Force as the Gradient of Potential Energy 116

4.4 The Second Condition that F be Conservative 118

4.5 Time-Dependent Potential Energy 121

4.6 Energy for Linear One-Dimensional Systems 123

4.7 Curvilinear One-Dimensional Systems 129

4.8 Central Forces 133

4.9 Energy of Interaction of Two Particles 138

4.10 The Energy of a Multiparticle System 144

Principal Definitions and Equations of Chapter 4 148

Problems for Chapter 4 150

CHAPTER 5 Oscillations 161

5.1 Hooke's Law 161

5.2 Simple Harmonic Motion 163

5.3 Two-Dimensional Oscillators 170

5.4 Damped Oscillations 173

5.5 Driven Damped Oscillations 179

5.6 Resonance 187

5.7 Fourier Series* 192

5.8 Fourier Series Solution for the Driven Oscillator* 197

5.9 The RMS Displacement; Parseval's Theorem' 203

Principal Definitions and Equations of Chapter 5 205

Problems for Chapter 5 207

* Sections marked with an asterisk could be omitted on a first reading.

Contents vii

CHAPTER 6 Calculus of Variations 215

6.1 Two Examples 216

6.2 The Euler—Lagrange Equation 218

6.3 Applications of the Euler—Lagrange Equation 221

6.4 More than Two Variables 226

Principal Definitions and Equations of Chapter 6 230

Problems for Chapter 6 230

CHAPTER 7 Lagrange's Equations 237

7.1 Lagrange's Equations for Unconstrained Motion 238

7.2 Constrained Systems; an Example 245

7.3 Constrained Systems in General 247

7.4 Proof of Lagrange's Equations with Constraints 250

7.5 Examples of Lagrange's Equations 254

7.6 Generalized Momenta and Ignorable Coordinates 266

7.7 Conclusion 267

7.8 More about Conservation Laws * 268

7.9 Lagrange's Equations for Magnetic Forces * 272

7.10 Lagrange Multipliers and Constraint Forces * 275

Principal Definitions and Equations of Chapter 7 280

Problems for Chapter 7 281

CHAPTER 8 Two-Body Central-Force Problems 293

8.1 The Problem 293

8.2 CM and Relative Coordinates; Reduced Mass 295

8.3 The Equations of Motion 297

8.4 The Equivalent One-Dimensional Problem 300

8.5 The Equation of the Orbit 305

8.6 The Kepler Orbits 308

8.7 The Unbounded Kepler Orbits 313

8.8 Changes of Orbit 315

Principal Definitions and Equations of Chapter 8 319

Problems for Chapter 8 320

CHAPTER 9 Mechanics in Noninertial Frames 327

9.1 Acceleration without Rotation 327

9.2 The Tides 330

9.3 The Angular Velocity Vector 336

9.4 Time Derivatives in a Rotating Frame 339

viii Contents

9.5 Newton's Second Law in a Rotating Frame 342

9.6 The Centrifugal Force 344

9.7 The Coriolis Force 348

9.8 Free Fall and the Coriolis Force 351

9.9 The Foucault Pendulum 354

9.10 Coriolis Force and Coriolis Acceleration 358

Principal Definitions and Equations of Chapter 9 359

Problems for Chapter 9 360

CHAPTER 10 Rotational Motion of Rigid Bodies 367

10.1 Properties of the Center of Mass 367

10.2 Rotation about a Fixed Axis 372

10.3 Rotation about Any Axis; the Inertia Tensor 378

10.4 Principal Axes of Inertia 387

10.5 Finding the Principal Axes; Eigenvalue Equations 389

10.6 Precession of a Top due to a Weak Torque 392

10.7 Euler's Equations 394

10.8 Euler's Equations with Zero Torque 397

10.9 Euler Angles * 401

10.10 Motion of a Spinning Top* 403

Principal Definitions and Equations of Chapter 10 407

Problems for Chapter 10 408

CHAPTER 11 Coupled Oscillators and Normal Modes 417

11.1 Two Masses and Three Springs 417

11.2 Identical Springs and Equal Masses 421

11.3 Two Weakly Coupled Oscillators 426

11.4 Lagrangian Approach: The Double Pendulum 430

11.5 The General Case 436

11.6 Three Coupled Pendulums 441

11.7 Normal Coordinates * 444

Principal Definitions and Equations of Chapter 11 447

Problems for Chapter 11 448

CHAPTER 12 Nonlinear Mechanics and Chaos 457

12.1 Linearity and Nonlinearity 458

12.2 The Driven Damped Pendulum DDP 462

12.3 Some Expected Features of the DDP 463

Contents ix

12.4 The DDP: Approach to Chaos 467

12.5 Chaos and Sensitivity to Initial Conditions 476

12.6 Bifurcation Diagrams 483

12.7 State-Space Orbits 487

12.8 Poincare Sections 495

12.9 The Logistic Map 498

Principal Definitions and Equations of Chapter 12 513

Problems for Chapter 12 514

CHAPTER 13 Hamiltonian Mechanics 521

13.1 The Basic Variables 522

13.2 Hamilton's Equations for One-Dimensional Systems 524

13.3 Hamilton's Equations in Several Dimensions 528

13.4 Ignorable Coordinates 535

13.5 Lagrange's Equations vs. Hamilton's Equations 536

13.6 Phase-Space Orbits 538

13.7 Liouville's Theorem* 543

Principal Definitions and Equations of Chapter 13 550

Problems for Chapter 13 550

CHAPTER 14 Collision Theory 557

14.1 The Scattering Angle and Impact Parameter 558

14.2 The Collision Cross Section 560

14.3 Generalizations of the Cross Section 563

14.4 The Differential Scattering Cross Section 568

14.5 Calculating the Differential Cross Section 572

14.6 Rutherford Scattering 574

14.7 Cross Sections in Various Frames * 579

14.8 Relation of the CM and Lab Scattering Angles * 582

Principal Definitions and Equations of Chapter 14 586

Problems for Chapter 14 587

CHAPTER 15 Special Relativity 595

15.1 Relativity 596

15.2 Galilean Relativity 596

15.3 The Postulates of Special Relativity 601

15.4 The Relativity of Time; Time Dilation 603

15.5 Length Contraction 608

15.6 The Lorentz Transformation 610

15.7 The Relativistic Velocity-Addition Formula 615

x Contents

15.8 Four-Dimensional Space—Time; Four-Vectors 617

15.9 The Invariant Scalar Product 623

15.10 The Light Cone 625

15.11 The Quotient Rule and Doppler Effect 630

15.12 Mass, Four-Velocity, and Four-Momentum 633

15.13 Energy, the Fourth Component of Momentum 638

15.14 Collisions 644

15.15 Force in Relativity 649

15.16 Massless Particles; the Photon 652

15.17 Tensors* 656

15.18 Electrodynamics and Relativity 660

Principal Definitions and Equations of Chapter 15 664

Problems for Chapter 15 666

CHAPTER 16 Continuum Mechanics 681

16.1 Transverse Motion of a Taut String 682

16.2 The Wave Equation 685

16.3 Boundary Conditions; Waves on a Finite String * 688

16.4 The Three-Dimensional Wave Equation 694

16.5 Volume and Surface Forces 697

16.6 Stress and Strain: The Elastic Moduli 701

16.7 The Stress Tensor 704

16.8 The Strain Tensor for a Solid 709

16.9 Relation between Stress and Strain: Hooke's Law 715

16.10 The Equation of Motion for an Elastic Solid 718

16.11 Longitudinal and Transverse Waves in a Solid 721

16.12 Fluids: Description of the Motion * 723

16.13 Waves in a Fluid* 727

Principal Definitions and Equations of Chapter 16 730

Problems for Chapter 16 732

APPENDIX Diagonalizing Real Symmetric Matrices 739

A.1 Diagonalizing a Single Matrix 739

A.2 Simultaneous Diagonalization of Two Matrices 743

Further Reading 747

Answers for Odd-Numbered Problems 749

Index 777

Preface

This book is intended for students of the physical sciences, especially physics, who have already studied some mechanics as part of an introductory physics course ("fresh- man physics" at a typical American university) and are now ready for a deeper look at the subject. The book grew out of the junior-level mechanics course which is offered by the Physics Department at Colorado and is taken mainly by physics majors, but also by some mathematicians, chemists, and engineers. Almost all of these students have taken a year of freshman physics, and so have at least a nodding acquaintance with Newton's laws, energy and momentum, simple harmonic motion, and so on. In this book I build on this nodding acquaintance to give a deeper understanding of these basic ideas, and then go on to develop more advanced topics, such as the Lagrangian and Hamiltonian formulations, the mechanics of noninertial frames, motion of rigid bodies, coupled oscillators, chaos theory, and a few more.

Mechanics is, of course, the study of how things move — how an electron moves down your TV tube, how a baseball flies through the air, how a comet moves round the sun. Classical mechanics is the form of mechanics developed by Galileo and Newton in the seventeenth century and reformulated by Lagrange and Hamilton in the eighteenth and nineteenth centuries. For more than two hundred years, it seemed that classical mechanics was the only form of mechanics, that it could explain the motion of all conceivable systems.

Then, in two great revolutions of the early twentieth century, it was shown that classical mechanics cannot account for the motion of objects traveling close to the speed of light, nor of subatomic particles moving inside atoms. The years from about 1900 to 1930 saw the development of relativistic mechanics primarily to describe fast- moving bodies and of quantum mechanics primarily to describe subatomic systems. Faced with this competition, one might expect classical mechanics to have lost much of its interest and importance. In fact, however, classical mechanics is now, at the start of the twenty-first century, just as important and glamorous as ever. This resilience is due to three facts: First, there are just as many interesting physical systems as ever that are best described in classical terms. To understand the orbits of space vehicles and of charged particles in modern accelerators, you have to understand classical xi

xii Preface

mechanics. Second, recent developments in classical mechanics, mainly associated with the growth of chaos theory, have spawned whole new branches of physics and mathematics and have changed our understanding of the notion of causality. It is these new ideas that have attracted some of the best minds in physics back to the study of classical mechanics. Third, it is as true today as ever that a good understanding of classical mechanics is a prerequisite for the study of relativity and quantum mechanics.

Physicists tend to use the term "classical mechanics" rather loosely. Many use it for the mechanics of Newton, Lagrange, and Hamilton; for these people, "classical mechanics" excludes relativity and quantum mechanics. On the other hand, in some areas of physics, there is a tendency to include relativity as a part of "classical me- chanics"; for people of this persuasion, "classical mechanics" means "non-quantum mechanics." Perhaps as a reflection of this second usage, some courses called "clas- sical mechanics" include an introduction to relativity, and for the same reason, I have included one chapter on relativistic mechanics, which you can use or not, as you please.

An attractive feature of a course in classical mechanics is that it is a wonderful opportunity to learn to use many of the mathematical techniques needed in so many other branches of physics — vectors, vector calculus, differential equations, complex numbers, Taylor series, Fourier series, calculus of variations, and matrices. I have tried to give at least a minimal review or introduction for each of these topics (with references to further reading) and to teach their use in the usually quite simple context of classical mechanics. I hope you will come away from this book with an increased confidence that you can really use these important tools.

Inevitably, there is more material in the book than could possibly be covered in a one-semester course. I have tried to ease the pain of choosing what to omit. The book is divided into two parts: Part I contains eleven chapters of "essential" material that should be read pretty much in sequence, while Part II contains five "further topics" that are mutually independent and any of which can be read without reference to the others. This division is naturally not very clear cut, and how you use it depends on your preparation (or that of your students). In our one-semester course at the University of Colorado, I found I needed to work steadily through most of Part I, and I only covered Part II by having students choose one of its chapters to study as a term project. (An activity they seemed to enjoy.) Some of the professors who taught from a preliminary version of the book found their students sufficiently well prepared that they could relegate the first four or five chapters to a quick review, leaving more time to cover some of Part II. At schools where the mechanics course lasts two quarters, it proved possible to cover all of Part I and much of Part II as well.

Because the chapters of Part II are mutually independent, it is possible to cover some of them before you finish Part I. For example, Chapter 12 on chaos could be covered immediately after Chapter 5 on oscillations, and Chapter 13 on Hamiltonian mechanics could be read immediately after Chapter 7 on Lagrangian mechanics. A number of sections are marked with an asterisk to indicate that they can be omitted without loss of continuity. (This is not to say that this material is unimportant. I certainly hope you'll come back and read it later!)

As always in a physics text, it is crucial that you do lots of the exercises at the end of each chapter. I have included a large number of these to give both teacher and

Preface xiii

student plenty of choice. Some of them are simple applications of the ideas of the chapter and some are extensions of those ideas. I have listed the problems by section, so that as soon as you have read any given section you could (and probably should) try a few problems listed for that section. (Naturally, problems listed for a given section usually require knowledge of earlier sections. I promise only that you shouldn't need material from later sections.) I have tried to grade the problems to indicate their level of difficulty, ranging from one star (*), meaning a straightforward exercise usually involving just one main concept, to three stars ( ***), meaning a challenging problem that involves several concepts and will probably take considerable time and effort. This kind of classification is quite subjective, very approximate, and surprisingly difficult to make; I would welcome suggestions for any changes you think should be made.

Several of the problems require the use of computers to plot graphs, solve differ- ential equations, and so on. None of these requires any specific software; some can be done with a relatively simple system such as MathCad or even just a spreadsheet like Excel; some require more sophisticated systems, such as Mathematica, Maple, or Matlab. (Incidentally, it is my experience that the course for which this book was written is a wonderful opportunity for the students to learn to use one of these fabu- lously useful systems.) Problems requiring the use of a computer are indicated thus: [Computer]. I have tended to grade them as *** or at least ** on the grounds that it takes a lot of time to set up the necessary code. Naturally, these problems will be easier for students who are experienced with the necessary software.

Each chapter ends with a summary called "Principal Definitions and Equations of Chapter xx." I hope these will be useful as a check on your understanding of the chapter as you finish reading it and as a reference later on, as you try to find that formula whose details you have forgotten.

There are many people I wish to thank for their help and suggestions. At the Uni- versity of Colorado, these include Professors Larry Baggett, John Cary, Mike Dubson, Anatoli Levshin, Scott Parker, Steve Pollock, and Mike Ritzwoller. From other institu- tions, the following professors reviewed the manuscript or used a preliminary edition in their classes:

Meagan Aronson, U of Michigan Dan Bloom, Kalamazoo College Peter Blunden, U of Manitoba Andrew Cleland, UC Santa Barbara Gayle Cook, Cal Poly, San Luis Obispo Joel Fajans, UC Berkeley Richard Fell, Brandeis University Gayanath Fernando, U of Connecticut Jonathan Friedman, Amherst College David Goldhaber-Gordon, Stanford Thomas Griffy, U of Texas Elisabeth Gwinn, UC Santa Barbara Richard Hilt, Colorado College George Horton, Rutgers Lynn Knutson, U of Wisconsin

xiv Preface

Jonathan Maps, U of Minnesota, Duluth John Markert, U of Texas Michael Moloney, Rose-Hulman Institute Colin Morningstar, Carnegie Mellon Declan Mulhall, Cal Poly, San Luis Obispo Carl Mungan, US Naval Academy Robert Pompi, SUNY Binghamton Mark Semon, Bates College James Shepard, U of Colorado Richard Sonnenfeld, New Mexico Tech Edward Stern, U of Washington Michael Weinert, U of Wisconsin, Milwaukee Alma Zook, Pomona College

I am most grateful to all of these and their students for their many helpful comments. I would particularly like to thank Carl Mungan for his amazing vigilance in catching typos, obscurites, and ambiguities, and Jonathan Friedman and his student, Ben Heidenreich, who saved me from a really embarassing mistake in Chapter 10. I am especially grateful to my two friends and colleagues, Mark Semon at Bates College and Dave Goodmanson at the Boeing Aircraft Company, both of whom reviewed the manuscript with the finest of combs and gave me literally hundreds of suggestions; likewise to Christopher Taylor of the University of Wisconsin for his patient help with Mathematica and the mysteries of Latex. Bruce Armbruster and Jane Ellis of University Science Books are an author's dream come true. My copy editor, Lee Young, is a rarity indeed, an expert in English usage and physics; he suggested many significant improvements. Finally and most of all, I want to thank my wife Debby. Being married to an author can be very trying, and she puts up with it most graciously. And, as an English teacher with the highest possible standards, she has taught me most of what I know about writing and editing. I am eternally grateful.

For all our efforts, there will surely be several errors in this book, and I would be most grateful if you could let me know of any that you find. Ancillary material, including an instructors' manual, and other notices will be posted at the University Science Books website, www.uscibooks.com .

John R. Taylor Department of Physics University of Colorado Boulder, Colorado 80309, USA John.Taylor@Colorado.edu

PART I

Essen ia s CHAPTER 1

CHAPTER 2

CHAPTER 3

CHAPTER 4

CHAPTER 5

CHAPTER 6

CHAPTER 7

CHAPTER 8

CHAPTER 9

CHAPTER 10

CHAPTER 11

Newton's Laws of Motion

Projectiles and Charged Particles

Momentum and Angular Momentum

Energy

Oscillations

Calculus of Variations

Lagrange's Equations

Two-Body Central-Force Problems

Mechanics in Noninertial Frames

Rotational Motion of Rigid Bodies

Coupled Oscillators and Normal Modes

Part I of this book contains material that almost everyone would consider essential knowledge for an undergraduate physics major. Part II contains optional further topics from which you can pick according to your tastes and available time. The distinction between "essential" and "optional" is, of course, arguable, and its impact on you, the reader, depends very much on your state of preparation. For example, if you are well prepared, you might decide that the first five chapters of Part I can be treated as a quick review, or even skipped entirely. As a practical matter, the distinction is this: The eleven chapters of Part I were designed to be read in sequence, and in writing each chapter, I assumed that you would be familiar with most of the ideas of the preceding chapters— either by reading them or because you had met them elsewhere. By contrast, I tried to make the chapters of Part II independent of one another, so that you could read any of them in any order, once you knew most of the material of Part I.

CHAPTER

Newton's Laws of. Motion

1.1 Classical Mechanics

Mechanics is the study of how things move: how planets move around the sun, how a skier moves down the slope, or how an electron moves around the nucleus of an atom. So far as we know, the Greeks were the first to think seriously about mechanics, more than two thousand years ago, and the Greeks' mechanics represents a tremendous step in the evolution of modern science. Nevertheless, the Greek ideas were, by modern standards, seriously flawed and need not concern us here. The development of the mechanics that we know today began with the work of Galileo (1564-1642) and Newton (1642-1727), and it is the formulation of Newton, with his three laws of motion, that will be our starting point in this book.

In the late eighteenth and early nineteenth centuries, two alternative formulations of mechanics were developed, named for their inventors, the French mathematician and astronomer Lagrange (1736-1813) and the Irish mathematician Hamilton (1805- 1865). The Lagrangian and Hamiltonian formulations of mechanics are completely equivalent to that of Newton, but they provide dramatically simpler solutions to many complicated problems and are also the taking-off point for various modern developments. The term classical mechanics is somewhat vague, but it is generally understood to mean these three equivalent formulations of mechanics, and it is in this sense that the subject of this book is called classical mechanics.

Until the beginning of the twentieth century, it seemed that classical mechanics was the only kind of mechanics, correctly describing all possible kinds of motion. Then, in the twenty years from 1905 to 1925, it became clear that classical mechanics did not correctly describe the motion of objects moving at speeds close to the speed of light, nor that of the microscopic particles inside atoms and molecules. The result was the development of two completely new forms of mechanics: relativistic mechanics to describe very high-speed motions and quantum mechanics to describe the motion of microscopic particles. I have included an introduction to relativity in the "optional" Chapter 15. Quantum mechanics requires a whole separate book (or several books), and I have made no attempt to give even a brief introduction to quantum mechanics. 3

4 Chapter 1 Newton's Laws of Motion

Although classical mechanics has been replaced by relativistic mechanics and by quantum mechanics in their respective domains, there is still a vast range of interesting and topical problems in which classical mechanics gives a complete and accurate description of the possible motions. In fact, particularly with the advent of chaos theory in the last few decades, research in classical mechanics has intensified and the subject has become one of the most fashionable areas in physics. The purpose of this book is to give a thorough grounding in the exciting field of classical mechanics. When appropriate, I shall discuss problems in the framework of the Newtonian formulation, but I shall also try to emphasize those situations where the newer formulations of Lagrange and Hamilton are preferable and to use them when this is the case. At the level of this book, the Lagrangian approach has many significant advantages over the Newtonian, and we shall be using the Lagrangian formulation repeatedly, starting in Chapter 7. By contrast, the advantages of the Hamiltonian formulation show themselves only at a more advanced level, and I shall postpone the introduction of Hamiltonian mechanics to Chapter 13 (though it can be read at any point after Chapter 7).

In writing the book, I took for granted that you have had an introduction to Newtonian mechanics of the sort included in a typical freshman course in "General Physics." This chapter contains a brief review of the ideas that I assume you have met before.

1.2 Space and Time

Newton's three laws of motion are formulated in terms of four crucial underlying concepts: the notions of space, time, mass, and force. This section reviews the first two of these, space and time. In addition to a brief description of the classical view of space and time, I give a quick review of the machinery of vectors, with which we label the points of space.

Space

Each point P of the three-dimensional space in which we live can be labeled by a position vector r which specifies the distance and direction of P from a chosen origin 0 as in Figure 1.1. There are many different ways to identify a vector, of which one of the most natural is to give its components (x, y, z) in the directions of three chosen perpendicular axes. One popular way to express this is to introduce three unit vectors, X, Si , i, pointing along the three axes and to write

r = ySr zi. (1.1)

In elementary work, it is probably wise to choose a single good notation, such as (1.1), and stick with it. In more advapced work, however, it is almost impossible to avoid using several different notations. Different authors have different preferences (another popular choice is to use i, j, k for what I am calling 1, ST, i) and you must get used to reading them all. Furthermore, almost every notation has its drawbacks, which can

Section 1.2 Space and Time 5

Figure 1.1 The point P is identified by its position vector r, which gives the position of P relative to a chosen origin 0. The vector r can be specified by its components (x, y, z) relative to chosen axes Oxyz.

make it unusable in some circumstances. Thus, while you may certainly choose your preferred scheme, you need to develop a tolerance for several different schemes.

It is sometimes convenient to be able to abbreviate (1.1) by writing simply

r = y, z)• (1.2)

This notation is obviously not quite consistent with (1.1), but it is usually completely unambiguous, asserting simply that r is the vector whose components are x, y, z. When the notation of (1.2) is the most convenient, I shall not hesitate to use it. For most vectors, we indicate the components by subscripts x, y, z. Thus the velocity vector v has components vx , vy , v z and the acceleration a has components ax , ay , a,.

As our equations become more complicated, it is sometimes inconvenient to write out all three terms in sums like (1.1); one would rather use the summation sign E followed by a single term. The notation of (1.1) does not lend itself to this shorthand, and for this reason I shall sometimes relabel the three components x, y, z of r as r 1 , r2 , r3 , and the three unit vectors X, ST, z as e l , e2 , e3 . That is, we define

r1 = x, r2 = y, r3 = z,

and

e l = e2 = ST, e3 = Z.

(The symbol e is commonly used for unit vectors, since e stands for the German "eins" or "one.") With these notations, (1.1) becomes

3

r = r ie i r2e2 r3e3 = E ri ei . i=1

(1.3)

For a simple equation like this, the form (1.3) has no real advantage over (1.1), but with more complicated equations (1.3) is significantly more convenient, and I shall use this notation when appropriate.

6 Chapter 1 Newton's Laws of Motion

Vector Operations

In our study of mechanics, we shall make repeated use of the various operations that can be performed with vectors. If r and s are vectors with components

r = (r 1 , r2 , r3) and s = (s 1 , s2 , s3),

then their sum (or resultant) r + s is found by adding corresponding components, so that

r + s = (r i + s i , r2 + S2, 7'3 S3). (1.4)

(You can convince yourself that this rule is equivalent to the familiar triangle and parallelogram rules for vector addition.) An important example of a vector sum is the resultant force on an object: When two forces Fa and Fb act on an object, the effect is the same as a single force, the resultant force, which is just the vector sum

F = Fa + Fb

as given by the vector addition law (1.4). If c is a scalar (that is, an ordinary number) and r is a vector, the product cr is

given by

cr = (cr1 , cr2 , cr3). (1.5)

This means that cr is a vector in the same direction ) as r with magnitude equal to c times the magnitude of r. For example, if an object of mass m (a scalar) has an acceleration a (a vector), Newton's second law asserts that the resultant force F on the object will always equal the product ma as given by (1.5).

There are two important kinds of product that can be formed from any pair of vectors. First, the scalar product (or dot product) of two vectors r and s is given by either of the equivalent formulas

r • s = rs cos 0 3

r2S2 r3s3 = E rns, n=1

where r and s denote the magnitudes of the vectors r and s, and 6 is the angle between them. (For a proof that these two definitions are the same, see Problem 1.7.) For example, if a force F acts on an object that moves through a small displacement dr, the work done by the force is the scalar product F • dr, as given by either (1.6) or (1.7). Another important use of the scalar product is to define the magnitude of a vector: The magnitude (or length) of any vector r is denoted by r I or r and, by Pythagoras's

theorem is equal to /r? r22 + r32 . By (1.7) this is the same as

r == Ill == .1% (1.8)

The scalar product r • r is often abbreviated as r 2 .

(1.6)

(1.7)

Although this is what people usually say, one should actually be careful: If c is negative, cr is in the opposite direction to r.

Section 1.2 Space and Time 7

The second kind of product of two vectors r and s is the vector product (or cross product), which is defined as the vector p = r x s with components

px = ryS, - rz Sy py = r,S, - rx s,

pz = r,Sy - ry Sx

or, equivalently

[X ST i r x s = det rx ry r, ,

s, s s

y z

where "det" stands for the determinant. Either of these definitions implies that r x s is a vector perpendicular to both r and s, with direction given by the familiar right-hand rule and magnitude rs sin B (Problem 1.15). The vector product plays an important role in the discussion of rotational motion. For example, the tendency of a force F (acting at a point r) to cause a body to rotate about the origin is given by the torque of F about 0, defined as the vector product 1" = r x F.

Differentiation of Vectors

Many (maybe most) of the laws of physics involve vectors, and most of these involve derivatives of vectors. There are so many ways to differentiate a vector that there is a whole subject called vector calculus, much of which we shall be developing in the course of this book. For now, I shall mention just the simplest kind of vector derivative, the time derivative of a vector that depends on time. For example, the velocity v(t) of a particle is the time derivative of the particle's position r(t); that is, v = dr/dt. Similarly the acceleration is the time derivative of the velocity, a = dv/dt.

The definition of the derivative of a vector is closely analogous to that of a scalar. Recall that if x (t) is a scalar function of t, then we define its derivative as

dx. Ax — = lim — dt At-0 At

where Ax = x (t + At) — x(t) is the change in x as the time advances from t to t + At. In exactly the same way, if r(t) is any vector that depends on t, we define its derivative as

where

dr,. Ar = um —

dt At->0 At (1.10)

Ar = r(t + At) — r(t) (1.11)

is the corresponding change in r(t). There are, of course, many delicate questions about the existence of this limit. Fortunately, none of these need concern us here: All of the vectors we shall encounter will be differentiable, and you can take for granted that the required limits exist. From the definition (1.10), one can prove that the derivative has all of the properties one would expect. For example, if r(t) and s(t)

(1.9)

8 Chapter 1 Newton's Laws of Motion

are two vectors that depend on t, then the derivative of their sum is just what you would expect:

d dr ds

dt = —dt + dt . (1.12)

Similarly, if r(t) is a vector and f (t) is a scalar, then the derivative of the product f (t)r (t) is given by the appropriate version of the product rule,

dr df

dt (f r) = f

dt +

dt r. (1.13)

If you are the sort of person who enjoys proving these kinds of proposition, you might want to show that they follow from the definition (1.10). Fortunately, if you do not enjoy this kind of activity, you don't need to worry, and you can safely take these results for granted.

One more result that deserves mention concerns the components of the derivative of a vector. Suppose that r, with components x, y, z, is the position of a moving particle, and suppose that we want to know the particle's velocity v = dr Idt. When we differentiate the sum

r = + + zi, (1.14)

the rule (1.12) gives us the sum of the three separate derivatives, and, by the product rule (1.13), each of these contains two terms. Thus, in principle, the derivative of (1.14) involves six terms in all. However, the unit vectors Sr', and i do not depend on time, so their time derivatives are zero. Therefore, three of these six terms are zero, and we are left with just three terms:

dr dx dy „ dz — = —x + —y + —z. dt dt dt dt

Comparing this with the standard expansion

V = v y ST

we see that

(1.15)

dx v = dY vx -= dt dt

and V, = dz

dt (1.16)

In words, the rectangular components of v are just the derivatives of the corresponding components of r. This is a result that we use all the time (usually without even think- ing about it) in solving elementary mechanics problems. What makes it especially noteworthy is this: It is true only because the unit vectors X, jr ', and i are constant, so that their derivatives are absent from (1.15). We shall find that in most coordinate systems, such as polar coordinates, the basic unit vectors are not constant, and the result corresponding to (1.16) is appreciably less transparent. In problems where we need to work in nonrectangular coordinates, it is considerably harder to write down velocities and accelerations in terms of the coordinates of r, as we shall see.

Section 1.3 Mass and Force 9

Time

The classical view is that time is a single universal parameter t on which all observers agree. That is, if all observers are equipped with accurate clocks, all properly syn- chronized, then they will all agree as to the time at which any given event occurred. We know, of course, that this view is not exactly correct: According to the theory of relativity, two observers in relative motion do not agree on all times. Nevertheless, in the domain of classical mechanics, with all speeds much much less than the speed of light, the differences among the measured times are entirely negligible, and I shall adopt the classical assumption of a single universal time (except, of course, in Chap- ter 15 on relativity). Apart from the obvious ambiguity in the choice of the origin of time (the time that we choose to label t = 0), all observers agree on the times of all events.

Reference Frames

Almost every problem in classical mechanics involves a choice (explicit or implicit) of a reference frame, that is, a choice of spatial origin and axes to label positions as in Figure 1.1 and a choice of temporal origin to measure times. The difference between two frames may be quite minor. For instance, they may differ only in their choice of the origin of time — what one frame labels t = 0 the other may label t' = to 0. Or the two frames may have the same origins of space and time, but have different orientations of the three spatial axes. By carefully choosing your reference frame, taking advantage of these different possibilities, you can sometimes simplify your work. For example, in problems involving blocks sliding down inclines, it often helps to choose one axis pointing down the slope.

A more important difference arises when two frames are in relative motion; that is, when one origin is moving relative to the other. In Section 1.4 we shall find that not all such frames are physically equivalent? In certain special frames, called inertial frames, the basic laws hold true in their standard, simple form. (It is because one of these basic laws is Newton's first law, the law of inertia, that these frames are called inertial.) If a second frame is accelerating or rotating relative to an inertial frame, then this second frame is noninertial, and the basic laws — in particular, Newton's laws — do not hold in their standard form in this second frame. We shall find that the distinction between inertial and noninertial frames is central to our discussion of classical mechanics. It plays an even more explicit role in the theory of relativity.

1.3 Mass and Force

The concepts of mass and force are central to the formulation of classical mechanics. The proper definitions of these concepts have occupied many philosophers of science and are the subject of learned treatises. Fortunately we don't need to worry much about

2 This statement is correct even in the theory of relativity.

10 Chapter 1 Newton's Laws of Motion

force

Figure 1.2 An inertial balance compares the masses m 1 and m 2

of two objects that are attached to the opposite ends of a rigid rod.

The masses are equal if and only if a force applied at the rod's

midpoint causes them to accelerate at the same rate, so that the

rod does not rotate.

these delicate questions here. Based on your introductory course in general physics, you have a reasonably good idea what mass and force mean, and it is easy to describe how these parameters are defined and measured in many realistic situations.

Mass

The mass of an object characterizes the object's inertia—its resistance to being accelerated: A big boulder is hard to accelerate, and its mass is large. A little stone is easy to accelerate, and its mass is small. To make these natural ideas quantitative we have to define a unit of mass and then give a prescription for measuring the mass of any object in terms of the chosen unit. The internationally agreed unit of mass is the kilogram and is defined arbitrarily to be the mass of a chunk of platinum—iridium stored at the International Bureau of Weights and Measures outside Paris. To measure the mass of any other object, we need a means of comparing masses. In principle, this can be done with an inertial balance as shown in Figure 1.2. The two objects to be compared are fastened to the opposite ends of a light, rigid rod, which is then given a sharp pull at its midpoint. If the masses are equal, they will accelerate equally and the rod will move off without rotating; if the masses are unequal, the more massive one will accelerate less, and the rod will rotate as it moves off.

The beauty of the inertial balance is that it gives us a method of mass comparison that is based directly on the notion of mass as resistance to being accelerated. In practice, an inertial balance would be very awkward to use, and it is fortunate that there are much easier ways to compare masses, of which the easiest is to weigh the objects. As you certainly recall from your introductory physics course, an object's mass is found to be exactly proportional to the object's weight 3 (the gravitational force on the object) provided all measurements are made in the same location. Thus two

3 This observation goes back to Galileo's famous experiments showing that all objects are accelerated at the same rate by gravity. The first modern experiments were conducted by the Hungarian physicist Eiityos (1848-1919), who showed that weight is proportional to mass to within

Section 1.3 Mass and Force 11

objects have the same mass if and only if they have the same weight (when weighed at the same place), and a simple, practical way to check whether two masses are equal is simply to weigh them and see if their weights are equal.

Armed with methods for comparing masses, we can easily set up a scheme to mea- sure arbitrary masses. First, we can build a large number of standard kilograms, each one checked against the original 1-kg mass using either the inertial or gravitational balance. Next, we can build multiples and fractions of the kilogram, again checking them with our balance. (We check a 2-kg mass on one end of the balance against two 1-kg masses placed together on the other end; we check two half-kg masses by verifying that their masses are equal and that together they balance a 1-kg mass; and so on.) Finally, we can measure an unknown mass by putting it on one end of the balance and loading known masses on the other end until they balance to any desired precision.