daniel kleppner robert kolenkow an introduction

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Daniel Kleppner of Professor Associate Physics Massach usetts Institute of Technology J Robert Formerly Associate Kolenkow Professor of Physics,Massachusetts Institute of AN INTRODUCTION Technology) TO MECHANICS) II) Boston, Massachusetts Burr Ridge, Illinois New Dubuque, Iowa Madison,Wisconsin San Francisco, California St Louis,Missouri))) York, New York McGraw-Hill) A AN Copyright All INTRODUCTION TO rights of The McGraw.HiU Companies Division @ 1973 by McGraw-Hill, Inc reserved Printed in the United States of America Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or in any form or by any means, or stored in a data base or retrieval distributed without the prior written of the publisher.) system, permission MECHANICS) Printed and by Book-mart bound 20 BKMBKM Press, Inc.) 998) This book was set in News Gothic by The Maple Press Company The editors were Jack L Farnsworth and J W Maisel; the was Edward A Butler; designer 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.) Mechanics QA805.K62 ISBN 0-07-035048-5) I Kolenkow, 531 Robert, joint 72-11770) author II Title.))) To our parents Beatrice and Otto Katherine and John))) OF EXAMPLES xv LIST xi PREFACE CONTENTS) TO THE 1.1 VECTORS AND KINEMATICS -A FEW TEACHER xix) INTRODUCTION 1.2 VECTORS of a Vector, The Algebra Definition 1.3 OF COMPONENTS MATHEMATICAL 1.4 BASE VECTORS PRELIMINARIES) 1.5 DISPLACEMENT 1.6 VELOCITY Motion in 10 AND THE POSITION 13 ACCELERATION AND FORMAL 1.8 MORE 1.9 MOTION Dimension, 14; Motion in Several 14; A Word Dimensions, Units, 18 SOLUTION OF KINEMATICAL EQUATIONS THE DERIVATIVE OF A VECTOR 23 ABOUT IN PLANE POLAR COORDINATES 27 27; Velocity in Coordinates, in Polar Coordinates, Acceleration The 1.1 Series, References PROBLEMS NEWTON'S LAWS 53 MECHANICS) First Law, MOMENTUM) 55; Second Newton's Law, 56; Newton's Third Law, 59 2.3 STANDARDSAND UNITS 64 67 The Fundamental 64; Systems of Units, Standards, 2.4 SOME APPLICATIONSOF NEWTON'S 68 LAWS 2.5 THE EVERYDAY OF PHYSICS 79 FORCES 80; The Electrostatic Force, 86; Field, Weight, and the Gravitational Gravity, and Atomic Contact Force of a String, 87; Tension-The 87; Tension Forces, Forces,91;The Normal Force, 92; Friction, 92; Viscosity, 95; The Linear Restoring 97 Force: Hooke's Law, the Spring, and Simple Harmonic Motion, Note 2.1 THE GRAVITATIONAL OF A SPHERICAL ATTRACTION 101 SHELL PROBLEMS 52 INTRODUCTION 2.2 NEWTONIAN dr/dt, 31; to Calculus Texts, 47 2.1 Newton's 27; Evaluating Coordinates, 36 47) NEWTON'S OF Polar APPROXIMATION METHODS 39 45 41; Taylor's Series,42;Differentials, LAWS-THE FOUNDATIONS about MATHEMATICAL Binomial Some VECTOR 11 and 1.7 Note One Dimensions Polar of Vectors, A VECTOR 3.1 3.2 112 INTRODUCTION DYNAM ICS OF A SYSTEM Center of 3.3 103) IMPULSE RELATION 3.5 PARTICLES OF MOMENTUM CONSERVATION Center of Mass Coordinates, 3.4 OF 113 Mass, 116 AND 122 127 A RESTATEMENT OF THE MOMENTUM 130 MOMENTUMAND THE FLOW OF MASS 133))) CONTENTS) viii) 3.6 Note MOMENTUM TRANSPORT 139 3.1 CENTER OF MASS 145 147) PROBLEMS WORK AND ENERGY) 4.1 INTRODUCTION 4.2 INTEGRATING 4.4 DI THE EQUATION OF MOTION IN ONE 153 DI M ENSION 4.3 152 THE THEOREM WORK-ENERGY INTEGRATING THE M ENS IONS 158 EQUATION IN OF 156 DIMENSION ONE MOTION IN SEVERAL 4.5 THE WORK-ENERGY 160 THEOREM 4.6 APPLYING THE WORK-ENERGY 162 THEOREM 4.7 POTENTJAL ENERGY 168 Illustrations of Potential Energy, 170 4.8 WHAT POTENTIAL ENERGY TELLS US ABOUT FORCE 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 4.13 173 184 186 POWER 187 CONSERVATION LAWS AND PARTICLE COLLISIONS Collisions and Conservation Collisions, 188; Elastic and Inelastic Laws, Collisions in One Dimension, 189; Collisions and Center of MassCoordinates, 4.14 PROBLEMS SOME MATHEMATICAL ASPECTS OF FORCE AND ENERGY) 194) INTRODUCTION 202 PARTIAL DERIVATIVES 202 5.3 HOW TO FIND THE FORCE IF YOU KNOW THE POTENTIAL 206 ENERGY 5.4 THE GRADIENT OPERATOR 207 5.5 THE PHYSICAL MEANING OF THE GRADIENT 210 Constant 211 Surfaces and Contour Lines, Energy 5.6 HOW TO FIND 215 OUT IF A FORCE IS CONSERVATIVE 5.7 STOKES' THEOREM 225 5.1 5.2 PROBLEMS ANGULAR 6.1 MOMENTUM 6.2 AND FIXED AXIS ROTATION) 228) INTRODUCTION 232 MOMENTUM ANGULAR 6.3 TORQUE OF A PARTICLE 233 238 ANGULAR MOMENTUMAND FIXED AXIS ROTATION 248 OF PURE ROTATION ABOUT AN AXIS 253 DYNAMICS 6.6 THE PHYSICALPENDULUM 255 The Simple Pendulum, 253; The Physical Pendulum, 257 6.7 MOTION INVOLVING ROTATION BOTH TRANSLATION AND The Work-energyTheorem, 267 6.8 THE BOHR ATOM 270 Note 6.1 CHASLES' THEOREM 274 Note 6.2 PENDULUM MOTION 276 6.4 6.5 PROBLEMS 279))) 260 188; 190 CONTENTS) BODY MOTION RIGID AND THE CONSERVATION OF ANGULAR MOMENTUM) ix) 7.1 INTRODUCTION 7.2 THE VECTOR NATURE 7.4 7.5 7.6 OF MOMENTUM ANGULAR 7.3 288 OF GYROSCOPE MOTION 300 ANGULAR MOMENTUM 305 OF A ROTATING RIGID BODY ANGULAR MOMENTUM 308 Axes, 313; Rotational Angular Momentum and the Tensor of Inertia, 308;Principal Kinetic 313; Rotation about a Fixed Point, 315 Energy, 7.7 ADVANCED TOPICS IN THE DYNAM ICS OF RIGID BODY ROTATION 316 Note APPLICATIONS SOME OF CONSERVATION 7.1 AND FICTITIOUS FORCES) Why the Precession: ROTATIONS INFINITESIMAL GYROSCOPES ABOUT Precession, Earth Wobbles, 317; Euler's 326 328 Torque-free Precession,331; 331; Case Case 334) PROBLEMS SYSTEMS AND FINITE Note 7.2 MORE Case Uniform Nutation, 331 NONINERTIAL AND 295 THE GYROSCOPE Introduction, 316; Torque-free Equations, 320 VELOCITY ANGULAR 288 8.1 INTRODUCTION 8.2 THE 8.3 UNIFORMLY 8.4 THE 8.5 PHYSICS 340 340 TRANSFORMATIONS GALILEAN SYSTEMS ACCELERATING OF EQUIVALENCE 346 IN A ROTATING COORDINATE 343 PRINCIPLE SYSTEM 355 and Rotating Coordinates, 356; Acceleration Relative to Rotating Coordinate System, 359 358; The Apparent Force in a Rotating Coordinates, THE EQUIVALENCE PRINCIPLE AND THE Note 8.1 RED SHIFT 369 GRAVITATIONAL Note 8.2 ROTATING COORDINATETRANSFORMATION 371 Time Derivatives PROBLEMS CENTRAL FORCE MOTION) 9.1 INTRODUCTION CENTRAL 9.4 FINDING 9.5 THE 9.7 Note FORCE THE ENERGY PLANETARY MOTION IN REAL AND EQUATION MOTION 390 KEPLER'S LAWS 400 9.1 PROPERTIES OF THE PROBLEMS OSCILLATOR) 378 MOTION AS A ONE BODY 378 PROBLEM 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 9.2 9.6 10 THE HARMONIC 372) PROBLEMS 382 ENERGY DIAGRAMS 383 ELLIPSE 403 406) 10.1 INTRODUCTION AND REVIEW 410 Standard Form of the Solution, 410; Nomenclature, 411; Energy 412; Time A verage Values, 413; A verage Energy, 413 10.2 THE DAMPED HARMONIC OSCILLATOR 414 418.))) 416; The Q of an Oscillator, Energy, Considerations, x) CONTENTS) FORCED HARMONIC OSCILLATOR 421 Forced Oscillator, 421; Resonance, 423; The Forced Damped Harmonic Oscillator, in a Lightly Damped System: The Quality 424; Resonance THE 10.3 The Undamped Factor Q, 426 RESPONSE 10.4 Note 10.1 432 VERSUS RESPONSE IN FREQUENCY FOR THE THE EQUATION OF MOTION 433 OSCILLATOR TIME IN OF SOLUTION DAMPED UNDRIVEN The Useof Complex Variables, Note 10.2 SOLUTION OF THE FORCED OSCILLATOR 437 THE 11 OF RELA TIVITY) 12 RELATIVISTIC THE 11.2 THE 11.3 THE 11.1 The Damped Oscillator, EQUATION 435 OF MOTION THE FOR 438) PROBLEMS SPECIAL THEORY 433; The MODE 442 445 450 OF SPECIAL RELATIVITY POSTULATES of Relativity, 451; The Universal Velocity, 451; The Principle FOR NEED A NEW 452 Special Relativity, 11.4 THE GALILEAN 11.5 THE LORENTZ PROBLEMS 459) TRANSFORMATIONS TRANSFORMATIONS INTRODUCTION 12.1 OF THOUGHT EXPERIMENT MICHELSON-MORLEY Postulates of 453 455 462 THE ORDER OF EVENTS 463 AND TIME DILATION CONTRACTION KINEMATICS) 12.2 SIMULTANEITYAND 12.3 LORENTZ THE 466 The Lorentz Contraction, 466; Time Dilation, 468 THE RELATIVISTIC TRANSFORMATIONOF VELOCITY 12.4 12.5 TH E DOPPLER EFFECT 475 The Doppler Shift in Sound, 475; Relativistic Doppler Effect, Effect for an Observer off the Line of Motion, 478 12.6 RELATIVISTIC MOMENTUM AND ENERGY) 14 FOU R- VECTORS AND RELATIVISTIC I NV ARIANCIE) 13.1 13.2 MOM 477; The Doppler 480 484) PROBLEMS 13 PARADOX TWIN THE 472 ENTUM 490 493 ENERGY MASSLESS PARTICLES 500 DOES LIGHT TRAVEL AT THE VELOCITY PROBLEMS 512) 13.3 13.4 14.1 INTRODUCTION OF LIGHT? 508 516 VECTORS AND TRANSFORMATIONS 516 Rotation about the z Axis, 517; Invariants of a Transformation, formation Properties of Physical Laws, 520; Scalar Invariants, 14.3 MINIKOWSKI SPACE AND 521 FOUR-VECTORS 14.4 THE MOMENTUM-ENERGY 527 FOUR-VECTOR 14.5 CONCLUDING REMARKS 534 14.2 536) PROBLEMS INDEX 539))) 520; 521 The Trans- EXAMPLES,CHAPTER OF LIST EXAMPLES) Wave on VECTORS AND KINEMATICS -A FEW MATHEMATICAL PRELl M INARI 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, 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, Motion in a Uniform Gravi20; 1.10 tational Effect of a Radio Field, 21; 1.11 Nonuniform Acceleration-The 1.12 an Motion 1.13 Circular 1.14 NEWTON'S LAWS-THE Vectors, 25 of a Velocity 38.) tion, CHAPTER EXAMPLES, 2.1 and Rotating and Straight Line in Polar Coordinates, 34; Motion Bead on a Spoke, 35; 1.15 Off-center Circle, 35; 1.16 Acof a Bead on a Spoke, 37; 1.17 Radial Motion without Accelera- Motion celeration ES) Electron, 22 Ionospheric Circular in Space-Inertial 60 Force, Systems and Fictitious The Astronauts' Tug-of-v.ar, 70; 2.3 Freight Train, 72; 2.4 Constraints, Block on String 1, 75; \037.6 Block on String 2, 76; 2.7 The Whirling 74; 2.5 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 Block and Wedge 88; 2.12 Whirling 89; 2.13 Pulleys, 90; 2.14 Rope, Rope, with in a Viscous 93; 2.15 The Spinning Friction, Terror, 94; 2.16 Free Motion for Simple Harmonic Medium, 96; 2.17 Spring and Block-The Equation Initial Conditions, Motion, 98; 2.18 The Spring Gun-An Example Illustrating Astronauts 2.2 FOUNDATIONS OF NEWTONIAN MECHANICS) 99.) MOMENTUM EXAMPLES, CHAPTER3 Drum Major's 115; 3.2 Nonuniform Rod, 119; 3.4 Center of 3.1 The Bola, Center of Mass Motion, 3.6 Spring Gun System, 125;3.8 3.9 Rubber Ball 3.3 117; Baton, Mass of a Center of Mass of a Sheet, 120; 3.5 Triangular 122 Recoil, 123; 3.7 Push Me-Pull Earth, Moon, and Sun-A Three Body You, 128 The Rebound, 131; 3.10 How to Avoid Broken Ankles, 132 Car and Hopper, 135; Momentum, 134; 3.12 Freight 3.13 Leaky Freight Car, 136; 3.14 Rocket in Free Space, 138; 3.15 Rocket in a Gravitational Field, 139 3.16 Momentum to a Surface, 141;3.17 A Dike at the Bend of a Transport Pressure of a Gas, 144.) River, 143; 3.18 3.11 Mass Flow WORK AND ENERGY) and EXAM PLES, CHAPTER the in a Uniform Gravitational Upward of Simple Harmonic Motion, in an Inverse Motion 156 Field, 4.1 MassThrown 4.3 4.4 Equation Vertical 154; 4.2 Solving Square The Conical Pend ulum, 161; 162 4.6 The Inverted 4.8 Work Done 167; 4.10 Field, 154 Parametric Pendulum, by a Central 4.5 Escape Velocity-The 164;4.7 Work Force, 167;4.9 Evaluation of a Line Done A by a Uniform Force, 165; Line Integral, Path-dependent Integral, General Case, 168.))) OF EXAMPLES) LIST xii) 4.11 Potential of an Inverse Potential Energy of a Uniform Force Field, 170; 4.12 172 and Spring, Force, 171;4.13 Bead,Hoop, Energy Square 4.14 Energy and Stability-The Teeter Toy, 175 4.15 Molecular Vibrations, 179;4.16 Small 181 Oscillations, 4.17 Block Sliding down Inclined Plane, 183 4.18 Elastic Collision of Two Balls, 190; 4.19 Limitations Scattering Angle, 193.) SOME MATHEMATICAL ASPECTS OF FORCE AND ENERGY) CHAPTER EXAMPLES, 5.1 Partial 5.3 Gravitational Attraction Field, 209; 5.5 5.6 5.7 220; Gravitational The Curl of the 5.9 A Most Using Stokes' of the a Gravitational How Theorem, Partial Uniform Derivative, 205 Gravitational Masses, 209 Star System, 212 Force, Unusual Force Function, 222;5.11 Energy 5.12 Applications 5.4 Particle, 208; Attraction by Two Point by for a Binary Contours Energy Laboratory 203; 5.2 Derivatives, on A Nonconservative Force, Construction of the Potential 219; 5.8 Field, 221; 5.10 the Curl Got Its Name, 224 227.) ANGULAR EXAM PLES, CHAPTER MOMENTUM 6.1 Angular Momentum Momentum of a Sliding Block, 236; 6.2 Angular AXIS FIXED of the Conical Pendulum, 237 ROTATION) and 6.3 Central Force Motion the Law of Equal 240; 6.4 Capture Areas, on a Sliding Cross Section of a Planet, Block, 244; 6.6 241; 6.5 Torque 247 Pendulum, 245; 6.7 Torque on the Conical Torque due to Gravity, The Parallel Axis 6.8 Moments of Inertia of Some Simple Objects, 250; 6.9 AND Theorem, 252 6.10 Atwood's Machine 6.11 Grandfather's step, 259 254 with a Massive Pulley, 6.12 Kater's Pendulum, Clock, 256; 258;6.13 The Door- 6.14 Angular Momentum of a Rolling 262; 6.15 Disk on Ice, 264; Wheel, down a Plane: a Plane, 265; 6.17 Drum 6.16 Drum Rolling down Rolling Energy Method, 268; 6.18 The Falling Stick, 269.) RIGID BODY MOTION AN D TH E CONSERVATION OF ANGULAR MOMENTUM) CHAPTER EXAMPLES, 7.1 Rotations through 289; 7.2 Rotation in the xy Plane, 291; Angles, of a Momentum of Angular Velocity, 291; 7.4 Angular Skew Rod, 293; 7.6 Rotating Skew Rod, 292; 7.5 Torque on the Rotating 294 Torque on the Rotating Skew Rod (GeometricMethod), 7.7 Gyroscope Precession, 298; 7.8 Why a Gyroscope Precesses, 299 7.9 Precessionof the Equinoxes, 300; 7.10 The Gyrocompass Effect, 301; 304 7.11 Gyrocompass Motion, 302; 7.12 The Stability of Rotating Objects, for a Rotating Skew 7.13 Rotating Dumbbell, 310; 7.14 The Tensor of Inertia Do Flying Saucers Make Better Spacecraft than 312; 7.15 Why Flying Rod, 7.3 Vector Ciga rs, 7.16 Euler's Finite Nature 314 Motion, 322; 7.17 The Equations and Torque-free Precession,324.))) Stability of Rotational Rotating Rod, 323; \037 18 532) FOUR-VECTORS RELATIVISTIC AND INVARIANCE) or) = (11flc2 )2 + 2E 1E E'2 we where and ')I11!1c2 have used E 1I1 2c 2, where total E = E' = (11f]2+ The 111 energy = (')1M + lIf the in 2)C E') (111 M + 2 = El2 ')I = - (1lf1c )2 - u 1/V1 /c Hence,) is) 6) initial the energy available for is) processes physical 2')11111111 2)2 + 7) M 2) case is that of eq ual 111 = masses Eq ua- 111 V2 V1 + ')I - E 1+')1 V2 - V1 + ')I) = and E'IE = At low speeds, low velocity limit, ')I is in rest mass energy and kinetic energy is relatively To discuss the high-speed limit, it is convenient to portant (8) in terms of the projectile energy El = ')IlIfc most of In the the unim- energy E' -E write Eq V2 V1 + E For E' = problem, El Fo r our 5) system laboratory An im porta nt practical tion (7) becomes) E' ,) ')IlI!l + E) Pl2C 2.) + 2')1111l1112)!c the fraction of a nd E 22,) + \037 111c \302\273 2) ElIMc , we have V\037 E V'E:) The fraction useful of energy ton synchrotron at the National can accelerate protons to the rest mass of the proton an decreases as Accelerator energy is about El-t For Laboratory of 300 GeV (1 GeV, we see GeV that example, in Batavia, = 10 eV) for protons the pro- Illinois, Since collid- \037 0.1 target, E' IE \037 v3/v200 Only 30 GeV is experiments frame beams colliding head on, the laboratory By using becomes the center of mass frame, and the total energy is available for inelastic events This technique of colliding has been used beams in electron accelerators and has proved feasible in proton extensively machines as well.))) ing with available a for hydrogen interesting identical 14.4 THE SEC Example 14.8 FOUR-VECTOR) MOMENTUM-ENERGY Pair Production Collisions) Electron-electron in 533) In Example 13.7 we analyzed the process by which a pair production, an electron to create an electron-positron photon collides with pair = 1.02 The threshold energy for the process was found to be E = 2moc2 MeV, where moc2 = 0.51 MeV is the rest energy of the electron or positron A related is the production of an electron-positron process pair by the collision of two electrons:) e- \037 e- e- + reaction The + e+).) satisfies conservation of charge for the process energy of the problem we introduce dynamics evidently the to find + (e- + e- threshold To describethe The problem the is following fou r-momenta:) P1: electron1 before -+) the collision the collision P2: electron before Pa: electron after the collision P4: electron after the collision Ps: electron created in e e+ pair P6: positron created in e-.e+ pair -+) -+) -+) -+) -+) Then conservation + Pl -+ P2 -+ = + -+ Pa P4 -+ = + (Pl -+ P2)2 -+ Since each terms + -+Ps + -+) P6 + -+ P4 (Pa -+ side of in whatever Let us compute particle Taking PI gives) we have) Squaring, = -+) of four-momentum Plr ( 1) P6)2 -+) is Lorentz invariant, we ca-n compute frame is most convenient the left hand side of Eq (1) in the laboratory to be initially at rest, we have) the equation the reference -El \037 C) + -+ps + P2 -+) ) frame = (O,imoc) and) + (PJ -+ P2)2 = -+P12 -+ = where + P2 -+ -2(moc)2 we have used \302\267 2Pl P2 -+-+) + p2 -+))) 2) 2moElr) = p2 - E2/ c2 = - mo2c2 , valid for any pa rticle 534) FOUR-VECTCRS RELATIVISTIC AND INVARIANCE) in the calculated The conveniently right hand side of Eq (1) is most all four particles are at rest (This center of mass frame At threshold, that the the minimizes the energy and is consistent with requirement Hence mass of zero in the center total momentum be frame.) spatial and the right hand side of (o,O,O,imoc), Pa, P4, Ps, P6 all have the form \037) \037 \037 Eq (1) becomes) \037 = (0,0,0, 4imoC)2 Eqs (2) and Substituting 2moEl = - -2(17'tOC)2 3) -16(moc)2.) (3) in (1) gives) Eq -16(moc)\037) or) = 7moc2.) El El includes of the = El Kl = - rest the at th projectile moc the kinetic energy is) 6moc2.) here can be applied The argument for projectile, so that of the energy reshold p+ + p+ \037 Sincethe + p+ p+ to the production of a to the instance, + (p+ + production proton negative of other particles, reaction) in the p-).) kinetic energy for mass is 0.94GeV, the threshold negative protons is 6(0.94) GeV = 5.64 GeV The Lawrence Radiation Laboratory, at the Bevatron California, Berkeley, to GeV to allow this process to be was designed to accelerate protons Prize Owen Chamberlain and Emilio Segre received the Nobel observed the in 1959 14.5 for rest proton of production negative producing The special theory of classical mechanics in its a complete from representing a heavy flavor of newtonian the equivalence of accord the with of work the frames inertial Newton req uirements by bringing of electro- theory however, Fundamentally, not has generalized classical mechanics into magnetic far relativity, physics, insistence on Einstein Essentially, antiprotons.) Remarks Concluding break with protons, or the same as those of unobservable insistence on of the emphases of newtonian concepts like operational physics absolute definitions space related special relativity are Einstein's rejection and time and his to observation were))) 14.5 SEC more much 535) REMARKS) CONCLUDING tion far-reaching Einstein laid the which was efforts Newton's were than direc- this in for the analysis of observables groundwork in the development of modern quantum essential In addition, he made significant contributions to our of the of how man obtains knowledge understanding mechanics philosophical world seen have we As was butions this in one of chapter, of the recognition power organizing principle in physics fies and simplifiesthe conceptsof as an of Einstein's great contri- transformation theory Transformation has served and relativity special uni- theory as a knowledgeable guide in the search for new laws is However,in spite of its power and harmony, relativity special deal with to uate not a complete dynamical it is since inadeq theory this was a fundamenTo Einstein reference frames accelerating it is of tal defect According to Mach's principle eq uivalence an inertial system in a between impossible to distinguish locally field and an accelerating coordinate system in free gravitational must frames the Therefore, space by the principle of relativity, Since of physical phenomena be equally valid for the description specialrelativity general theory of with deals of dealing with of dealing is incapable frames, it is inherently went far Einstein incapable toward relativity, transformations It is in 1916 between essentially a theory of gravitational field coordinate systems From this is rega rded as a local distortion in the field of the gravitational to general the deflection are relativity of starlight coordinate sun, however, by the sun, one gravitation, effects the since by changing effect of gravity of space Even detect of his not systems, merely point of view the in the geometry and difficult to small predicted, amounts all with The general theory published just systems it is possible to \"produce\"a inertial fields gravitational difficulties these removing reference accelerating with attributable For example, most dramatic of arc General relativity's has been on cosmology, since greatest impact Its is the at large gravity only important force in the universe role in terrestrial because has been minor, however, partly physics the effects are small and partly because so far it has not been I n contrast, extended to include special relaelectromagnetism of applications and is part of the working has a multitude tivity of every physicist knowledge to assess in Einstein'simpact on the twentieth century is difficult its entirety He altered and enlarged our of the perceptions and in this respect he ranksamongthe greatfigures natural world, effects of Western thought.))) to only 1.7 seconds 536) FOUR-VECTORS Problems 1r INVARIANCE) A neutral pi meson, rest mass 135 MeV, decays symmetrically The energy of each photon two photons while at high speed moving the laboratory system is 100 MeV into 14.1 Find the meson's b Find the angle a 0) RELATIVISTIC AND \037) of each photon and speed the in V motion.) Ans A high 14.2 p \037 'Y + 0 (7r 7r ) is with a proton at collides ray) ('Y according produced rest A reaction) the to \037 42\302\260 ) minimum the energy The rest mass of occur? 7r p + is the What photon energy pi meson neutral V Ie ratio between the momentum system laboratory initial line of the as a answer your Express in a must have 'Y ray is 938 MeV, and proton to mass of a reaction this for rest the is 135 MeV.) Ans 14.3 A photon energy high 'Y + e-\037e- What is the + (e- + an electron hits ray) ('Y pair according to electron-positron 145 MeV Approximately and an produces reaction) the e+).) minimum the energy must have 'Y ray to reaction the for occur? 14.4 A of rest mass Jl1 spontaneously rest masses m1 and m2 particle particles with pa rticl es are) E1 (M2 + m1 = K with collides - + E j2Af \037 nucleus nucleus M 4)c2 = O tion in terms A rocket ward along a that Show rest system P = -+))) 'Y M Find the ca n the be jv(-l,O,O,i), + 2 + m2 )c j2Jl1 kinetic speed with rest A nuclear reaction to make the reac- energy nucleus are M minimum value of of M 1, ]lf , and Q Ans clue If Jl1 of initial the x axis 14.6 - m12 the and J11 + M )e + Q, (M] where Q > occur, (M2 two scheme) the nuclei and rest masses a re related by) (J11 + = mass M] moving at high of rest mass J11 at The rest massesof The rest into energies of from that the a nucleus occurs accordingto nucleus m22)c of rest A nucleus 14.5 decays Show mass by Mo starts emitting four-momentum written) photons = K1 required JlI1 = from rest Qjc and , then K = itself propels 5Q/2 for- backward of the rocket's exhaust in the initial PROBLEMS) 537) M j is where as a b Show the that rocket of the mass final the the exhaust for even whole (Note valid are Doppler-shifted.) of the rocket relative final velocity is result this that the photons though initial frame) to the is) v = - x2 x2 + the x is where 14.7 C') line of freq uency v a sent a satisfy frequency X = AV x axis at the For simplicity, instantaneous The - - U /C )3, AV du/dt a sine wave propagates = = where a represents vt] wave frequency along can f(x,t) corresponds to some component of the the light signal, a nd the wavelength x the repre- electro- constitutes the same Consider a /(1 21r[(X/A) wavelength which field sin A A then wave; light magnetic A nd wavelength with velocity = axis along f(x,t) = function norm = clue Ans The Mo/M j mass, be \037) u fo u r-velocity 14.8 to final mass initial acceleration representing motion along the x axis Letthe a four-vector only straight consider rocket's the of ratio Construct and c coordinate system x', y', z', reference frame the wave in the wave v velocity t' this In moving the has form) = A' f'(x',t') sin - 271\" P't} G: a and k = \037 that Show v' are -, \" 27r ( the velocity of components of a '\\ 1\\ iV - Extend k given given provided in the x, y, z, t that l/A' system by) C) ) b Using the result of Doppler shift by evaluating c is correctly light four-vector the verse Doppler shift analysis by part the a, of part considering derive frequency the result in a moving the for longitudinal system b to find the expression a wave propagating for the along the transy axis.))) INDEX))) 540) INDEX) c, role in relativity, 107 Capstan, M., 492 Abraham, 13 Acceleration, Capture cross sectionof centripetal, 36, 359 Coriolis,36, Newtonian of, in Centerof 36 coordinates, collisions of, 486, 533 37 332 gyroscope, suspension Centimeter, 67 Central force harmonic freq uency, Angular momentum, and fixed potential, 385 axis and kinetic rotation, 248 spin, 262 nates, 190 Apogee, 396 Approximations,numerical, Atomic clock, 470 Atwood's machine, 104,254 value, Balmer, J., 271 39 and Bounded 496, 533 41 Conical 433 variables, pendulum, 503 503 77, 161, 237, 245 orbits, 386 H., 322 rotations, operations, Compton, A H., Com pton effect, 270 Bucherer, A infinitesimal Complex 212 Bohr, N., 270 r atom, 490, of vector system, Binomial series, Boh 188 Com m utativity: Base vectors,10 Binary laws, 188 inelastic, 188 of star conservation elastic, relativistic, 413) Average coordi- of mass center and 533 Antiprotons, D., 498 Collisions,187 533 506, Antiparticles, 274 17,25 of friction, 93 Coefficient 289 Angular velocity, J Cockroft, of, 288 534 theorem, Circular motion, 381 36 67 of units, system Chasle's of, 272 nature 382 energy, potential Centripetal acceleration, Chamberlain, 0., 262 vector 380 382 problem, two-body Centrifugal cgs energy, 314,370 quantization areas, Centrifugal force, 359 233 of, and effective eq ual 378 and of, 305 definition 378 motion, reduction to one-body problem, 411 233ff conservation 260 of motion, of law 411 motion, Angular orbital, 531 percussion, constants of simple 127 190 Center of Air track, 53 Amplitude coordinates, and, relativistic, 358 coordinates, rotating tangential, Air of, 264 energy Center of mass relativistictransformation in momentum of, 261 kinetic radial, 36 116, 145 mass, angular 22 in polar H., 81 Cavendish, mechanics, 454 nonuniform, a planet, 241 359 invariance 455 492) Conservation: of angular momentum, 305))) INDEX) 541) Conservation: syste m, of four-momentum, 529 of of Constraints, 380 motion, Contour i nates: Electrostatic 359 a falling mass, 278 integral, Elliptic Elliptic orbit, 398 362 and weather systems, 364 Energy, 152ff law of, conservation ray, 512 Coulomb's law, Crab nebula, Curl, 218) of, 86 ki 156 netic, mathematical 510 493 414, 435 oscillator, a nd theorem, 434 Derivative: vector, 15, 23 Diatomic Dicke, Dimension of a physical quantity, 18 475, Doppler navigation, Dot product, principle, 346, velocity,157,162 446 eq uations, 320 Euler's theorem,232 462 function, Exponential Doorstop, 259 Doppler effect, Euler's Event, 11 Displacement, 240, 382 185 Ether, 143 Dike, 44) 507, 530 479 67 175, 322 Eq uivalence Esca pe 204 law of, areas, uilibrium, Erg, R H., 354 Differentials,45, of units, system Eotvos, R., 354 Eq 179 molecule, work, 160 English Eq ual 202 partial, 169 total, 207 (operator), De Moivre's 174 surface, 211 Damping time, 418 of a of, 202 aspects potential, 168 relativistic, Del 184, 495 176, 383 diagrams, and stability, Damped 22 86 force, Ellipse, 392,403 359 force, and deflection of Cosmic to radio wave, motion due Coriolis acceleration, 36, Cosines, of, 492 mass polar, 27 Coriolis 381 451, 534 Electron: 12 cartesian, 392 potential, Einstein, A., 272, Electric field, 87 211 lines, Coord 296 eq ui noxes, Effective Contact forces,87 the and tide, 348 Eccentricity, 70, 74 125 system, and precessionof Conservative force, 163,215 Constant energysurface,211 Constants 366 362, Earth-moon-sun 122 momentum, reference as a rotating Earth, 169, 184 of energy, Fictitious force, 62, 344 Field: Dredl, 290 electric, 87 Dyne, 67) gravitational, 85))) 369 542) INDEX) G F., FitzGerald, H Fizeau, 450, 459 Foot (unit), Gravity: 475 L., 460, and 163 conservative, 352 tides, and weight, 84 criteria for, 215 Gyrocompass, 301 contact, 87 Gyroscope,295, 68 diagram, Hall, D B., 469 gravitational, 80 Halley's comet, Force, (see damped, 414,435 528 relativistic, and transport of 421, 436 forced, momentum, Hertz, H., 502 139 Hertz (unit), of, 67 units vector nature Foucault pendulum, 366 Hooke's 527 525 I nelastic 469) altitude, variation with latitude 340, 80 relativity, 535 motion under, 67 Grandfather's clock, 256 Gravitational mass, 81, 352 Gravitational red Graviton, 501 Gravity, 86 electric, gravitational, 389) operator, 207, 210 Gradient Gram, of units, Inverse square law: 453 of, 144 of 99 system Invariants, 520 Galileantransformations, pressure 81, 356 55, 340, 455 conditions, I nternational constant), 80 General theory mass, system, Initial 83 374 (problem), I nertial I nertial Infinitesimal rotations, 322 83 of gravity), with G (gravitational (see Force, fictitious) variation Gas, force I nertial 95 g (acceleration 188 collisions, Inertia, 372 coefficient of, 93 Frisch, 392) 130 Impulse, 460 Friction, 92 D H., 186 Hyperbola, coefficient, drag 97 law, Horsepower, Four-velocity, 525 Freq uency, 411 fluid, orbit, 408 Hooke, R., 97 Four-momentum, Four-vector, 411 transfer Hohmann 524 59, of, 95 viscous, Fresnal 407 oscillator, 410 Harmonic fictitious) 328 of, 331) nutation electric, 86 fictitious, 344 of friction, 92 inertial shell, 101 of spherical 58 Force, field, 85 gravitational 67 80) shift, Joule, Joule J P., (unit), 185 156) 369 Kater's pendulum, Kaufmann, W., 23))) 258 67 INDEX) 543) experiment, Kennedy-Thorndike Meson decay, 468 Kepler's laws, 240, 400 Meter 66 definition, Kilogram, Kinematical eq uations, and center of mass of 264 313 R A., Minkowski force, Minkowski space, 521 Moment of Laws of motion, 53 and 466 of, of, 445 theory model of, 501 of, 508 medium, moving Line integral, 450 F W., Morley, in 450, 457 Lorentz transformations, 347 circular, 17, 25, 34 487 invariant, coordinate accelerating system, Lorentz contraction, 466 Lorentz and four-vectors, four vector, 527 Motion, 19 53 Linear restoring force, 97 H A., 490 139 Momentum-energy 451 159,166 air track, conservation of, 122, 490,529 a nd the flow of mass, 133 transport, in Lorentz, Angular relativistic, 451 445, constancy Linear 313 axes, momentum) electromagnetic of, axis theorem,252 (see angular Light: particle 249, 309 inertia, principal unit of, 66 speed 67 units, Momentum, 112 Length: contraction of and parallel 336 trick, 528 diatomic, 179 Molecule, Lariat 502 Millikan, mks system 383) system, two-body 445, 448 A., 445 motion, rigid body, of rotating A Michelson- Morley experiment, 188 collisions, 67 system, Michelson, formal 156 Kinetic energy, 66 (unit), Metric 19 solution, in 445 J C., Maxwell, 459 in conservative 455 523) 168 systems, laws of, 53 in nonconservative systems, 182 in Mach, Mach's principle, 369 angular momentum momentum of, 113 Mascons,390 Mass, on rotating along a system: Many-particle polar plane 27 coordinates, relation to acceleration,20 E., 369, 443 of, 305 earth, 368 line, straight 34 in uniform gravitational field, 21 Muon decay, 468) 56 gravitational, 353 inertial, 353 relativistic,490 of, 66 standard unit of, 67) 501 Newton, I., 52, 353, Newton (unit), 67 Neutrino, Newton's law of 368, 442 gravitation, 80))) 544) INDEX) laws Newton's of motion, 52ff., 55 relation to force, 173,206, 59 law, force, 182 Pound, R V., 370 332) Operational definition, Orbital 262 394 393 rbed, Pair 388) 296, 331 Principle of eq uivalence, Principle of relativity, 346, 369 451 of superposition, 58 Products of inertia, 309 Parabola, 392 468, 524 time, Proper Parallel axistheorem,252 Pulleys, 90 202 derivatives, Pri nci pal axes, 313 Principia, 440, 452n Principle 505, 533 production, Partial 300 noxes, Pressure under inverse square force,385 pertu of eq ui torq ue-free, 317,331 of a gas, 144 bounded, 386 hyperbolic, 67 186 of gyroscope, Orbits, 382 elli ptic, Pound, Power, Precession: 57 momentum, angular 214 211 surface, 92 force, Nutation, 385 of, 211 gradient N onconservative Normal 486 paradox, effective, law, 56 second third vaulter Pole Potential energy, 168 442 first law, Pulsar, 510) Pendulum: inverted, 164 Kater's,258 Q (q physical, 257 Schuler, 373 simple, 255 periodversus amplitude, 396 Perigee, Period of motion, Perturbedorbit, 388 Photoelectric effect, 502 501 Photon, mass of, 512 Physical pendulum, 257 Planck, M., 272 Planets: pressure, Radius of gyration, Relative 502 257 mass, 179, 191,379 Red uced velocity, 48 general theory, 535 special theory, orbits of, table, curve, 427 Rest energy,491 395 Polar coordinates,27 in, 36 acceleration in, 30) 491 mass, Rigid body motion, of, 391 perturbation 450 423 Resonance, Rest motion of, 390 velocity Radiation 418) Relativity: 411 Phase, rest 411 256 uality factor), Rocket, 288,308 136 relativistic, 536 Rossi, B., 465 Rotating bucket Rotating coordinate experiment, system, 368 355))) 545) INDEX) transforma- coordinate Rotating and atomic forces,91 Tensor, 25, 294, 297 vectors, Rotating Rotations, noncommutativity 520 of, E., 271) Rutherford, 44 dilation, 468, 524 396 unit Scalar, 308, 520 Schuler Second, definition E., 534 system of units), 523 a four-vector, of 42 SI (international laws, 520 of physical 516 a vector, of Transformations: 67 harmonic Simple 97, 154, motion, 410 255 Simultaneity, 463 Skew rod, 292-294,312 theory of 480) 296 precession, 3, 10 vectors, Units, 18,67 Universal gravitation, 466 interval, Special Unit H.,469 Spacelike 453 455, 523 paradox, Uniform 178 oscillations, Smith, J Twin 67 (unit), Galilean, 340, Lorentz, Simplependulum, constant of, 80 81) 451 relativity, of light: Speed in em in a moving pty space, 445 medium, Spin angular Vector 474 262 momentum, 174 of rotating objects, 304,322 units, 64 and Standards Stokes' theorem, 225 of forces, Superposition 58, 82 Synchronous satellite, 104 of units, addition, 67) base acceleration, 36 Taylor's series, 42 toy, 175, 181) 10, 28 vectors, components, derivative of, 15, displacement vector, 11 23 four-dimensional, 523 of cross multiplication: product, of scalar Tangential 207 operator, Vectors, and area, Sta bility, Teeter 169 Transformation properties: Taylor's, System energy, 21 Trajectory, 41 binomial, Small mechanical Total Series: Slug precession, 317,324 Torque-free 66 of, 466 238 Torque, 373 pendulum, Segre, 418 Timelike interval, uct, 66 of, Time constant, invariants, 521 Scalarprod J., 271 348 Tide, Time, orbit, 311 of inertia, Tensor Thomson, J 285, 322 Satellite 87 Tension, 371 tion, (dot) product,5 10 orthogonal, positi on rotating, vector, 11 25))) 546) INDEX) Vibration eliminator, Vectors: subtraction, transformation properties 427 95) Viscosity, of, 516 unit, Watt angular, 289 four-, in systems, Weight, 68, 84 525 polar 156, Work, coordinates, 30 in one theorem, dimension, for rotation, 267 relative, 48 relativistictransformation 472, 526 33) of, Work World World 502 function, Ii 364 160 Work-energy radial, 33 tangential, 186 (unit), Weather 13 average, S., 498 E T Walton, Velocity, 13 ne, poi nt, 524 524))) 160 156 [...]... has Daniel Kleppner Robert J Kolenkow) )) AN INTRODUCTION TO MECHANICS))) VECTORS AND KINEMATICS- A FEW MATHEMATICAL PRELIMINARIES))) 2) VECTORS 1.1 I KINEMATICS-A AND the MATHEMATICAL help you PRELIMINARIES) uction ntrod The goal of this of FEW book is to of mechanics principles the very heart of physics; its the everyday physical standing acquire a deep understanding is at The subject of mechanics... E we thank Professors George B Benedekand ticular, should We Pritchard for a number of examples and problems for their also like to thank Lynne Rieck and Mary Pat Fitzgerald It gives this cheerful fortitude Daniel Kleppner Robert J Kolenkow) )) in typing the manuscript.) to form a comprehensive introduction chapters and constitute the heart of a one-semester mechanics classical the covered In a 12-week... angle 8 = 0 We a plane determine they when is zero a is called hand right triple to tail perpendicular to the plane to be what they are drawn taken as the B when and ambiguity, A, B, and C form B and x ) and When we draw define the between A Note 7r.) if ) of A I mag- coordinate system with A and B in the xy plane as A lies on the x axis and B lies toward shown in the sketch the e If form a C and... courses systems in advanced mathematics noninertial of subject such speculative and and the principle of theory to point of view, the tant techniq ue Chapters 9 and for use many solving offers a natural interesting topics as transequivalence From a more noninertial of physical systems is an impor- problems chapters the principles developed applied important problems, central force motion and are generally... vecThe is a mathematical introduction, chiefly first chapter of a vector,) tors and kinematics The concept of rate of change Our book 1 The is written some knowing primarily calculus, background provided John Wiley & Sons, Ramsey, for students enough in \"Quick Calculus\" by Daniel Kleppner New York, 1965, is adequate.))) and Norman xvi) PREFACE) plays most the probably an role important this topic is developed... in an introd course in ever, presenting uctory which mechanics is both exciting and intellectually rewarding Mechanics is a mature science and a satisfying of its discussion is lost a in At the treatment other superficial principles easily and extreme, advanced \"enrich\" to attempts can produce topics the subject by emphasizing a false sophistication which empha- techniq ue rather than understanding... make an intelligent estimate, for instance, the from the momentum approach or from and that he will know how to set off on a new start students is unsuccessful report first approach Many skills these sense of satisfaction from acquiring than a numerical rather Many of the problems requirea symbolic the importance of numerito minimize solution This is not meant cal work but to reinforce the habit of analyzing... SPECIAL 11.1 The Galilean THEORY the CHAPTER EXAMPLES, Galilean 11 453; 11.2 Transformations, A Light Pulse as Described t;>y 455.) Transformations, OF RELATIVITY) 12 RELATIVISTIC 12 CHAPTER EXAMPLES, KINEMATICS) 12.1 Simultaneity, 463; of 12.2 An Application and Timelike of the Lorentz Transformations, Intervals, 465 12.4 The Orientation of a Moving Rod, 467; 12.5 Time Dilation and Meson Decay,468;... mechanics Consequently, throughout care, both analytically and geometrically in particular, later proves to be invalumathematical difficult with approach, geometrical momentum visualizing the dynamics of angular 2 discusses inertial Newton's laws, and some Chapter systems, able for common forces of the Much ton's laws, since analyzing general principles can be a a complex system in terms inertial and... symbolia numerica! Answers are given to some problems; in others, cally to allow the student to check his sym\"answer clue\" is provided bolic Some of the problems are challengingand req uire result such serious problems thought and discussion Sincetoo many should have a each at once can result in frustration, assignment of easier mix and harder problems we would prefer to start a course in mechanChapter1