Physicswith a Strange Sense of Humor, Some Moralizing, and a Few Duly Cynical Observations About HumanityJohn Ganey Draft Edition of 23 August 2011

ii Copyright c 2003-2011 John Ganey This book is a work of ction. Any truths or resemblance to reality are entirely coincidental.

This completely gratuitous plot of z = cos(xy) has absolutely nothing to do with anything.

3 1

3

z 2 dz cos

3 = ln 3 e 9

Neither does the above limerick (author unknown).

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ContentsLicense Other Stu iii xvii

I

Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 3 3 5 7 11 16 19 19 21 21 23 24 25 27 29 29 32 32 35 37 42 43

1 Preliminaries 1.1 About the Course . . . . . . . . . . . 1.2 About This Book . . . . . . . . . . . 1.3 Good Karma . . . . . . . . . . . . . . 1.4 The Zen of Problem Solving . . . . . 1.4.1 An Unfortunate Example . . . 1.5 Signicant Figures . . . . . . . . . . . 1.6 Units & Conversions . . . . . . . . . . 1.7 Conversion Factors & Constants . . . 1.8 Order-of-Magnitude Estimates . . . . 1.8.1 An Example . . . . . . . . . . 1.8.2 General Points . . . . . . . . . 1.8.3 A Brief Discourse on Malarkey 1.9 Problems . . . . . . . . . . . . . . . . 1.10 Sketchy Answers . . . . . . . . . . . . 0 Optics 0.1 Light Waves . . . . . . . . . . . . . . 0.2 Geometrical Optics . . . . . . . . . . 0.2.1 Reection & Refraction . . . . 0.2.2 Ray Diagrams & Images . . . 0.2.3 Thin Lenses . . . . . . . . . . 0.2.4 Optical Instruments . . . . . 0.2.5 The Eye & Corrective Lenses vii

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viii

CONTENTS 0.3 Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3.1 Interference, Diraction, Dispersion, & Polarization 0.3.2 Why the Sky is Blue . . . . . . . . . . . . . . . . . 0.4 Parabolic Mirrors . . . . . . . . . . . . . . . . . . . . . . . 0.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.6 Sketchy Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 49 52 59 64 65 69 71 71 73 78 79 83 85 85 90 92 94 96 100 106 107 107 109 112 115 119

1 Vectors 1.1 Unit Vectors . . . . . . . . 1.2 Dot & Cross Products . . 1.2.1 The Dot Product . 1.2.2 The Cross Product 1.2.3 Some Special Cases 1.3 Problems . . . . . . . . . . 1.4 Sketchy Answers . . . . .

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2 Vector Calculus 2.1 Line Elements & Integrations . 2.2 Surface Elements & Integrations 2.3 Volume Elements & Integrations 2.4 The Gradient . . . . . . . . . . 2.5 Divergence & Gausss Theorem 2.6 Curl & Stokess Theorem . . . . 2.6.1 An Important Result . . 2.7 A Few More Important Results 2.7.1 B = 0 B = A 2.7.2 The Behavior of 2 1 . . r 2.7.3 Helmholtzs Theorem . . 2.8 Problems . . . . . . . . . . . . . 2.9 Sketchy Answers . . . . . . . .

II

Basic Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121. . . . . . . . 123 123 129 132 135 136 136 139 144

3 Kinematics 3.1 Location, Velocity, & Acceleration . 3.2 One-Dimensional Motion . . . . . . 3.2.1 Constant Acceleration . . . 3.2.2 Vertical Free-Fall . . . . . . 3.3 Two-Dimensional Motion . . . . . . 3.3.1 Projectile Motion . . . . . . 3.3.2 Uniform Circular Motion . . 3.3.3 Nonuniform Circular Motion

CONTENTS 3.3.4 General Motion in Polar Coordinates 3.3.5 Two-Dimensional Relative Velocities Problems . . . . . . . . . . . . . . . . . . . . Sketchy Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix 147 151 153 182

3.4 3.5

4 Dynamics 4.1 Newtons Laws . . . . . . . . . . . 4.2 Special Forces . . . . . . . . . . . . 4.3 Force Diagrams . . . . . . . . . . . 4.4 Circular Motion . . . . . . . . . . . 4.4.1 Road Banking . . . . . . . . 4.5 Newtons Law of Gravity & Orbits 4.6 Perceived Weight . . . . . . . . . . 4.7 Semi- & Almost Nonbogus Friction 4.8 The Catenary . . . . . . . . . . . . 4.9 Problems . . . . . . . . . . . . . . . 4.10 Sketchy Answers . . . . . . . . . .

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189 . 189 . 192 . 195 . 202 . 205 . 206 . 210 . 211 . 213 . 218 . 245 249 . 250 . 256 . 260 . 262 . 265 . 268 . 284 287 . 287 . 294 . 295 . 298 . 299 . 302 . 303 . 307 . 310 . 313 . 320 . 339

5 Work & Energy 5.1 The Bogonics of Work & Power . . . . . . . 5.2 Potential Energy & Energy Conservation . . 5.3 A Practical Example of Energy Conservation 5.4 Results for Potential Energy . . . . . . . . . 5.5 How to Beat a Dead Horse . . . . . . . . . . 5.6 Problems . . . . . . . . . . . . . . . . . . . . 5.7 Sketchy Answers . . . . . . . . . . . . . . .

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6 Center of Mass & Momentum 6.1 Center of Mass . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Dynamics of the Center of Mass . . . . . . . . . . . 6.3 Momentum & Momentum Conservation . . . . . . . . . . 6.4 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Center-of-Mass & Relative Coordinates . . . . . . . . . . 6.6 Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Two-Body Collisions . . . . . . . . . . . . . . . . . . . . 6.7.1 The One-Dimensional Two-Body Elastic Collision 6.8 Summary of Important Points . . . . . . . . . . . . . . . 6.9 Some Gravitational Yawing . . . . . . . . . . . . . . . . 6.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Sketchy Answers . . . . . . . . . . . . . . . . . . . . . .

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x 7 Rotational Dynamics 7.1 Two-Dimensional Rotations . . . . . . . . . . . . Table of Translational-Rotational Analogies . . . 7.2 Three-Dimensional Rotations . . . . . . . . . . . Weird Properties of Three-Dimensional Rotations 7.3 Coriolis Eects . . . . . . . . . . . . . . . . . . . 7.4 Constant Angular Acceleration . . . . . . . . . . 7.5 Moments of Inertia . . . . . . . . . . . . . . . . . 7.6 Conservation of Angular Momentum . . . . . . . 7.7 Kinetic Energy . . . . . . . . . . . . . . . . . . . 7.8 Torque Due to Gravity . . . . . . . . . . . . . . . 7.9 Rolling . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 All Good Things Must Come to an End . 7.10 Massive Pulleys . . . . . . . . . . . . . . . . . . . 7.11 The Parallel-Axis Theorem . . . . . . . . . . . . . 7.12 Gyroscopes & Tops . . . . . . . . . . . . . . . . . 7.13 Summary of Important Points . . . . . . . . . . . 7.14 Problems . . . . . . . . . . . . . . . . . . . . . . . 7.15 Sketchy Answers . . . . . . . . . . . . . . . . . . 8 Static Equilibria 8.1 The Conditions of Equilibrium 8.2 Stable & Unstable Equilibria . 8.3 Problems . . . . . . . . . . . . 8.4 Sketchy Answers . . . . . . .

CONTENTS 341 . 341 . 350 . 351 . 358 . 360 . 367 . 368 . 373 . 382 . 384 . 385 . 391 . 392 . 393 . 395 . 397 . 401 . 430 435 . 435 . 439 . 442 . 451 453 . 453 . 459 . 462 . 464 . 469 . 472 . 477 . 479 . 479 . 480 . 482 . 483 . 486 . 499

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9 Harmonic Motion 9.1 The First Refrain . . . . . . . . . . . . 9.2 Once Again, with Feeling . . . . . . . . 9.3 Some Practical Considerations . . . . . 9.4 Pendula . . . . . . . . . . . . . . . . . 9.5 Damped Harmonic Oscillations . . . . 9.6 Driven Damped Harmonic Oscillations 9.7 Small Oscillations . . . . . . . . . . . . 9.8 Wave Eects . . . . . . . . . . . . . . . 9.8.1 Traveling Waves . . . . . . . . . 9.8.2 Standing Waves . . . . . . . . . 9.8.3 Sound Intensity & Decibels . . 9.8.4 Doppler Shift . . . . . . . . . . 9.9 Problems . . . . . . . . . . . . . . . . . 9.10 Sketchy Answers . . . . . . . . . . . .

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CONTENTS

xi

III

Beyond Basic Mechanics

501

10 Relativity 503 10.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . 503 10.2 Einstein Disses Newton . . . . . . . . . . . . . . . . . . . . . 505 10.3 The Lorentz Transform . . . . . . . . . . . . . . . . . . . . . 508 10.3.1 Special Case: A Mirror & Light Pulse . . . . . . . . . 509 10.3.2 Derivation of the Lorentz Transform . . . . . . . . . . 512 10.3.3 A Nicer Derivation of the Lorentz Transform . . . . . 516 10.3.4 A More Modern Derivation of the Lorentz Transform 519 10.3.5 Some Observations & Notation . . . . . . . . . . . . . 523 10.3.6 The Inverse Lorentz Transform . . . . . . . . . . . . . 527 10.3.7 The Lorentz Transform from Symmetry . . . . . . . . 529 10.3.8 Lorentz Transforms as Rotations . . . . . . . . . . . . 540 10.4 Time Dilation & Length Contraction . . . . . . . . . . . . . . 546 10.4.1 Time Dilation . . . . . . . . . . . . . . . . . . . . . . 546 10.4.2 Length Contraction . . . . . . . . . . . . . . . . . . . 549 10.4.3 When to Use What . . . . . . . . . . . . . . . . . . . 551 10.5 The Invariant Interval & Proper Time . . . . . . . . . . . . . 552 10.6 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . 556 10.7 Momentum, Energy, & Stu . . . . . . . . . . . . . . . . . . 558 10.7.1 A Nicer Derivation of Momentum & Energy . . . . . . 564 10.8 The Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . 566 10.9 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . 568 10.9.1 The Field Equations . . . . . . . . . . . . . . . . . . . 572 10.9.2 Gravitational Time Dilation . . . . . . . . . . . . . . 575 10.10 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . 577 10.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 10.12 Sketchy Answers . . . . . . . . . . . . . . . . . . . . . . . . . 603 11 Fluid Dynamics 11.1 The Bernoulli Equation . . . . 11.2 Archimedess Principle . . . 11.3 Frisbees & Airplanes . . . . . . 11.4 Brazilian Soccer . . . . . . . . 11.5 Why Golf Balls Have Dimples 11.6 Problems . . . . . . . . . . . . 11.7 Sketchy Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 605 608 609 610 611 613 619

12 Things Nuclear 12.1 The Composition of Nuclei . . . . . . . . . . . . . . . . . . 12.2 Types of Nuclear Decay . . . . . . . . . . . . . . . . . . . . 12.3 Decay Rates & Constants . . . . . . . . . . . . . . . . . . .

621 . 621 . 623 . 625

xii 12.4 Fission & Fusion . . . . . . . . . . . . . . . 12.5 Dosimetry & Biological Eects . . . . . . . 12.6 One More Reason New Jersey Is Disgusting 12.7 Problems . . . . . . . . . . . . . . . . . . . 12.8 Sketchy Answers . . . . . . . . . . . . . . . Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 632 640 642 646 647 649 649 653 654 655 660 669 672 678 681 686 688 688 698 701 709 713 718 728

13 Thermal Physics 13.1 Statistical Mechanics Versus Thermodynamics . . 13.2 Some (ugh!) Chemistry . . . . . . . . . . . . . . . 13.3 Temperature Scales . . . . . . . . . . . . . . . . . 13.4 Heat Energy & Changes of Temperature & Phase 13.5 Ideal Gases . . . . . . . . . . . . . . . . . . . . . . 13.6 Processes, Cycles, & the First Law . . . . . . . . . 13.6.1 A Painfully Long Example . . . . . . . . . 13.6.2 A Mercifully Short Example . . . . . . . . 13.6.3 Adiabatic Processes . . . . . . . . . . . . . 13.6.4 Some General Observations . . . . . . . . . 13.7 Heat Engines, & Refrigerators . . . . . . . . . . . 13.7.1 The Carnot Cycle . . . . . . . . . . . . . . 13.7.2 Air Conditioning & Refrigeration . . . . . . 13.8 Reversibility, Entropy, & the Second Law . . . . . 13.8.1 Entropy in Statistical Mechanics . . . . . . 13.8.2 Some Examples & Observations . . . . . . 13.9 Problems . . . . . . . . . . . . . . . . . . . . . . . 13.10 Sketchy Answers . . . . . . . . . . . . . . . . . . .

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IV

Electromagnetism for Big People. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

731. . . . . . . . . . 733 733 734 737 739 740 742 745 746 748 749

14 The 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

Maxwell Equations: An Overview The Maxwell Equations . . . . . . . . . . . . . Charge & Current . . . . . . . . . . . . . . . . Gausss Law & Electric Fields & Forces . . . . Magnetic Gausss Law & Magnetic Fields . . . Faradays Law . . . . . . . . . . . . . . . . . . Ampres Law & Magnetic Forces . . . . . . . Superposition . . . . . . . . . . . . . . . . . . The Potential Functions . . . . . . . . . . . . . 14.8.1 Gauge Transforms & Gauge Symmetry 14.9 Light, Locality, & Relativity . . . . . . . . . .

CONTENTS 15 Electrostatics 15.1 Applications of Gausss Law . . . . . . 15.1.1 Spherical Charge Distributions . 15.1.2 Cylindrical Charge Distributions 15.1.3 Planar Charge Distributions . . 15.1.4 Superposition . . . . . . . . . . 15.2 Coulombs Semibogus Law . . . . . . . 15.3 Electric Fields by Direct Integration . . 15.3.1 Rings of Charge . . . . . . . . . 15.3.2 Disks of Charge . . . . . . . . . 15.3.3 Finite Line Segments of Charge 15.4 Electric Field Lines . . . . . . . . . . . 15.5 Electric Dipoles . . . . . . . . . . . . . 15.6 Electrostatic Potential & Voltage . . . . 15.7 Equipotential Lines & Surfaces . . . . . 15.8 Electrostatic Potential Energy . . . . . 15.9 Conductors . . . . . . . . . . . . . . . . 15.10 The Method of Images . . . . . . . . . 15.11 Problems . . . . . . . . . . . . . . . . . 15.12 Sketchy Answers . . . . . . . . . . . . . 16 DC Circuits 16.1 Resistance & Power . . . . . . . . . 16.2 Series & Parallel Connections . . . . 16.3 Loop & Junction Rules . . . . . . . 16.4 Capacitance . . . . . . . . . . . . . 16.4.1 Parallel-Plate Capacitors . . 16.4.2 Cylindrical Capacitors . . . . 16.4.3 Spherical Capacitors . . . . . 16.4.4 A Few Observations . . . . . 16.4.5 Dielectrics . . . . . . . . . . 16.5 Capacitors in Circuits . . . . . . . . 16.5.1 Energy Stored in a Capacitor 16.5.2 Games People Play . . . . . 16.6 RC Circuits . . . . . . . . . . . . . 16.7 Treating AC As DC . . . . . . . . . 16.8 Problems . . . . . . . . . . . . . . . 16.9 Sketchy Answers . . . . . . . . . . . 17 Magnetostatics 17.1 Magnetic Forces . . . . . 17.2 Ampres Law . . . . . . 17.2.1 Field of an Innite 17.2.2 Field of a Solenoid . . . . . . . . Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii 757 . 757 . 758 . 762 . 764 . 767 . 768 . 770 . 770 . 772 . 775 . 778 . 781 . 783 . 786 . 791 . 793 . 796 . 799 . 818 . . . . . . . . . . . . . . . . . . . . 821 821 824 831 834 834 835 836 838 839 840 843 844 847 850 853 868 871 873 875 875 879

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xiv 17.3 The Biot-Savart Law . . . . . . . . 17.3.1 Derivation of the Biot-Savart 17.3.2 Field of an Innite Wire . . 17.3.3 Field of a Circular Arc . . . 17.3.4 Field of a Ring of Current . 17.3.5 Field of a Solenoid . . . . . . 17.4 Magnetic Dipoles . . . . . . . . . . 17.5 Problems . . . . . . . . . . . . . . . 17.6 Sketchy Answers . . . . . . . . . . . 18 Electrodynamics 18.1 Faradays Law . . . . . . . . . . 18.2 Ye Olde Sliding Bar . . . . . . . 18.3 Generators & Motors . . . . . . 18.4 On the Issue of Time Derivatives 18.5 Problems . . . . . . . . . . . . . 18.6 Sketchy Answers . . . . . . . . . 19 More DC Circuits 19.1 Inductance . . . . . . 19.2 LR Circuits . . . . . 19.3 Energy Density of the 19.4 Problems . . . . . . . 19.5 Sketchy Answers . . . . . . . . . . . . . . . . . . Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 883 888 889 890 891 892 897 907 909 909 917 919 921 926 934

. . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

935 . 935 . 938 . 940 . 945 . 949 . . . . . . 951 951 960 963 967 967 970

20 AC Circuits 20.1 Fourier Transforms . . . . . 20.2 Impedance . . . . . . . . . . 20.2.1 The RC Circuit . . . 20.2.2 The LR Circuit . . . 20.2.3 The RLC Circuit . . 20.3 Delta Functions for Dummies

V And Now for Something Completely Dierent . . . 97321 Lagrangian Dynamics 21.1 The Calculus of Variations . . . . . . . . . . 21.2 The Brachistochrone . . . . . . . . . . . . . . 21.2.1 A Brief Digression, For Those Inclined 21.3 Lagrangian Dynamics . . . . . . . . . . . . . 21.4 Lagrange Multipliers & Constraints . . . . . 21.5 Forces of Constraint . . . . . . . . . . . . . . 21.6 Problems . . . . . . . . . . . . . . . . . . . . 21.7 Sketchy Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . to Digress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975 . 975 . 979 . 984 . 987 . 993 . 997 . 1002 . 1013

CONTENTS 22 Real 22.1 22.2 22.3 22.4 22.5 Physics The Early Days . . . . . . . . . . . . . . . . . . . . Newton . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell & Others . . . . . . . . . . . . . . . . . . . Relativity . . . . . . . . . . . . . . . . . . . . . . . . Quantum Mechanics . . . . . . . . . . . . . . . . . . 22.5.1 Wave Functions & Operators . . . . . . . . . 22.5.2 The Schrdinger Equation & Discrete States 22.5.3 Quantum Tunneling . . . . . . . . . . . . . . 22.5.4 The Quantum Harmonic Oscillator . . . . . . 22.5.5 Path Integrals . . . . . . . . . . . . . . . . . Quantum Field Theory . . . . . . . . . . . . . . . . 22.6.1 The Electromagnetic Force from Symmetry . Unication Theories . . . . . . . . . . . . . . . . . . 22.7.1 Kaluza-Klein Theories . . . . . . . . . . . . . 22.7.2 Grand Unied Theories . . . . . . . . . . . . 22.7.3 Supersymmetry & Supergravity . . . . . . . 22.7.4 String Theory . . . . . . . . . . . . . . . . . 22.7.5 The Empirical Myth? . . . . . . . . . . . . . Our Amazing & Expanding Universe . . . . . . . . . 22.8.1 The Robertson-Walker Metric & Ination . . Problems . . . . . . . . . . . . . . . . . . . . . . . .

xv 1015 . 1018 . 1018 . 1019 . 1020 . 1021 . 1027 . 1032 . 1037 . 1039 . 1044 . 1046 . 1057 . 1073 . 1073 . 1075 . 1075 . 1077 . 1080 . 1081 . 1085 . 1090 1091 . 1093 . 1101 . 1107 1113 1119 . 1119 . 1124 . 1125 1127 1133 1141

22.6 22.7

22.8 22.9

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A Projectile Motion with Air Resistance A.1 Solution by First Approximation . . . . . . . . . . . . . . . A.2 Solution by Series Expansion . . . . . . . . . . . . . . . . . . A.3 Solution by Numerical Integration . . . . . . . . . . . . . . . B Energy & Momentum Conservation C Proofs of Keplers Laws C.1 The First Law . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 The Second Law . . . . . . . . . . . . . . . . . . . . . . . . C.3 The Third Law . . . . . . . . . . . . . . . . . . . . . . . . . D The Linearity of Lorentz Transforms1 E Proof That 1 + 2 + 3 + 4 + = 12

F Gratuitous Pictures of Field Lines

xvi Afterword

CONTENTS 1149

A Bibliography of Sorts 1151 Books About Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 1151 Real Physics Books . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152 Benediction Index 1155 1157

CONTENTS

xvii

AcknowledgementsA This text was written in LTEX 2 in the Emacs editor on boxes running Linux and OpenBSD. Figures were done with Xg and gnuplot. Some calculations were checked by the symbolic-manipulation program Maxima running on CLISP. Octave was used for numerical integrations. All of these applications are free and open-source software. No part of this text has been tainted by the proprietary software of evil monopolists.

Spiritual Canon Though a physics text provides only very limited scope for exercises of the soul, we have tried to write this text in the spirit of Aristophanes, one of the very few able to transcend the egotism and petty delusions of the human species and see things for what they really are.

Spiritual Cannon A physics text does, however, provide great scope for ring cats and babies out of cannons, and we will be doing these sorts of things at every opportunity.

xviii

CONTENTS

Part I Preliminaries

1

Chapter 1 PreliminariesYesterday, I awoke with the whole day ahead of me, so I rolled over and went back to sleep. I mean, who needs that kind of pressure? J.B. (otherwise unknown)

1.1

About the Course

You are born. Then for a long period of time nothing makes any sense at all. Then you die. Try to look upon this course as a small but integral part of that experience.

1.2

About This Book

The principal criteria by which any introductory physics textbook is judged are, of course, its logically disciplined approach to the subject; its full and easily understandable explanations, discussions, and step-by-step derivations of everything from fundamental principles; its profusion of lucid examples and clarifying illustrations; and its abundance of enlightening exercises, at all levels of diculty, to strengthen and extend the students grasp of the physics. As the reader will quickly discover, what distinguishes this book from the myriad others already available is its wantonly perverse disregard for all of these virtues. But then you cant have everything. We think the important thing is that weve had fun writing it. Its subject matter being full as weighty as, and not dissimilar to, that of Lucretiuss didactic epic De Rerum Natura, we had considered composing this entire volume in pentameter, along the lines of 3

4

CHAPTER 1. PRELIMINARIES No matter how two meeting masses swerve, They take due care momentum to conserve.

and so on. The violent exertion of producing even this modest couplet has, however, left our ushed and sweaty Muse too winded and exhausted for us to further contemplate such a grand endeavor a lack of stamina that may perhaps not occasion unendurable disappointment in the reader. Yet even as a prose work this book is not without artistic virtue. Indeed, if one were to judge from what one sees in museums of modern art, apparently anything qualies as art these days. And so we suppose that we could, in this modern spirit, nonetheless still declare this humble tome to be a work of art. Though you could also consider it a labor of love, if youre willing to count love of evil. But let us turn from questions of art and love to reality for a moment. This book is very much a work in progress; it is still in a very nascent and primitive state. But, hey, you try nding the time and energy to write all this stu. And dont even get us started about how long it takes to do even simple diagrams. Woof. Anyway, we hope that in conjunction with what we do in class this book will nonetheless serve your needs well enough that you wont need any other text. At any rate, this is all you will have to y on, so youll have to make do.1 If you nd errors or have suggestions for improvements, by all means let us know.2 At the end of each chapter are two sections, one consisting of problems, followed by another consisting of unelaborated, bottom-line answers to selected problems, against which you can check your own results. Be aware, however, that Sometimes it is possible to get the correct numerical or algebraic answer to a problem by the wrong logic. If your answer agrees with the given one but you are not completely condent of your understanding of the problem, be sure to come for help or ask about it in class. These answers are still new enough that there are going to be typos and various other oversights. If your answer disagrees with a given answer, you may well be the one whos right.3Actually, if you feel the need for greater elucidation or a more detailed exposition on a particular topic, you can certainly check out one or more of the introductory physics texts in the library, where you should be able to nd quite a number of texts written for courses that use calculus. 2 This is, of course, a trivially easy matter if we have you in class, but we can also be contacted at jbg52@columbia.edu. If you do send an email, be sure that the word PHYSICS (all uppercase) appears somewhere in the subject line or your words of wisdom will irretrievably vanish into the spam lter. 3 Developing condence in your logic and reasoning is an important part of your scientic1

1.3. GOOD KARMA

5

Two very important aspects of fully understanding the physics aspects neglected by all the physics texts we have seen are the abilities to recognize nonsense and to ascertain when a problem is insoluble. To help you develop these abilities, there are a few problems in the text that either cannot be solved or ask something that doesnt make sense. If you really understand the physics, you should be able to recognize these problems. Many problems oer hints in footnotes. Ideally, you will have done your best with a problem before you resort to looking at those hints the more you grapple with a problem on your own, the more you will improve your understanding. And now, before we embark on our grand adventure, one nal note. While we have, of course, done our best to do justice to the subject matter, we would be unconscionably remiss were we not to point out at the outset that those in search of truth and wisdom, not to mention wit and entertainment, would be far better o reading Henry Fieldings Tom Jones. We heartily recommend it.

1.3

Good Karma

Vanity of vanities, saith the Preacher, vanity of vanities; all is vanity. Ecclesiastes 1:2. How can I be successful in this course? you ask. Do the reading. Sometimes the reading will make sense immediately, sometimes it will make sense only after we have talked about the physics it covers in class. Should you feel the need or the inclination, feel free to use other physics texts as well. There are, of course, many demands on your time and energy. One consequence of this is that typically The homework for tonight is to read pages . . . " is translated into Woohoo! We dont have any homework tonight!" Neglect the reading at your own peril: to do well in the course, you need to understand the physics, and the point of the reading is to help you gain that understanding. We know that you have the best intentions in this regard, and to help you realize those good intentions, there may well be pop quizzes to check whether youve done the reading.education. Contrary to certain fashionable political views, all points of view are not equally valid, and it is actually a virtue in science to be able to assert politely, of course, and in the interest of furthering everyones understanding , Im right and youre wrong.

6

CHAPTER 1. PRELIMINARIES

Do the homework problems. The homework problems are intended not only to give you practice, but also to make you think about the physics. It is only by thinking about the physics, by struggling with it, that you will develop an understanding and a working knowledge of the concepts. Some problems will be straightforward, others will be more challenging. On those occasions when you nd yourself stuck on a problem, do not simply give up: make as much progress as you can with the part that you are stuck on and then make note (that is, actually write down) what it is that has you stuck. Then try to continue with the remaining parts of the problem. In the past, weve often found ourselves in a vicious cycle: since the homework makes important points about the physics, we need to discuss it in class. The world being the place it is, people then often gure that they can blow o the homework hes going to go over it in class, anyway, right? which means we have to spend even more time on the homework in class, so that people are even less inclined to do it, and so on. To prevent this from happening, we will be regularly checking and sometimes even collecting the homework. You may even be called to the board to demonstrate your solutions to homework problems. And we will also be getting a very good idea of how well you are keeping up by observing you contributions to group work. Always aim for understanding the concepts. This is not a course in plug-and-chug or monkey-see-monkey-do. To the extent possible, tests will ask you to demonstrate your understanding of the principles by asking you to explain them or to solve problems that require you to apply them in ways dierent from what you will have seen in homework and examples. If you really understand the physics, you should be able to reason your way successfully through these novel applications.4 And your grades on the problems will correspondingly reect our assessment of your understanding of and ability to apply the physics: to get credit, you need always to show your work or indicate your reasoning clearly. Arithmetic or algebraic errors are relatively minor; logic errors are relatively serious. You wont get much credit for pulling the right numerical answers out of the sky or arriving at them by incorrect reasoning.Most of the time, of course; no one is always going to get everything right, especially under time pressure. Thats why tests are curved. (Curiously, it seems to have escaped the notice of many people that all tests are always curved: even if the test is graded on a traditional 100-point scale, someone has to decide what constitutes 100 points of work and how many points each part of the test is worth.)4

1.4. THE ZEN OF PROBLEM SOLVING

7

Use consultation. With rare exception, we are in the classroom and available for consultation every consultation period that we do not have a lab. When you come for help, we will try to help you think your own way through whatever is causing you diculty, as opposed to simply telling you the answer. This process can sometimes be a bit painful (for both of us), but just telling you the answer isnt really going to help you understand the physics. And, historically, people who have come to consultation have realized improvement in their performance. Our hope is that you will leave the course with a meaningful understanding of some physics. And even in the unlikely event that you never have occasion to make direct use any of the physics we cover, we hope that the course will have helped develop your general ability to reason logically and analytically. And with that, you can better gure out for yourself what is really meaningful in life and live accordingly.5

.6

1.4

The Zen of Problem Solving

Physicists dont express physical relations mathematically just out of spite; they do it because math is the natural language of physics.7 If a picture is worth 10,000 words, then an equation is worth 10,000 pictures. Consider, for example, string theory: the action for the heterotic string can be written S=5

1 2

d2

h h X X + i

This might seem self-evident, and yet it is shocking how little thought many people give to what they do with their lives. Too many blindly pursue the vainest and most supercial ambitions. They might as well be roaches or even stones living by blind instinct, insensible of those things that truly are meaningful and worthwhile. Why so much importance placed on going to Harvard, Princeton, or Yale? On driving a Mercedes or a BMW? On becoming a doctor, lawyer, or CEO? If you are worried about college admissions, maybe you have some thinking to do. 6 The unexamined life is not worth living, Platos Apology of Socrates, 38a which we would strongly recommend everyone read (though its hard to nd a translation that does justice to the original). And while were at it, wed also strongly recommend Voltaires Candide. 7 This is why all nonmathematical expositions written for people on the street, even those written by very capable and knowledgeable authors, do not convey any real understanding of physics; words are simply inadequate for the task. If you dont understand the math, you dont understand the physics.

8

CHAPTER 1. PRELIMINARIES

If string theory is correct, this single equation accounts for all of the physics of our universe. Now try summing up all of the physics of our universe in words. Understanding the physics boils down to understanding the mathematical relations by which it is expressed: you need to understand what physical quantity is represented by each of the variables and what each equation tells you about how those quantities are related. And the best way to develop this understanding is by solving problems. Sometimes you will intuitively know how to do a problem as soon as you look at it. Sometimes, you may initially be more or less at a loss. In this latter case, you need to think your way through the problem logically and methodically, and there is a general procedure that will help guide your thinking: Draw a picture. Often making a simple sketch will help you visualize whats going on. List the quantities involved. Making a simple table of the variables involved and the physical quantities to which they correspond, and noting which are known and unknown, will help you see at a glance what you have to work with. If you dene any variables of your own, be sure to make your notation clear. If, for example, you use vc to represent the velocity of a cat, then you should say something like vc = velocity of cat. Doing so is not only essential if your work is to be intelligible to someone else such as, say, the person grading it , but will also help clarify your own thinking.8 Work from the relevant general principles and relations. It may not be immediately clear to you precisely which principles and relations a solution requires, but the fundamental physical principles and the equations that express them are few in number, and listing those that may be relevant will often help you see which will enable you to make progress. A common error (especially under the pressure of an exam) is to start wildly thrashing about, writing down more or less loosely connectedThe reader will here note that we habitually not only put a space on either side of a dash, but also retain any punctuation marks that would have been present without the dash. We do this because we believe it a Right and Good Thing. We also habitually punctuate lists as A, B, and C and see nothing wrong with splitting innitives. Those who disagree with these usages are invited to inquire where they can go. We may, however, justly be faulted for irregularities in capitalization and a frustratingly incorrigible tendency to hyphenate what should properly be either a single or two separate words. Notice of particular instances of such errors is welcome.8

1.4. THE ZEN OF PROBLEM SOLVING

9

heaps of numbers or algebraic expressions, hoping that the solution will y out from them like partridges ushed out from the bushes. Dont do this. Never start by writing down numbers or secondary expressions; always rst write down, in terms of algebraic variables, the general relations relevant to the problem. You will be far less likely to make mistakes, and far more likely to nd the path to enlightenment, if you work from general principles than you will if you try to pull a solution out of the sky.9 Work in terms of variables. As we get further into the course, you will increasingly nd that the problems dont even give you any numbers to work with; rather, you will have to keep track in your mind of which variables represent known quantities and which unknown quantities, and solving the problems will involve arriving at expressions for the latter in terms of the former. Developing the ability to solve problems algebraically rather than numerically is a large and critical step toward doing big-people physics. Even when you have numerical values for known quantities, it is highly desirable to solve problems algebraically and postpone the numerical substitutions until the very last step. In addition to even more compelling reasons that will be explained shortly, it is much easier to go back and x mistakes when you have worked in terms of variables rather than numbers: working in terms of numbers is like having shag carpeting in the kitchen. Keep your eye on the target. Another large and critical step toward doing big-people physics is learning how to make incremental progress even when you cant see all the way through to the ultimate solution. Having recognized the relevant general principles and written down the corresponding relations, you should try to chip away at the problem: what new expressions can you derive and what calculations can you do that will bring you closer to where you want to go? Developing your ability to make such incremental progress, more or less in the dark but toward a clear goal, is by far the most important and meaningful thing we will do all year. It is as much an art as a skill. This is where the real Zen comes in. Fortunately, however, this ability can, like any art or skill, be cultivated through earnest practice. And there are even a few things you can do to make the process easier for yourself: Be rigorously logical and methodical. Be disciplined in your thinking. Do not write down expressions for quantities in the hope that9

Or whatever you are in the habit of pulling your solutions out of.

10

CHAPTER 1. PRELIMINARIES they might be true; work things out so that you know that they are true. A common error is to engage in random algebra, aimlessly doing whatever calculations come to mind in the almost always vain hope that some good will come out of it. Never do algebra unless you have a plan. You can mess around all you want with three equations and four unknowns, but it wont get you anywhere. Even when the goal is clear and denite, often there are easier and harder ways to do the math. Dont just start ailing away at the calculation with a stick; the time taken to think about the easiest, most ecient way to do the calculation will almost always more than pay for itself even (in fact, especially) on tests. Structure your work on the page to reect the logic. The same people who do random algebra also tend to splatter their work over the page like a plate of spaghetti thrown against a wall. Your work should never squirm and writhe around the page, nor should it pass through wormholes from one part of the page to another. The more logically structured your work, the less likely you will make mistakes and the more likely you will nd the path of enlightenment. Do some struggling. Youll never develop your ability if you just throw your hands up and give up when youre at a loss. To really learn, you have to spend some time scratching your head over the dicult problems.

Stop and think. Dont go gleefully skipping o when you have arrived at a result for the unknown. Think about what that result means physically and how it behaves: Does it make sense? Is it acceptable to have gotten a negative answer? Are the dimensions and units correct? Reecting on what your result means is critical to your understanding of the physics and often will also help you catch errors. This is the principal reason why its highly desirable to work in terms of variables rather then numbers: only when your result is in terms of variables can you ascertain how your solution will behave for certain values or limits of the parameters. Just how you go about analyzing your result depends on the problem and will, we hope, be more clear after you read the example that follows. In general you want to ask yourself questions like, What happens when one of the masses is really large or really small? Or when two masses are equal? Or when an angle is 0 or ? To help you develop this ability 2 to analyze your results, many of the problems will explicitly ask you what happens in various cases.

1.4. THE ZEN OF PROBLEM SOLVING

11

We cannot emphasize enough how benecial adherence to the above precepts will be to developing both your understanding of the physics and your ability to apply it. And we say this not only because of the importance of these precepts, but because long experience has shown that no advice is more likely to fall on deaf ears. At times, teaching can be like reading Shakespeare to your dog. Its depressing, really. But you can make a dierence: heed these precepts, and you will go a long way toward restoring our faith in humanity. Not to mention developing both your understanding of the physics and your ability to apply it.

1.4.1

An Unfortunate Example

What we need now is a simple, easy example that doesnt involve any physics you dont yet know and that illustrates everything we just said about the Zen of problem solving. I wish I could think of a problem that satises all those constraints. I really do. But the only problem I can think of that doesnt involve any physics you dont yet know and that illustrates everything we just said about the Zen of problem solving is not quite as simple as one would like. While this is unfortunate, we trust that you will be able to keep your focus on the techniques that the example is intended to illustrate and not let its apparent complexity cause you to stampede to the Registrars Oce like panicked wildebeest. And so: Suppose you want to get to a point directly across a conveniently rectilinear river of width , from point A to point B, as shown, rather melodramatically, in g. (1.1). You can row a boat at speed vr , the rivers current ows to the right at speed vc > vr (that is, the current is faster than you can row), and you can walk along the shore of the river at speed vw . We will try to determine the angle at which you should row (that is, aim the boat) in order to get from A to B in the shortest time. Because the current will be carrying you downstream as you row, we expect that the fastest way will not be to row straight toward B ( = 0) but rather to row somewhat A

vc

B Figure 1.1: The River of Death

12 A

CHAPTER 1. PRELIMINARIES

vr

vr cos

B

vr sin

Figure 1.2: Your Actual Path and the Components of vr into the current, thereby partially canceling out its tendency to carry you downstream. Because vc > vr , this cancellation can only be partial; you will be carried somewhat to the right and thus have to walk back to point B along the rivers shore, so that your actual path will be more or less as shown on the left side of g. (1.2). The logic is thus: we are given three speeds (vc , vr , and vw ), and speed is distance over time: distance speed = time Together with our knowledge of the distance across the river, this basic relation for speed should enable us to express the time to get from A to B in terms of the angle at which you row. Our rst task is to work this relation out more precisely. The time to get from A to B will be the sum of the time to reach the far shore and the time to walk along the shore from your landing point to point B. We know the distance perpendicularly across the river is , so if we can get a result for your speed in the direction perpendicularly across the river we can gure out the time it will take to reach the far shore. As you can see from the right side of g. (1.2), the component of your rowing speed that is carrying you perpendicularly across the river is vr cos . The time tcross that it will take you to reach the far side will therefore be time = tcross distance speed = vr cos

If we knew how far you will have to walk along the shore, we could, knowing that the speed at which you can walk along the shore is vw , gure out the time it will take you to get from your landing point to point B. And now that we know how long it will take to cross the river, we can gure out how far you will have to walk along the shore: from the right side of g. (1.2), you can see that the component of your rowing speed that is working against the current is vr sin . This means that speed at which you are being carried

1.4. THE ZEN OF PROBLEM SOLVING

13

to the right is not the full vc but rather vc vr sin . During the time that it will take you to reach the other side, the distance ddrift that you will be carried downstream is therefore distance = speed time ddrift = (vc vr sin ) tcross = (vc vr sin ) vr cos vc vr sin = vr cos Walking at speed vw , the corresponding time tw that you will have to spend walking along the shore back to point B will thus be time = distance speed ddrift tw = vw vc vr sin vr cos = vw vc vr sin = vw vr cos

The total time tAB to go from A to B will therefore be tAB = tcross + tw vc vr sin = + vr cos vw vr cos (vw + vc vr sin ) = vw vr cos where we have neatened up a bit by pulling out an overall factor of /vw vr cos . Our rst task is now complete: we have an expression for the time to get from A to B in terms of and given quantities. Next we want to minimize this time tAB with respect to , which is a straightforward if somewhat tedious maxima-minima problem: 0= dtAB d d (vw + vc vr sin ) = d vw vr cos d 1 (vw + vc vr sin ) vw vr d cos

=

14

CHAPTER 1. PRELIMINARIES

If we divide both sides by the annoying factor /vw vr and do out the derivative, we have 0= d 1 (vw + vc vr sin ) d cos 1 d 1 d (vw + vc vr sin ) + (vw + vc vr sin ) = d cos cos d sin 1 (v + vc vr sin ) + (vr cos ) 2 w cos cos

=

Solving this for might look awful, even hopeless, but a miracle occurs when you pull out an overall factor of 1/ cos2 and simplify: 10 0= sin 1 (v + vc vr sin ) + (vr cos ) 2 w cos cos 1 (vw + vc vr sin ) sin vr cos2 = 2 cos 1 = (vw + vc ) sin vr (sin2 + cos2 ) cos2 1 ((vw + vc ) sin vr ) = cos2 sin =

which yields

vr (1.1) vw + vc This is our result for the that will minimize the time to get from A to B. As a quick check on our algebra, we can look at the dimensions in our result: since angles are dimensionless quantities,11 the and hence sin on the left side of eq. (1.1) are dimensionless. In the denominator on the right side, we are adding two speeds, which is ne if we instead had something like vw + , that would indicate an error somewhere earlier in the calculation, because it doesnt make any sense to add a length to a speed (which is length/time). And overall on the right side of eq. (1.1) we have the ratio of two speeds, which, being dimensionless, matches the left side of eq. (1.1).12 So everything is okey-dokey dimension-wise. This does not prove that our10 11

Whether this constitutes proof of a divine being is left for the reader to decide. Degrees, radians, grads, etc., are not physical units; theyre just reminders of how youre measuring your angles. Angles represent ratios of arc length to radius (s = r) and are therefore dimensionless; properly they should always be measured in what we call radians, because C = 2r gives 2 as the angle corresponding to a full circle. But again, radians do not have dimensions: since the length of the circumference C and the radius r in C = 2r cancel out, the 2 radians in a full circle have no dimensions. 12 It turns out that there are only three fundamental physical dimensions: length, mass, and time. The dimensions of all physical quantities are some combination of these basic three. When doing a dimensional analysis, it is customary to denote length, mass, and

1.4. THE ZEN OF PROBLEM SOLVING

15

result is correct, but had we found a mismatch in dimensions, that would denitely have indicated an error. Now that we have a result in which we are fairly condent, we need to try to understand it. The rst thing to note is that, of the givens, does not depend on the width of the river. This should seem sensible when you consider that, since is the only distance in the problem, and since time = distance speed

all times must be proportional to , which therefore constitutes merely an overall factor that divides out when we minimize the time. That is, changing the width of the river by a factor will scale all the distances and times, including the time to get from A to B, by that same factor, but it will not change the directions and angles of the path that minimizes tAB . What our result for does depend on are the speeds vc , vr , and vw , in the form of the ratio vr vw + vc This suggests that it might be interesting to look at what happens when vc , vr , or vw is very small or very large.13 If vw , then sin 0 and hence 0, which corresponds to rowing (aiming) straight across the river. This what we would have expected: when you can walk really fast, it doesnt matter how far along the shore you have to walk, so you should simply set the course that will minimize the time to reach the other side, without worrying about how far you will drift because of the current. When vr is very small (vr 0) or vc very large (vc ), we again get 0: when your rowing speed is very small or the current is very swift, you arent going to save yourself any time by trying to oset the current; youre better o just rowing (aiming) directly across and then walking however far you have to along the shore to get back to point B. We could look at other limits, but these are sucient to illustrate how you go about interpreting and understanding your nal result when solving problems.14time by [], [m], and [t]. Thus a speed would have dimensions []/[t], and the dimensions of eq. (1.1) would be 1=13 [] [t] [] [t]

+

[] [t]

=1

If the denominator were vw vc instead of vw + vc , the case vw = vc would also be interesting to consider. 14 For those who are curious, we cannot look at the limit vw 0 without reworking the problem: although this limit seems to yield sin vr /vc , on our way to our result for sin we divided by vw . We also cannot look at vc 0 or vr without reworking

16

CHAPTER 1. PRELIMINARIES

If you go back through the Zen principles of problem solving, you will see that the above solution adheres to them all: we drew pictures to help visualize the situation, made note of known and unknown quantities, and clearly dened our notation. We used the denition of speed and a little trigonometry to obtain a result for the time to get from A to B in terms of the known quantities, then minimized this time with respect to by doing a maxima-minima calculation. We also looked at the quantities on which our result for depended and how that result behaved in various limits of the given velocities and saw that everything made sense. Also note that we used notation that reected the kinds of quantities represented: tcross for the time to cross, ddrift for the distance you drift downstream, etc. If you are in the habit of labeling every unknown quantity x, you should break that habit.

1.5

Signicant Figures

Now, no one really enjoys signicant gures, but then that really isnt the point, is it? No measured quantity is ever known with perfect accuracy, and signicant gures allow us, at least in some primitive, rudimentary way, to take this into account. Suppose, for example, that you were so bored that you were reduced to trying to determine the area of a piece of xerox paper by measuring its length and width. If your ruler allows you to measure down to a millimeter and indicates that the length and width are 27.9 cm and 21.6 cm, for the area your calculator will give you 27.9 21.6 = 602.64 cm2 . But quoting the ve-gure result 602.64 cm2 would imply that you were able to determine the area to an accuracy of about 1 part in 100,000, when in fact this result was derived from measurements each of only three gures and therefore accurate to only about 1 part in 1000. Since your result for the area is only as accurate as the data that went into calculating it, in this case you should quote an area accurate only to 1 part in a 1000 that is, you should round o your result for the area to three gures: 603 cm2 .the problem because we set everything up on the assumption that vc > vr . If we were to rework things for the case vr sin > vc , our speed parallel the river would instead be vr sin vc , and our solution for would become sin = vr vc vw

This would seem to yield sin vr /vw when vc 0 and sin when vr , but we would have to remember that in maxima-minima calculations it is possible that the maximum or minimum is at the endpoints of the domain of the parameter: in this case is restricted to 0 < , and the minimum in these two limits would actually be 2 at = 0, as expected.

1.5. SIGNIFICANT FIGURES

17

The rules and uses of signicant gures should already be familiar to you from previous courses in science; here we will merely refresh your memory a bit: For all mathematical operations except addition and subtraction, the number of signicant gures in a nal result is the number in the least accurately known data. (The annoying case of addition and subtraction is dealt with below.) Thus 1.23 4.5 = 5.5 as a nal result. Keep one extra gure in intermediate results, to avoid rounding error.15 Thus 1.23 4.5 = 5.54 as an intermediate result that you are going to use in a subsequent calculation. Zeros on the left (leading zeros) dont count toward signicant gures; zeros on the right (trailing zeros) do. Thus 0.00123 has three signicant gures: a length of 1.23 mm is accurate to 1 part in 1000 and does not become more or less accurate simply by being expressed in other units, such as 0.00123 m. On the other hand, 0.4500 has four signicant gures: 0.45 would mean a value 45 of 100 to an accuracy of 1 part in 100, while 0.4500 means a value of 45 to an accuracy of 1 part in 10,000. 100 Pure numbers (like or 2) are exact and do not aect the number of signicant gures in the calculation. Alternatively, you can think of pure numbers as having an innite number of signicant gures. In practice, you simply express them to as many digits as necessary to prevent them from limiting the signicant gures in your calculation. For example, in a calculation with four signicant gures, it is enough to use 3.1416 for . Dont count the digits in the power of ten in scientic notation, for example, the 106 in 4.32 106 ; these powers of ten are equivalent to leading zeros. If a number lacks a decimal point, there are passionately diering opinions about the signicance of trailing zeros: some would argue vehemently that 500 has three signicant gures, others equally vehemently that it has only one. All of this just goes to show how silly people canHumanity, which distinguishes itself from the lower animals chiey by its capacity for irrationality, seems to persist in the belief that the rounding used with signicant gures causes rounding error. Be sure to remember (especially when doing lab work) that the rules for signicant gures do not cause rounding error; they prevent it.15

18

CHAPTER 1. PRELIMINARIES sometimes be. Our convention will be to be reasonable about it and show the decimal point when it matters. And of course you can always avoid this ambiguity by using scientic notation: the number of signicant gures in 5 102 , 5.0 102 , and 5.00 102 are clear.

Signicant gures are determined dierently in the case of addition and subtraction. One complication is that a subtraction may wipe out some of what would otherwise have been signicant gures by making them into leading zeros. Suppose, for example, that you were trying to determine the length of an object and found that its start and end were at 12.6 cm and 11.8 cm along a meter-stick. Since each of these measurements has three signicant gures, you might be tempted to give 12.6 11.8 = 0.800 cm for the objects length. This would, however, imply that you had determined to objects length down to 0.001 cm, when in fact your measurements were only accurate to 0.1 cm. The problem is that the subtraction has made leading zeros out of the rst two digits in the 12.6 and 11.8. So your result for the objects length really has only one surviving signicant gure and should be quoted as 0.8 cm. If only that were the sole problem presented by addition and subtraction. But, alas, there is a second complication. Suppose, for example, that you are taking the dierence between 2.45 cm and 0.011 cm. Your calculator will give you 2.45 0.011 = 2.439. Since the 2.45 has three gures and the 0.011 two, you might reckon that the dierence has two gures and should be quoted as 2.4 cm. The problem with this reasoning is that the 0.011 is so much smaller than the 2.45 that subtracting it aects the 2.45 only starting at the third gure, so that all three of the original signicant gures in the 2.45 remain signicant. The dierence should therefore be quoted as 2.44 cm. The same sort of argument would lead you to quote 2.46 for the result of 2.45 + 0.011. Both kinds of complications can be taken into account by means of the following (admittedly rather opaquely phrased) prescription: When numbers are added or subtracted, the position of the leftmost of the last signicant gures of these numbers is the position of the last signicant gure in the result. Say what? you say. Some examples: 23.6 23.6 2.3 23.6 21.4 22.8 0.284 3.88 2.2 0.8 2.0 19.7 And when exponents are involved, just express the numbers in terms of a common power of ten: 2.34 108 3.6 106 = (2.34 0.036) 108 = 2.30 108 So much for signicant gures.

1.6. UNITS & CONVERSIONS

19

1.6

Units & Conversions

Dont forget to indicate the units on any data or result that has physical dimensions. For example, simply reporting 4 for the height of an object is meaningless: 4 what? Inches? Meters? Miles? We will usually stick with the MKS (SI) system of units, so that all of the physical quantities we deal with will be measured in meters, kilograms, seconds, or some combination of these basic three units.16 In conversions, do multiplications and divisions of units just as you do ordinary arithmetic with numbers. For example, to convert 18 inches into feet, do something like 1 ft = 1.5 ft 18 in 12 in Likewise, to convert cubic feet to cubic inches, 1 ft3 12 in ft3

= 1728 in3

Doing conversions in this kind of explicit detail might feel really MickeyMouse, but it will save you from the dreaded and all too common error of getting the conversion factors upside down. Conversion factors can usually be found in the jackets or appendices of textbooks; in the next section you will nd the values of some of the more common ones.

1.7Length 1 km 1m 1 in 1 mi 1 ly16

Conversion Factors, Prexes, & Physical Constants17

= = = = =

0.6214 mi 3.281 ft 2.540 cm 5280 ft 9.461 1015 m

= =

39.37 in 1.609 km

MKS stands for (doh!) meter-kilogram-second, and SI for the French phrase for International System. 17 Values of the physical constants and astronomical values are mostly from the various documents available through Lawrence Berkeley National Laboratory (http://pdg.lbl.gov/2005/reviews/contents_sports.html). Conversion factors are from the freely redistributable open-source Units program (http://www.gnu.org/ software/units/units.html), with some rounding. A few odd values we just made up.

20 Volume 1 gallon = 1 liter = Speed 1 mph 60 mph Force 1N 1 lb = = 0.4470 m/s 88 ft/sec

CHAPTER 1. PRELIMINARIES

3.785 liters 1000 cm3 =

1 1000

m3

= =

0.2248 lb 4.448 N

Pressure 1 Pa 1 atm 1 mm Hg

= 1.450104 lb/in2 = 1.01325105 Pa = 133.3 Pa

= 760 mm Hg =

14.70 lb/in2

Energy & Power 1 cal 1 BTU 1 eV 1 hp = = = = 4.1868 J 1055 J 1.602176531019 J 745.7 W

Prex k (kilo) M (mega) G (giga) T (tera) P (peta) E (exa)

Factor 103 106 109 1012 1015 1018

Prex c (centi) m (milli) (micro) n (nano) p (pico) f (femto) a (atto)

Factor 102 103 106 109 1012 1015 1018

Body Sun Earth Mars Moon

Mass (kg) 1.98844 1030 5.9723 1024 6.4185 1023 7.347673 1022

Radius (m) 6.961 108 6.378140 106 3.3774 106 1.7360 106

Mean Orbital Radius (m) 1.49597870660 1011 2.279 1011 3.84403 108

Period (days) 365.24219 686.9600 27.32166155

1.8. ORDER-OF-MAGNITUDE ESTIMATES Physical Constants Speed of light Electron charge Universal gravitational constant Plancks constant Electron mass Proton mass Neutron mass Electric force constant Electric permittivity of vacuum Magnetic permeability of vacuum Boltzmann constant Avogadros number Gas constant Atomic mass unit (u) Absolute zero Acceleration due to gravity c e G h = h/2 me mp mn k 0 0 k N0 R = = = = = = = = = = = = = =

21

g

2.99792458 108 m/s 1.60217653 1019 C 6.6742 1011 Nm2 /kg2 6.6260693 1034 Js 1.05457168 1034 Js 9.1093826 1031 kg 1.67262171 1027 kg 1.67492728 1027 kg 8.987551788 109 Nm2 /C2 8.854187817 1012 C2 /Nm2 4 107 N/A2 1.3806505 1023 J/K 6.0221415 1023 8.3144727 J/molK 1.66053886 1027 kg 273.15 C = 9.80665 m/s2

1.81.8.1

Order-of-Magnitude EstimatesAn Example

It is possible to make rough but meaningful estimates of all sorts of quantities things you might at rst think you were clueless about with just common knowledge and common sense. All of these order-of-magnitude estimates18 are pretty much the same, so the procedure is most easily illustrated by example: Suppose (to take a decidedly lame but classic example) you want to estimate the number of jelly beans in a jar maybe a Kewpie doll is at stake, or tickets to the Final Four. You could simply directly estimate the number of jelly beans in the jar, but that would be just a wild guess; theres no way to get a good feel for such a large number: a hundred thousand? A million? A zillion? Who could say? To make a reliable estimate, you need to break the problem down into smaller pieces, small enough that you can easily visualize or get a feel for them. If you knew the volume of the interior of the jar and the average volume taken up by a jelly bean, all youd have to do is divide the former by the18

We use this coloring for technical terms when they appear for the rst time.

22

CHAPTER 1. PRELIMINARIES

latter and youd have the number of jelly beans. So you need to come up with estimates for these two volumes. First, the jelly bean: You could try to estimate the volume taken up by a single jelly bean directly, but for most people this would still be rather hard to get a good feel for: a cubic centimeter? A half an ounce? A tenth of a milliliter? You need to break the problem down still further: to derive a reliable result for the volume of a single jelly bean, you need to estimate its geometry and its corresponding dimensions. Now, a jelly bean is actually a kind of balloon shape, but thats a complicated shape to deal with. You could approximate that the jelly bean is a little cylinder, in which case youll need to estimate its diameter and length in order to get a result for its volume. Or you could approximate that the jelly bean is a little rectangular box, in which case youll need to estimate a length, a width, and a height. Which choice is best? And in any case, what about the air space between beans? This is where the order-of-magnitude part of order-of-magnitude estimate comes in: order of magnitude refers to the power of ten (like the 4 in 104 ); in an order-of-magnitude estimate, youre only trying to get within a factor of ten of the actual answer. In the present example, that means it doesnt matter whether you treat the jelly bean as a balloon shape, a cylinder, or a box, or even whether you account for the (relatively) small air space between beans the dierences between the resulting volume estimates are not going to be signicant. So you might as well choose the simplest shape to deal with: the box. If the jar of jelly beans is right in front of you, you can just look at the beans when estimating their dimensions. But even if you dont have any beans to look at when youre working through the estimate, you can still visualize one in your mind. Lets suppose that, one way or the other, 1 your estimate of the dimensions is 1 cm 3 cm 1 cm = 1 cm3 0.1 cm3 . 3 9 Someone else might have somewhat dierent estimates for the length, width, and height and instead get 0.4 cm3 . Another person might get 0.08 cm3 . All of these estimates would be correct in the sense that they are based on reasonable estimates of the dimensions of a jelly bean; there is no one right answer for such a rough estimate, and you wouldnt even consider two estimates to dier signicantly unless that dierence was by a factor of ten or more. For this reason also we rounded o our result to one signicant gure which is really all we can justify in such a rough estimate.19 Even if the jar were opaque and you had no prior experience with jelly beans, you could still make an estimate of the volume of a jelly bean based only on the knowledge that jelly beans are eaten by the handful. RememberThere is, of course, such a thing as a bad estimate. To say that a jelly bean is like a cube 1 inch on a side would be a bad estimate because 1 inch 1 inch 1 inch is clearly unreasonably large.19

1.8. ORDER-OF-MAGNITUDE ESTIMATES

23

ing that you are only trying to get within a factor of ten in your estimates, you could reason as follows: If we assume, for lack of any better information, that the geometry of a jelly bean is cubic, that wont be too far o even if it is in fact spherical or oblong or some other reasonable geometry. And each side of this box would reasonably be on the order of 1 cm: 10 cm on a side would clearly be too large (youd have to take bites out of one); 0.1 cm = 1 mm on a side seems too small (itd be like eating poppy seeds). Thus you would arrive at an estimate of 1 cm 1 cm 1 cm = 1 cm3 for the volume of each jelly bean not very dierent from the estimates obtained by someone who had actually seen jelly beans.20 Now for the jar: If you knew that the jar was a one-gallon jar, it would be both accurate and correct to use 1 gal = 3785 cm3 for its volume. Otherwise, you have to do for the jar what you did for the jelly bean: estimate its geometry and dimensions. Suppose the jar looks cylindrical and that you 1 estimate it has a diameter of 15 cm and a height of 3 m 30 cm. Your 2 2 estimate for its volume would then be r h 3 (15) 30 2 104 cm3 . Your result for the number of jelly beans in the jar would then be 2 104 0.1cm3 jar

cm3 bean

= 2 105

beans jar

Bear in mind that this is, of course, only a very rough estimate. The actual number of beans could easily be anywhere within about a factor of ten of this: 80,000, or 400,000, or even 345,262.

1.8.2

General Points

The key to making good order-of-magnitude estimates is breaking the problem down into pieces small enough that you can easily visualize or get a feel for them. When it is beyond your experience to give a knowledgeable estimate of a quantity, you can fall back on the 1, 10, 100 reasoning. (See footnote 20.) The point of the homework involving estimates is to give you practice making estimates. (Duh!) You may be able to nd some of the results you need in books or, of all places, on the Internet, but that would be defeating the purpose of the homework so dont do it.Similarly, suppose you needed an estimate of how long a walrus lives. You dont need to have any specialized knowledge of walruses: for a large mammal, one year is clearly too short, and for any mammal 100 years would be a very long time, so anything in between would, for lack of any better information, be reasonable: 10 years, 30 years, 45 years, whatever these numbers are all of the same order of magnitude.20

24

CHAPTER 1. PRELIMINARIES

1.8.3

A Brief Discourse on Malarkey

To some of you this estimate business may seem like malarkey.21 Believe it or not there are, however, uses for it in physics and engineering: a quick order-of-magnitude estimate can often save you a great deal of design time by making immediately clear whether a project is feasible or not. From the jellybean estimate, for example, you would immediately conclude that it would not really be feasible to count the exact number of jelly beans in the jar by hand 200,000 is just too many to count. You would also be skeptical of anyone who claimed to have counted all the beans by hand. More generally, if you are able to make your own order-of-magnitude estimates, you will often be able to test the validity of assertions made in news articles by reporters and politicians who think youre the most gullible animal in the barnyard. And still more generally, making estimates is good practice in basic problem solving and nothing is more certain than that life will require you to do a great deal of problem solving.

Then again, to some of you, this whole course probably seems like total malarkey. To the extent that this is true, however, the course is excellent preparation for life.

21

1.9. PROBLEMS

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1.9

Problems

1. Be obsessively careful of signicant gures in all of the following. (a) Determine the circumference of a circle of radius 3.5 cm. (b) Determine the volume of a rectangular box that is 4.6 cm by 12.3 cm by 5.70 cm. (c) Determine the total mass of three objects whose individual masses are 1.23 102 kg, 3.4 kg, and 9.87 101 kg. (d) Three points A, B, and C lie along a straight line. Determine the distance between points A and B if the distance between points A and C is 14.7 m and the distance between points B and C is 3.62 m. (Dude: that means take the dierence.) 2. The density of water is 1.0 g/cm3 . What is it in (a) kg/m3 ? (b) lb/ft3 ? Note: a pound is actually a unit of force (weight), not mass, but there is a correspondence 1 kg 2.2 lb. (Well discuss this in more detail when we get to forces.) 3. (a) Determine your age in seconds. (b) How exact is this result? That is, by how much could your result for the number of seconds be o? You should take into account how accurately you know when you were born (which certainly isnt likely be down to the precise second) and also how accurately you dealt with leap years. (c) How many signicant gures does your result for your age in seconds therefore have? 4. Estimate the percent error in measuring (a) a mass of 20 g on a scale that gives a reading out to a hundredth of a gram. (b) a length of 20 cm with an ordinary meter stick (which, you will recall, has gradations of millimeters). 5. Suppose that for some silly reason you wanted to determine the volume of a circular cylinder of diameter 1.23 102 mm and height 0.025 m. What do you suppose that volume would be?

26

CHAPTER 1. PRELIMINARIES

6. People who apparently are in a position to know tell us that the mass and radius of the Earth are 5.9723 1024 kg and 6.378140 103 km, respectively. Determine the Earths average density in g/cm3 , explicitly noting any assumptions you are making. 7. How much cow op does a typical cow op over its lifetime? 8. (a) How many drops of water are there in all the oceans combined? (b) While youre at it, estimate the number of water (H2 O) molecules currently in your body that were in the body of Socrates at the moment he died, assuming that the water molecules that were in Socrates are now evenly mingled among all the water molecules on Earth. (If you want, you can use Confucius or Cleopatra or whatever other ancient personage you regard as awesomely righteous.) 9. How many golf balls would be needed to completely ll the average Pizza Hut? 10. How much toothpaste will you use over the course of your lifetime? 11. How much water does the average family use per week? 12. One year it was proposed that the Bowl 22 be lled with Jello as the senior prank. How much would it cost (in terms of dollars spent for the Jello at the supermarket) to do this? 13. How many rolls of toilet paper are consumed (in the broader sense of that word) on campus each academic year? 14. Estimate the number of toilets needed for