Eng Course- Introductory Physics I- Download Free PDF

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Contents
Preface xiii
Textbook Layout and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
I: Getting Ready to Learn Physics 3
Preliminaries 3
See, Do, Teach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Other Conditions for Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Your Brain and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
How to Do Your Homework Eﬀectively . . . . . . . . . . . . . . . . . . . . . . . 23
The Method of Three Passes . . . . . . . . . . . . . . . . . . . . . . . . . 27
Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Homework for Week 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
II: Elementary Mechanics 37
Week 1: Newton’s Laws 39
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.1: Introduction: A Bit of History and Philosophy . . . . . . . . . . . . . . . . 45
1.2: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3: Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4: Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.5: Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iii CONTENTS
1.5.1: The Forces of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.5.2: Force Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.6: Force Balance – Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . 63
Example 1.6.1: Spring and Mass in Static Force Equilibrium . . . . . . . 63
1.7: Simple Motion in One Dimension . . . . . . . . . . . . . . . . . . . . . . . 64
Example 1.7.1: A Mass Falling from Height H . . . . . . . . . . . . . . . 65
Example 1.7.2: A Constant Force in One Dimension . . . . . . . . . . . . 71
1.7.1: Solving Problems with More Than One Object . . . . . . . . . . . . 74
Example 1.7.3: Atwood’s Machine . . . . . . . . . . . . . . . . . . . . . . 75
Example 1.7.4: Braking for Bikes, or Just Breaking Bikes? . . . . . . . . . 77
1.8: Motion in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
1.8.1: Free Flight Trajectories – Projectile Motion . . . . . . . . . . . . . 81
Example 1.8.1: Trajectory of a Cannonball . . . . . . . . . . . . . . . . . 81
1.8.2: The Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Example 1.8.2: The Inclined Plane . . . . . . . . . . . . . . . . . . . . . . 85
1.9: Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1.9.1: Tangential Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
1.9.2: Centripetal Acceleration . . . . . . . . . . . . . . . . . . . . . . . . 89
Example 1.9.1: Ball on a String . . . . . . . . . . . . . . . . . . . . . . . . 91
Example 1.9.2: Tether Ball/Conic Pendulum . . . . . . . . . . . . . . . . 92
1.9.3: Tangential Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 93
1.10: Conclusion: Rubric for Newton’s Second Law Problems . . . . . . . . . . 94
Homework for Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Week 2: Newton’s Laws: Continued 111
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.1: Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Example 2.1.1: Inclined Plane of Length L with Friction . . . . . . . . . . 115
Example 2.1.2: Block Hanging oﬀ of a Table via a Pulley Pulls Another Block, with Friction118
Example 2.1.3: Find The Minimum Braking Distance for a Car Stopping With and Without Skidding120
Example 2.1.4: Car Rounding a Banked Curve with Friction . . . . . . . . 123
2.2: Drag Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125CONTENTS iii
2.2.1: Stokes, or Laminar Drag . . . . . . . . . . . . . . . . . . . . . . . . 128
2.2.2: Rayleigh, or Turbulent Drag . . . . . . . . . . . . . . . . . . . . . . 129
2.2.3: Terminal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Example 2.2.1: Falling From a Plane and Surviving . . . . . . . . . . . . . 132
Example 2.2.2: Solution to Equations of Motion for Stokes’ Drag . . . . . 133
2.2.4: Advanced: Solution to Equations of Motion for Turbulent Drag . . 134
Example 2.2.3: Dropping the Ram . . . . . . . . . . . . . . . . . . . . . . 135
2.3: Inertial Reference Frames – the Galilean Transformation . . . . . . . . . . 138
2.3.1: Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.3.2: Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2.4: Non-Inertial Reference Frames – Pseudoforces . . . . . . . . . . . . . . . . 143
2.4.1: Identifying Inertial Frames . . . . . . . . . . . . . . . . . . . . . . . 145
Example 2.4.1: Weight in an Elevator . . . . . . . . . . . . . . . . . . . . 147
Example 2.4.2: Pendulum in a Boxcar . . . . . . . . . . . . . . . . . . . . 148
2.4.2: Advanced: General Relativity and Accelerating Frames . . . . . . . 151
2.4.3: Just For Fun: Hurricanes . . . . . . . . . . . . . . . . . . . . . . . . 152
Homework for Week 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Week 3: Work and Energy 167
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.1: Work-Kinetic Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.1.1: Units of Work and Energy . . . . . . . . . . . . . . . . . . . . . . . 171
3.1.2: Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.1.3: Derivations of the Work-Kinetic Energy Theorem . . . . . . . . . . 173
Example 3.1.1: Pulling a Block . . . . . . . . . . . . . . . . . . . . . . . . 177
Example 3.1.2: Range of a Spring Gun . . . . . . . . . . . . . . . . . . . . 178
3.2: Conservative Forces: Potential Energy . . . . . . . . . . . . . . . . . . . . 179
3.2.1: Force from Potential Energy . . . . . . . . . . . . . . . . . . . . . . 180
3.2.2: Potential Energy Function for Near-Earth Gravity . . . . . . . . . . 183
3.2.3: Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.3: Conservation of Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . 185
3.3.1: Force, Potential Energy, and Total Mechanical Energy . . . . . . . 186iv CONTENTS
Example 3.3.1: Falling Ball Reprise . . . . . . . . . . . . . . . . . . . . . . 186
Example 3.3.2: Block Sliding Down Frictionless Incline Reprise . . . . . . 187
Example 3.3.3: A Simple Pendulum . . . . . . . . . . . . . . . . . . . . . 187
Example 3.3.4: Looping the Loop . . . . . . . . . . . . . . . . . . . . . . . 188
3.4: Generalized Work-Mechanical Energy Theorem . . . . . . . . . . . . . . . 191
Example 3.4.1: Block Sliding Down a Rough Incline . . . . . . . . . . . . 191
Example 3.4.2: A Spring and Rough Incline . . . . . . . . . . . . . . . . . 192
3.4.1: Heat and Conservation of Energy . . . . . . . . . . . . . . . . . . . 193
3.5: Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Example 3.5.1: Rocket Power . . . . . . . . . . . . . . . . . . . . . . . . . 194
3.6: Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.6.1: Energy Diagrams: Turning Points and Forbidden Regions . . . . . . 198
Homework for Week 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Week 4: Systems of Particles, Momentum and Collisions 213
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.1: Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.1.1: Newton’s Laws for a System of Particles – Center of Mass . . . . . 219
4.1.2: Coarse Graining: Continuous Mass Distributions . . . . . . . . . . . 222
Example 4.1.1: Center of Mass of a Continuous Rod . . . . . . . . . . . . 224
Example 4.1.2: Center of mass of a circular wedge . . . . . . . . . . . . . 226
Example 4.1.3: Breakup of Projectile in Midﬂight . . . . . . . . . . . . . . 227
4.2: Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.2.1: The Law of Conservation of Momentum . . . . . . . . . . . . . . . . 229
4.3: Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Example 4.3.1: Average Force Driving a Golf Ball . . . . . . . . . . . . . 233
Example 4.3.2: Force, Impulse and Momentum for Windshield and Bug . 234
4.3.1: The Impulse Approximation . . . . . . . . . . . . . . . . . . . . . . 234
4.3.2: Impulse, Fluids, and Pressure . . . . . . . . . . . . . . . . . . . . . 236
4.4: Center of Mass Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . 238
4.5: Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4.5.1: Momentum Conservation in the Impulse Approximation . . . . . . 240CONTENTS v
4.5.2: Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4.5.3: Fully Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . 241
4.5.4: Partially Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . 241
4.5.5: Dimension of Scattering and Suﬃcient Information . . . . . . . . . 241
4.6: 1-D Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4.6.1: The Relative Velocity Approach . . . . . . . . . . . . . . . . . . . . 245
4.6.2: 1D Elastic Collision in the Center of Mass Frame . . . . . . . . . . 246
4.6.3: The “BB/bb” or “Pool Ball” Limits . . . . . . . . . . . . . . . . . . 249
4.7: Elastic Collisions in 2-3 Dimensions . . . . . . . . . . . . . . . . . . . . . . 251
4.8: Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Example 4.8.1: One-dimensional Fully Inelastic Collision (only) . . . . . . 254
Example 4.8.2: Ballistic Pendulum . . . . . . . . . . . . . . . . . . . . . . 256
Example 4.8.3: Partially Inelastic Collision . . . . . . . . . . . . . . . . . 257
4.9: Kinetic Energy in the CM Frame . . . . . . . . . . . . . . . . . . . . . . . 259
Homework for Week 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Week 5: Torque and Rotation in One Dimension 277
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
5.1: Rotational Coordinates in One Dimension . . . . . . . . . . . . . . . . . . 279
5.2: Newton’s Second Law for 1D Rotations . . . . . . . . . . . . . . . . . . . . 281
5.2.1: The r-dependence of Torque . . . . . . . . . . . . . . . . . . . . . . 283
5.2.2: Summing the Moment of Inertia . . . . . . . . . . . . . . . . . . . . 285
5.3: The Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Example 5.3.1: The Moment of Inertia of a Rod Pivoted at One End . . . 287
5.3.1: Moment of Inertia of a General Rigid Body . . . . . . . . . . . . . . 288
Example 5.3.2: Moment of Inertia of a Ring . . . . . . . . . . . . . . . . . 288
Example 5.3.3: Moment of Inertia of a Disk . . . . . . . . . . . . . . . . . 289
5.3.2: Table of Useful Moments of Inertia . . . . . . . . . . . . . . . . . . 290
5.4: Torque as a Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Example 5.4.1: Rolling the Spool . . . . . . . . . . . . . . . . . . . . . . . 292
5.5: Torque and the Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . 293
Example 5.5.1: The Angular Acceleration of a Hanging Rod . . . . . . . . 294vi CONTENTS
5.6: Solving Newton’s Second Law Problems Involving Rolling . . . . . . . . . 295
Example 5.6.1: A Disk Rolling Down an Incline . . . . . . . . . . . . . . . 296
Example 5.6.2: Atwood’s Machine with a Massive Pulley . . . . . . . . . . 297
5.7: Rotational Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.7.1: Work Done on a Rigid Object . . . . . . . . . . . . . . . . . . . . . 300
5.7.2: The Rolling Constraint and Work . . . . . . . . . . . . . . . . . . . 301
Example 5.7.1: Work and Energy in Atwood’s Machine . . . . . . . . . . 302
Example 5.7.2: Unrolling Spool . . . . . . . . . . . . . . . . . . . . . . . . 304
Example 5.7.3: A Rolling Ball Loops-the-Loop . . . . . . . . . . . . . . . 305
5.8: The Parallel Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Example 5.8.1: Moon Around Earth, Earth Around Sun . . . . . . . . . . 309
Example 5.8.2: Moment of Inertia of a Hoop Pivoted on One Side . . . . 309
5.9: Perpendicular Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Example 5.9.1: Moment of Inertia of Hoop for Axis in the Plane of the Hoop312
Homework for Week 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Week 6: Vector Torque and Angular Momentum 329
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
6.1: Vector Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
6.2: Total Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
6.2.1: The Law of Conservation of Angular Momentum . . . . . . . . . . . 333
6.3: The Angular Momentum of a Symmetric Rotating Rigid Object . . . . . . 334
Example 6.3.1: Angular Momentum of a Point Mass Moving in a Circle . 337
Example 6.3.2: Angular Momentum of a Rod Swinging in a Circle About One End337
Example 6.3.3: Angular Momentum of a Rotating Disk . . . . . . . . . . 338
Example 6.3.4: Angular Momentum of Rod Sweeping out Cone around its CM338
6.4: Angular Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . 338
Example 6.4.1: The Spinning Professor . . . . . . . . . . . . . . . . . . . . 339
6.4.1: Radial Forces and Angular Momentum Conservation . . . . . . . . 340
Example 6.4.2: Mass Orbits On a String . . . . . . . . . . . . . . . . . . . 342
6.5: Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
Example 6.5.1: Fully Inelastic Collision of Ball of Putty with a Free Rod . 346CONTENTS vii
Example 6.5.2: Fully Inelastic Collision of Ball of Putty with Pivoted Rod 351
6.5.1: More General Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 353
6.6: Angular Momentum of an Asymmetric Rotating Rigid Object . . . . . . . 353
Example 6.6.1: Rotating Your Tires . . . . . . . . . . . . . . . . . . . . . 356
6.7: Precession of a Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Example 6.7.1: Finding ωp From ∆L/∆t (Average) . . . . . . . . . . . . . 360
Example 6.7.2: Finding ωp from ∆L and ∆t Separately . . . . . . . . . . 360
Example 6.7.3: Finding ωp from Calculus . . . . . . . . . . . . . . . . . . 362
Homework for Week 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Week 7: Statics 373
Statics Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
7.1: Conditions for Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 374
7.2: Static Equilibrium Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Example 7.2.1: Balancing a See-Saw . . . . . . . . . . . . . . . . . . . . . 377
Example 7.2.2: Two Saw Horses . . . . . . . . . . . . . . . . . . . . . . . 378
Example 7.2.3: Hanging a Tavern Sign . . . . . . . . . . . . . . . . . . . . 379
7.2.1: Equilibrium with a Vector Torque . . . . . . . . . . . . . . . . . . . 381
Example 7.2.4: Building a Deck . . . . . . . . . . . . . . . . . . . . . . . . 381
7.3: Tipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Example 7.3.1: Tipping Versus Slipping . . . . . . . . . . . . . . . . . . . 383
Example 7.3.2: Tipping While Pushing . . . . . . . . . . . . . . . . . . . . 385
7.4: Force Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
Example 7.4.1: Rolling the Cylinder Over a Step . . . . . . . . . . . . . . 388
Homework for Week 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
III: Applications of Mechanics 401
Week 8: Fluids 401
Fluids Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
8.1: General Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
8.1.1: Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404viii CONTENTS
8.1.2: Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
8.1.3: Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
8.1.4: Viscosity and ﬂuid ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . 408
8.1.5: Properties Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Static Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
8.1.6: Pressure and Conﬁnement of Static Fluids . . . . . . . . . . . . . . 409
8.1.7: Pressure and Conﬁnement of Static Fluids in Gravity . . . . . . . . 411
8.1.8: Variation of Pressure in Incompressible Fluids . . . . . . . . . . . . 414
Example 8.1.1: Barometers . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Example 8.1.2: Variation of Oceanic Pressure with Depth . . . . . . . . . 416
8.1.9: Variation of Pressure in Compressible Fluids . . . . . . . . . . . . . 417
Example 8.1.3: Variation of Atmospheric Pressure with Height . . . . . . 418
8.2: Pascal’s Principle and Hydraulics . . . . . . . . . . . . . . . . . . . . . . . 419
Example 8.2.1: A Hydraulic Lift . . . . . . . . . . . . . . . . . . . . . . . 421
8.3: Fluid Displacement and Buoyancy . . . . . . . . . . . . . . . . . . . . . . . 423
8.3.1: Archimedes’ Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Example 8.3.1: Testing the Crown I . . . . . . . . . . . . . . . . . . . . . 426
Example 8.3.2: Testing the Crown II . . . . . . . . . . . . . . . . . . . . . 427
8.4: Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
8.4.1: Conservation of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 431
8.4.2: Work-Mechanical Energy in Fluids: Bernoulli’s Equation . . . . . . 434
Example 8.4.1: Emptying the Iced Tea . . . . . . . . . . . . . . . . . . . . 436
8.4.3: Fluid Viscosity and Resistance . . . . . . . . . . . . . . . . . . . . . 436
8.4.4: A Brief Note on Turbulence . . . . . . . . . . . . . . . . . . . . . . 439
8.5: The Human Circulatory System . . . . . . . . . . . . . . . . . . . . . . . . 440
Example 8.5.1: Atherosclerotic Plaque Partially Occludes a Blood Vessel . 445
Example 8.5.2: Aneurisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Example 8.5.3: The Giraﬀe . . . . . . . . . . . . . . . . . . . . . . . . . . 448
Homework for Week 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
Week 9: Oscillations 459
Oscillation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459CONTENTS ix
9.1: The Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 461
9.1.1: The Archetypical Simple Harmonic Oscillator: A Mass on a Spring 461
9.1.2: The Simple Harmonic Oscillator Solution . . . . . . . . . . . . . . . 467
9.1.3: Plotting the Solution: Relations Involving ω . . . . . . . . . . . . . 468
9.1.4: The Energy of a Mass on a Spring . . . . . . . . . . . . . . . . . . . 469
9.2: The Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
9.2.1: The Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 472
9.3: Damped Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
9.3.1: Properties of the Damped Oscillator . . . . . . . . . . . . . . . . . . 477
Example 9.3.1: Car Shock Absorbers . . . . . . . . . . . . . . . . . . . . . 480
9.4: Damped, Driven Oscillation: Resonance . . . . . . . . . . . . . . . . . . . 481
9.4.1: Harmonic Driving Forces . . . . . . . . . . . . . . . . . . . . . . . . 483
9.4.2: Solution to Damped, Driven, Simple Harmonic Oscillator . . . . . . 485
9.5: Elastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . 489
9.5.1: Simple Models for Molecular Bonds . . . . . . . . . . . . . . . . . . 489
9.5.2: The Force Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
9.5.3: A Microscopic Picture of a Solid . . . . . . . . . . . . . . . . . . . . 493
9.5.4: Shear Forces and the Shear Modulus . . . . . . . . . . . . . . . . . 496
9.5.5: Deformation and Fracture . . . . . . . . . . . . . . . . . . . . . . . 497
9.6: Human Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
Example 9.6.1: Scaling of Bones with Animal Size . . . . . . . . . . . . . 502
Homework for Week 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
Week 10: The Wave Equation 517
Wave Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
10.1: Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
10.2: Waves on a String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
10.3: Solutions to the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . 522
10.3.1: An Important Property of Waves: Superposition . . . . . . . . . . 522
10.3.2: Arbitrary Waveforms Propagating to the Left or Right . . . . . . . 522
10.3.3: Harmonic Waveforms Propagating to the Left or Right . . . . . . 523
10.3.4: Stationary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524x CONTENTS
10.4: Reﬂection of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
10.5: Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Homework for Week 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Week 11: Sound 543
Sound Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
11.1: Sound Waves in a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
11.2: Sound Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
11.3: Sound Wave Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
11.3.1: Sound Displacement and Intensity In Terms of Pressure . . . . . . 548
11.3.2: Sound Pressure and Decibels . . . . . . . . . . . . . . . . . . . . . 550
11.4: Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
11.4.1: Moving Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
11.4.2: Moving Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
11.4.3: Moving Source and Moving Receiver . . . . . . . . . . . . . . . . . 555
11.5: Standing Waves in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
11.5.1: Pipe Closed at Both Ends . . . . . . . . . . . . . . . . . . . . . . . 556
11.5.2: Pipe Closed at One End . . . . . . . . . . . . . . . . . . . . . . . . 556
11.5.3: Pipe Open at Both Ends . . . . . . . . . . . . . . . . . . . . . . . 557
11.6: Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
11.7: Interference and Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . 559
Homework for Week 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
Week 12: Gravity 571
Gravity Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
12.1: Cosmological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
12.2: Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
12.2.1: Ellipses and Conic Sections . . . . . . . . . . . . . . . . . . . . . . 583
12.3: Newton’s Law of Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . 585
12.4: The Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
12.4.1: Spheres, Shells, General Mass Distributions . . . . . . . . . . . . . 594
12.5: Gravitational Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . 595
12.6: Energy Diagrams and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 596CONTENTS xi
12.7: Escape Velocity, Escape Energy . . . . . . . . . . . . . . . . . . . . . . . 598
Example 12.7.1: How to Cause an Extinction Event . . . . . . . . . . . . 599
12.8: Bridging the Gap: Coulomb’s Law and Electrostatics . . . . . . . . . . . 600
Homework for Week 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601