THE
MECHANICAL
UNIVERSE
MECHANICS AND HEAT,
ADVANCED EDITION
STEVEN С FRAUTSCHI
PROFESSOR OF THEORETICAL PHYSICS
CALIFORNIA INSTITUTE OF TECHNOLOGY
RICHARD P. OLENICK
ASSISTANT PROFESSOR OF PHYSICS, UNIVERSITY OF DALLAS
VISITING ASSOCIATE, CALIFORNIA INSTITUTE OF TECHNOLOGY
TOM M. APOSTOL
PROFESSOR OF MATHEMATICS, CALIFORNIA INSTITUTE OF TECHNOLOGY
DAVID L. GOODSTEIN
PROFESSOR OF PHYSICS AND APPLIED PHYSICS, CALIFORNIA INSTITUTE OF TECHNOLOGY
AND PROJECT DIRECTOR, THE MECHANICAL UNIVERSE
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The University has printed
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CAMBRIDGE UNIVERSITY PRESS
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CONTENTS
Preface
xiii
Chapter 1 INTRODUCTION TO THE MECHANICAL UNIVERSE
1.1 The Copernican Revolution
1.2 Units and Dimensions
5
1.3 A Final Word
9
Problems
10
1
Chapter 2 THE LAW OF FALLING BODIES
2.1
2.2
2.3
2.4
11
Historical Background
11
Galileo's Law
12
Do Heavy Bodies Fall Faster Than Light Ones?
Medieval Laws of Falling Bodies
14
12
1
2.5
2.6
2.7
2.8
2.9
Experimental Determination of the Law of Falling Bodies
The Average Velocity of a Falling Body
18
Instantaneous Velocity: The Derivative
18
Acceleration
20
A Final Word
22
Problems
23
15
Chapter 3 THE LANGUAGE OF NATURE: DERIVATIVES AND INTEGRALS
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
The Development of Differential Calculus
27
Derivatives and Slopes of Tangent Lines
28
Leibniz's Notation. Analytic Definition of the Derivative
31
Rules of Differentiation and Derivatives of Special Functions
32
Antidifferentiation, the Reverse of Differentiation
38
Antidifferentiation and Quadrature
39
The Leibniz Integral Notation
44
Applications of the Second Fundamental Theorem to Physics
48
A Final Word
51
Problems
52
Chapter 4 INERTIA
4.1
4.2
4.3
4.4
4.5
4.6
If the Earth Moves: Aristotelian Objections
The Earth Moves: Galileo's Law of Inertia
Relative Motion
62
Projectile Motion: A Consequence of Inertia
Calculating a Particular Trajectory
65
A Final Word
68
Problems
68
Chapter 5 VECTORS
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
57
57
60
63
75
Coordinate Systems
75
Vectors
77 .
Addition and Subtraction of Vectors, and Multiplication by a Scalar
The Scalar Product of Vectors
88
The Cross Product of Vectors
93
Derivatives of Vector Functions in a Fixed Coordinate System
96
Position Vector Expressed in Polar Coordinates
100
Uniform Circular Motion
101
A Final Word
105
Problems
106
Chapter 6 NEWTON'S LAWS AND EQUILIBRIUM
6.1 The End of the Confusion
6.2 Newton's Laws of Motion
7 73
774
773
CONTENTS
6.3
6.4
6.5
6.6
6.7
Units of Mass, Momentum, and Force
117
Projectile Motion as an Application of Newton's Second Law
Equilibrium: Balance of Forces
720
Equilibrium: Balance of Torques
725
A Final Word
734
Problems
734
7 79
Chapter 7 UNIVERSAL GRAVITATION AND CIRCULAR MOTION
7.1
7.2
7.3
7.4
7.5
7.6
7.7
The Genesis of an Idea
747
The Law of Universal Gravitation
742
Acceleration of Gravity on the Earth
749
Why the Moon Doesn't Fall to the Earth
757
Circular Orbits
754
Other Examples of Uniform Circular Motion
756
A Final Word
759
Problems
760
Chapter 8 FORCES
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
763
The Fundamental Forces: Classification and Unification
The Strength of Gravitational and Electric Forces
765
Contact Forces
768
Application of Newton's Laws
772
Friction
779
Driving on Curved Roadways
785
Motion in a Resistive Medium
787
The Oil-Drop Experiment
792
A Final Word
796
Problems
797
Chapter 9 FORCES IN ACCELERATING REFERENCE FRAMES
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
747
Inertial and Noninertial Reference Frames
203
Galilean Relativity
204
Inertial Forces
208
Inertial Forces in a Linearly Accelerating Frame
Centrifugal Force
270
Effect of the Earth's Rotation on g
211
Centrifuges
273
A Final Word
275
Problems
276
279
227
203
208
Chapter 10 ENERGY: CONSERVATION AND CONVERSION
10.1 Toward an Idea of Energy
10.2 Work and Potential Energy
763
279
10.3
10.4
10.5
10.6
10.7
10.8
10.9
Kinetic Energy and the Conservation of Energy
228
Gravitational Potential Energy
236
Potential Energy and Stability
240
Heat and Energy
246
Mechanical Advantage and Efficiency of Machines
250
Power
252
A Final Word
253
Problems
254
Chapter 11 THE CONSERVATION OF MOMENTUM
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
The Universe as a Machine
263
Newton's Laws in Retrospect
265
The Center of Mass
266
The Law of Conservation of Momentum
269
Rocket Propulsion
273
Energy and Momentum Conservation in Collisions
Center-of-Mass Coordinates
282
Impulse: Collision Forces and Times
286
A Final Word
288
Problems
289
Chapter 12 OSCILLATORY MOTION
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
275
295
Finding a Clock That Wouldn't Get Seasick
295
Simple Harmonic Motion
297
The General Solution of the Differential Equation of Simple
Harmonic Motion
307
Examples of Simple Harmonic Oscillators
303
Energy Conservation and Simple Harmonic Motion
306
The Simple Pendulum
309
Gaining Insight Through Approximations
312
Damped Oscillations
313
Forced Oscillations
316
Describing Resonance
377
Damped Forced Oscillations
327
Swinging and Singing Wires in the Wind
326
A Final Word
327
Problems
330
Chapter 13 ANGULAR MOMENTUM
13.1
13.2
13.3
13.4
263
Rotary Motion
335
Torque and Angular Momentum
Angular Momentum Conservation
Force and Torque
342
335
336
338
13.5
13.6
13.7
13.8
Kepler's Law of Equal Areas
344
Vortices and Firestorms
348
Conservation of Angular Momentum and Energy
A Final Word
355
Problems
356
Chapter 14 ROTATIONAL DYNAMICS FOR RIGID BODIES
14.1
14.2
14.3
14 A
14.5
14.6
14.7
14.8
14.9
14.10
14.11
14.12
14.13
14.14
473
An Ancient Question
473
The Gyroscope
475
Angular Velocity of Precession
422
The Earth as a Gyroscope
425
A Final Word
426
Problems
427
Chapter 16 KEPLER'S LAWS AND THE CONIC SECTIONS
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
363
Rotation of a Rigid Body About a Fixed Axis
363
Center of Mass of a Continuous Mass Distribution
364
Moment of Inertia
371
Calculation of Moments of Inertia
374
The Parallel-Axis Theorem
378
Energy and Work in Rigid-Body Rotation
381
Analogies Between Rotational and Translational Motion
The Physical Pendulum
384
The Torsion Pendulum
387
Combined Translations and Rotations
388
Kinematics of the Rolling Wheel
393
Rolling Down an Inclined Plane
394
Coriolis Forces
396
A Final Word
407
Problems
407
Chapter 15 GYROSCOPES
15.1
15.2
15.3
15.4
15.5
350
The Quest for Precision
437
Kepler's Laws
433
Conic Sections
437
The Ellipse
439
The Conies and Eccentricity
439
Properties of the Ellipse
442
Cartesian Equations for Conic Sections
A Final Word
448
Problems
448
446
437
CONTENTS
X
Chapter 17 SOLVING THE KEPLER PROBLEM
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
17.10
17.11
Setting the Stage
451
Polar Coordinates and the Unit Vectors г andO
452
Solution of the Kepler Problem
457
Celestial Omens: Comets
459
Energy and Eccentricity
460
Orbits and Eccentricity
467
Kepler's Third Law
463
Planetary Motion and Effective Potential
464
Applications of Orbital Dynamics
466
Calculating the Orbit From Initial Conditions
469
A Final Word
472
Problems
473
Chapter 18 NAVIGATING IN SPACE
18.1
18.2
18.3
18.4
18.5
477
Freeways in the Sky
477
Navigating in Space
478
Transfer Orbits
480
Gravity Assist
485
A Final Word
487
Problems
489
Chapter 19 TEMPERATURE AND THE GAS LAWS
19.1
19.2
19.3
19.4
19.5
497
Temperature and Pressure
497
The Gas Laws of Boyle, Charles, and Gay-Lussac
The Ideal-Gas Law
499
Temperature and Energy
500
A Final Word
507
Problems
503
Chapter 20 THE ENGINE OF NATURE
20.1
20.2
20.3
20.4
20.5
20.6
20.7
457
505
The Age of Steam
505
Work and the Pressure-Volume Diagram
507
The First Law of Thermodynamics
572
Adiabatic and Isothermal Processes
575
The Second Law of Thermodynamics
579
The Carnot Engine
527
A Final Word
525
Problems
526
497
Chapter 21 ENTROPY
21.1
21.2
21.3
21.4
21.5
537
Toward an Understanding of Entropy
531
Engines and Entropy
534
Entropy and the Second Law of Thermodynamics
An Implication of the Entropy Principle
547
A Final Word
544
Problems
544
Chapter 22 THE QUEST FOR LOW TEMPERATURE
22.1
22.2
22..3
22.4
22.5
22.6
539
547
Cooling Off
547
The States of Matter
548
Behavior of Water
557
Liquefaction of Gases
553
The Joule-Thomson Effect
557
A Final Word
559
Problems
560
Appendix A THE INTERNATIONAL SYSTEM OF UNITS
Appendix В CONVERSION FACTORS
563
565
Appendix С FORMULAS FROM ALGEBRA, GEOMETRY, AND TRIGONOMETRY
Appendix D ASTRONOMICAL DATA
577
Appendix E PHYSICAL CONSTANTS
573
SELECTED BIBLIOGRAPHY
Index
579
575
569