# An introduction to mechanics Kleppner Kolenkow

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CONTENTS
LIST OF EXAMPLES xi PREFACE xv TO THE TEACHER xix
1.1 INTRODUCTION 2

1.2 VECTORS 2

Definition of a Vector, The Algebra of Vectors, 3.

1.3 COMPONENTS OF A VECTOR 8

1.4 BASE VECTORS 10

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 EQUATIONS 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 dr/dt, 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
2.1 INTRODUCTION 52

2.2 NEWTON’S LAWS 53

Newton’s First Law, 55; Newton’s Second Law, 56; Newton’s Third Law, 59.

2.3 STANDARDS AND UNITS 64

The Fundamental Standards, 64; Systems of Units, 67.

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
3.1 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 MOMENTUM RELATION 130

3.5 MOMENTUM AND THE FLOW OF MASS 133

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 Illustrations of Potential Energy, 170.

4.8 WHAT POTENTIAL ENERGY TELLS 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
5.1 INTRODUCTION 202

5.2 PARTIAL DERIVATIVES 202

5.3 HOW TO FIND THE FORCE IF YOU KNOW THE POTENTIAL ENERGY 206

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
6.1 INTRODUCTION 232

6.2 ANGULAR MOMENTUM OF A PARTICLE 233

6.3 TORQUE 238

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

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7 RIGID BODY MOTION AND THE CONSERVATION OF  ANGULAR  MOMENTUM 7.1 INTRODUCTION 288  7.2 THE VECTOR NATURE OF ANGULAR VELOCITY AND ANGULAR MOMENTUM 288  7.3 THE GYROSCOPE 295  7.4 SOME APPLICATIONS OF GYROSCOPE MOTION 300  7.5 CONSERVATION OF ANGULAR MOMENTUM 305  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 SYSTEMS AND FICTITIOUS FORCES 8.1 INTRODUCTION 340  8.2 THE GALILEAN TRANSFORMATIONS 340  8.3 UNIFORMLY ACCELERATING SYSTEMS 343  8.4 THE PRINCIPLE OF EQUIVALENCE 346  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 FORCE MOTION 9.1 INTRODUCTION 378  9-2 CENTRAL FORCE MOTION AS A ONE BODY PROBLEM 378  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 HARMONIC OSCILLATOR 10.1 INTRODUCTION AND REVIEW 410  Standard Form of the Solution, 410; Nomenclature, 411; Energy Considerations, 412; Time Average Values, 413; Average Energy, 413.  10.2 THE DAMPED HARMONIC ORCII 1 АТПР Д14
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
11.1 THE NEED FOR A NEW MODE OF THOUGHT 442

11.2 THE MICHELSON-MORLEY EXPERIMENT 445

11.3 THE POSTULATES OF SPECIAL RELATIVITY 450

The Universal Velocity, 451; The Principle of Relativity, 451; The Postulates of Special Relativity, 452.

11.4 THE GALILEAN TRANSFORMATIONS 453

11.5 THE LORENTZ TRANSFORMATIONS 455 PROBLEMS 459
12.1 INTRODUCTION 462

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; Relaiivistic Doppler Effect, 477; The Doppler Effect for an Observer off the Line of Motion, 478.

12.6 THE TWIN PARADOX 480 PROBLEMS 484
13.1 MOMENTUM 490

13.2 ENERGY 493

13.3 MASSLESS PARTICLES 500

13.4 DOES LIGHT TRAVEL AT THE VELOCITY OF LIGHT? 508 PROBLEMS 512
14.1 INTRODUCTION 516

14.2 VECTORS AND TRANSFORMATIONS 516

Rotation about the z Axis, 517; Invariants of a Transformation, 520; The Transformation Properties of Physical Laws, 520; Scalar Invariants, 521.

14.3 MINIKOWSKI SPACE AND FOUR-VECTORS 521

14.4 THE MOMENTUM-ENERGY FOUR-VECTOR 527

14.5 CONCLUDING REMARKS 534 PROBLEMS 536
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.
EXAMPLES 1.5 Vector Algebra, 9; 1.6 Construction of a Perpendicular Vector, 10.  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-
1 VECTORS AND KINEMATICS —A FEW MATHEMATICAL PRELIMINARIES tational Field, 21; 1.11 Nonuniform Acceleration—The Effect of a Radio Wave on an Ionospheric Electron, 22. §§  1.12 Circular Motion and Rotating Vectors, 25. Ц  1.13 Circular Motion and Straight Line Motion in Polar Coordinates, 34;  1.14 Velocity of a Bead on a Spoke, 35; 1.15 Off-center Circle, 35; 1.16 Acceleration of a Bead on a Spoke, 37; 1.17 Radial Motion without Acceleration, 38. Щ
2 NEWTON’S LAWS—THE FOUNDATIONS OF  NEWTONIAN  MECHANICS EXAMPLES, CHAPTER 2 Ш  2.1 Astronauts in Space—Inertial Systems and Fictitious Force, 60.  2.2 The Astronauts' Tug-of-v,ar, 70; 2.3 Freight Tpiin, 72; 2.4 Constraints, 74; 2.5 Block on String 1, 75; 2.6 Block on String, 76; 2.7 The Whirling Block, 76; 2.8 The Conical Pendulum, 77. *  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 'Tree Motion in a Viscous Medium, 96; 2.17 Spring and Block—The Equating- for Simple Harmonic Motion, 98; 2.18 The Spring Gun—An Example Illustrating Initial Conditions, 99. ;
3 MOMENTUM EXAMPLES, CHAPTER 3  3.1 The Boia, 115; 3.2 Drum Major’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 Ме-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 AND ENERGY EXAMPLES, CHAPTER 4  4.1 Mass Thrown Upward in a Uniform Gravitational Field, 154; 4.2 Solving 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.
4.11 Potential Energy of a Uniform Force Field, 170; 4.12 Potential Energy of an Inverse Square Force, 171; 4.13 Bead, Hoop, and Spring, 172.

4.14 Energy and Stability—The Teeter Toy, 175.

4.15 Molecular Vibrations, 179; 4.16 Small Oscillations, 181.

4.17 Block Sliding down Inclined Plane, 183.

4.18 Elastic Collision of Two Balls, 190; 4.19 Limitations on Laboratory Scattering Angle, 193.
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.
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 Torque on the Conical Pendulum, 245; 6.7 Torque due to Gravity, 247.

6.8 Moments of Inertia of Some Simple Objects, 250; 6.9 The Parallel Axis Theorem, 252.

6.10 Atwood's Machine with a Massive Pulley, 254.

6.11 Grandfather's Clock, 256; 6.12 Kater’s Pendulum, 258; 6.13 The Doorstep, 259.

6.14 Angular Momentum of a Rolling Wheel, 262; 6.15 Disk on Ice, 26 ,

6.16 Drum Rolling down a Plane, 265; 6.17 Drum Rolling down a Plane: 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, 29 ,

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, 30 ,

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. 1Q

7.16 Stability of Rotational Motion, 322; 7.17 The Rotating Rod, 323; 7. ° Fiilar'c Pnilaftrmc anrl Tnmup-frpe Precession. 324.
8 NONINERTIAL SYSTEMS AND FICTITIOUS FORCES EXAMPLES, CHAPTER 8  8.1 The Apparent Force of Gravity, 346; 8.2 Cylinder on an Accelerating Plank, 347; 8.3 Pendulum in an Accelerating Car, 347.  8.4 The Driving Force of the Tides, 350; 8.5 Equilibrium Height of the Tide, 352.  8.6 Surface of a Rotating Liquid, 362; 8.7 The Coriolis Force, 363; 8.8 Deflection 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 FORCE MOTION EXAMPLES, CHAPTER 9  9.1 Noninteracting Particles, 384; 9.2 The Capture of Comets, 387; 9.3 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 HARMONIC OSCILLATOR EXAMPLES, CHAPTER 10  10.1 Initial Conditions and the Frictionless Harmonic Oscillator, 411.  10.2 The 0 of Two Simple Oscillators, 419; 10.3 Graphical Analysis of a Damped Oscillator, 420.  10.4 Forced Harmonic Oscillator Demonstration, 424; 10.5 Vibration Eliminator, 428.
11 THE SPECIAL THEORY OF  RELATIVITY EXAMPLES, CHAPTER 11  11.1 The Galilean Transformations, 453; 11.2 A Light Pulse as Described by the Galilean Transformations, 455.
12 RELATIVISTIC KINEMATICS EXAMPLES, CHAPTER 12  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 MOMENTUM AND ENERGY EXAMPLES, CHAPTER 13  13.1 Velocity Dependence of the Electron's Mass, 492.  13.2 Relativistic Energy and Momentum in an Inelastic Collision, 496; 13.3 The Equivalence of Mass and Energy, 498.  13 Л Tkn ctt i cm. 11 r n -X , ,_LX pm.
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.
EXAMPLES, CHAPTER 14

14.1 Transformation Properties of the Vector Product, 518; 14.2 A Nonvector, 519.

14.3 Time Dilation, 524; 14.4 Construction of a Four-vector: The Four-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.
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