THÔNG TIN TÀI LIỆU
General Physics I:
Classical Mechanics
D.G. Simpson, Ph.D.
Department of Physical Sciences and Engineering
Prince George’s C ommunity College
Largo, Maryland
Fall 2013
Last updated: December 16, 2013
Contents
Acknowledgments 11
1WhatisPhysics? 12
2Units 14
2.1 SystemsofUnits 14
2.2 SIUnits 15
2.3 CGSSystemsofUnits 18
2.4 British Engineering Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 UnitsasanError-CheckingTechnique 18
2.6 UnitConversions 19
2.7 CurrencyUnits 20
2.8 OddsandEnds 21
3 Problem-Solving Strategies 22
4Density 24
4.1 SpecificGravity 25
4.2 DensityTrivia 25
5 Kinematics in One Dimension 27
5.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Velocity 27
5.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.4 HigherDerivatives 29
5.5 DotNotation 29
5.6 InverseRelations 29
5.7 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.8 Summary 32
5.9 GeometricInterpretations 33
6 Vectors 35
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.2 Arithmet i c: Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3 Arithmetic: Algebraic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.4 Derivatives 40
6.5 Integrals 40
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6.6 OtherVectorOperations 40
7 The Dot Product 42
7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2 Component Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3 Properties 43
7.4 MatrixFormulation 44
8 Kinematics in Two or Three Dimensions 46
8.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2 Velocity 46
8.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.4 InverseRelations 47
8.5 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.6 Verticalvs.HorizontalMotion 48
8.7 Summary 49
9 Projectile Motion 51
9.1 Range 52
9.2 Maximum Altitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.3 Shape of the Projectile Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.4 Hitting a Target on the Ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.5 Hitting a Target on a Hill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.6 OtherConsiderations 57
9.7 The Monkey and the Hunter Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.8 Summary 59
10 Newton’s Method 60
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
10.2 TheMethod 60
10.3 Example:SquareRoots 60
10.4 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
11 Mass 63
12 Force 64
12.1 TheFourForcesofNature 64
12.2 Hooke’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.3 Weight 65
12.4 NormalForce 65
12.5 Tension 65
13 Newton’s Laws of Motion 66
13.1 FirstLawofMotion 67
13.2 SecondLawofMotion 67
13.3 ThirdLawofMotion 67
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14 The Inclined Plane 68
15 Atwood’s Machine 69
16 Statics 73
16.1 MassSuspendedbyTwoRopes 73
16.2 ThePulley 76
16.3 TheElevator 76
17 Friction 78
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
17.2 StaticFriction 78
17.3 KineticFriction 79
17.4 Rolling Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
17.5 The CoefficientofFriction 79
18 Resistive Forces in Fluids 81
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
18.2 Model I: F
R
/ v 81
18.3 Model II: F
R
/ v
2
83
19 Circular Motion 86
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
19.2 CentripetalForce 87
19.3 CentrifugalForce 88
19.4 RelationsbetweenCircularandLinearMotion 89
19.5 Examples 89
20 Work 90
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
20.2 Case I: Constant F k r 90
20.3 Case II: Constant F ¬ r 91
20.4 Case III: Variable F k r 91
20.5 Case IV (General Case): Variable F ¬ r 91
20.6 Summary 92
21 Energy 93
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
21.2 KineticEnergy 93
21.3 PotentialEnergy 94
21.4 OtherFormsofEnergy 97
21.5 ConservationofEnergy 97
21.6 TheWork-EnergyTheorem 98
21.7 TheVirialTheorem 98
22 Conservative Forces 100
23 Power 101
23.1 Energy Conversion of a Falling Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
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23.2 RateofChangeofPower 102
23.3 VectorEquation 103
24 Linear Momentum 104
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
24.2 ConservationofMomentum 104
24.3 Newton’sSecondLawofMotion 104
25 Impulse 106
26 Collisions 108
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
26.2 The Coefficient of Restitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
26.3 Perfectly Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
26.4 Perfectly Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
26.5 Newton’sCradle 111
26.6 Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
26.7 Collisions in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
27 The Ballistic Pendulum 114
28 Rockets 116
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
28.2 TheRocketEquation 116
28.3 MassFraction 117
28.4 Staging 118
29 Center of Mass 119
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
29.2 DiscreteMasses 119
29.3 Continuous Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
30 The Cross Product 123
30.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
30.2 Component Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
30.3 Properties 124
30.4 MatrixFormulation 126
30.5 Inverse 126
31 Rotational Moti on 128
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
31.2 Translationalvs.RotationalMotion 128
31.3 ExampleProblems 130
32 Moment of Inertia 132
32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
32.2 ParallelAxisTheorem 136
32.3 PlaneFigureTheorem 138
32.4 Routh’sRule 138
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32.5 Lees’Rule 138
33 Torque 140
33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
33.2 RotationalVersionsofNewton’sLaws 141
33.3 Rotational Version of Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
34 The Pendulum 142
34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
34.2 The Simple Plane Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
34.3 The Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
34.4 The Conical Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
34.5 The Torsional Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
34.6 The Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
34.7 Other Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
35 Simple Harm onic Motion 148
35.1 Energy 150
35.2 FrequencyandPeriod 152
35.3 MassonaSpring 152
36 Rolling Bodies 154
36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
36.2 Velocity 154
36.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
36.4 KineticEnergy 156
36.5 TheWheel 157
36.6 Ball Rolling in a Bowl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
37 Galileo’s Law 160
37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
37.2 ModernTreatment 160
38 The Coriolis Force 162
38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
38.2 Examples 163
39 Angular Momentum 164
39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
39.2 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
40 Conservation Laws 166
41 The Gyroscope 167
41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
41.2 Precession 167
41.3 Nutation 168
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42 Roller Coasters 169
42.1 GeneralPrinciples 169
42.2 LoopShapes 169
43 Elasticity 170
43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
43.2 Longitudinal(Normal)Stress 170
43.3 Transverse(Shear)Stress—Translational 171
43.4 Transverse(Shear)Stress—Torsional 172
43.5 VolumeStress 172
43.6 ElasticLimit 173
43.7 Summary 173
44 Fluid Mechanics 174
44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
44.2 Archimedes’Principle 174
44.3 FloatingBodies 174
44.4 Pressure 175
44.5 ChangeinFluidPressurewithDepth 176
44.6 Pascal’sLaw 177
44.7 FluidDynamics 177
44.8 TheContinuityEquation 178
44.9 Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
44.10 Torricelli’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
44.11 The Siphon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
44.12 Viscosity 182
44.13 The Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
44.14 Stokes’sLaw 184
44.15 Fluid Flow through a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
44.16 Superfluids 185
45 Hydraulics 188
45.1 TheHydraulicPress 188
46 Pneumatics 190
47 Gravity 191
47.1 Newton’sLawofGravity 191
47.2 GravitationalPotential 191
47.3 TheCavendishExperiment 192
47.4 Helmert’sEquation 192
47.5 EscapeVelocity 193
47.6 Gauss’sFormulation 193
47.7 GeneralRelativity 197
47.8 BlackHoles 198
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48 Earth Rotation 199
48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
48.2 Precession 199
48.3 Nutation 199
48.4 PolarMotion 201
48.5 RotationRate 201
49 Geodesy 203
49.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
49.2 TheCosineFormula 203
49.3 Vincenty’s Formulæ: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
49.4 Vincenty’sFormulæ:DirectProblem 204
49.5 Vincenty’sFormulæ:InverseProblem 206
50 Celestial Mechanics 209
50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
50.2 Kepler’sLaws 209
50.3 Time 210
50.4 OrbitReferenceFrames 210
50.5 OrbitalElements 211
50.6 RightAscensionandDeclination 212
50.7 Computing a Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
50.8 TheInverseProblem 214
50.9 CorrectionstotheTwo-BodyCalculation 214
50.10 Bound and Unbound Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
50.11 The Vis Viva Equation 215
50.12 Bertrand’sTheorem 216
50.13 DifferentialEquationforanOrbit 216
50.14 LagrangePoints 217
50.15 TheRingsofSaturn 218
50.16 HyperbolicOrbits 220
50.17 ParabolicOrbits 221
51 Astrodynamics 222
51.1 CircularOrbits 222
51.2 Geosynchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
51.3 Elliptical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
51.4 TheHohmannTransfer 228
51.5 GravityAssistManeuvers 229
51.6 TheInternationalCometaryExplorer 231
52 Partial Derivatives 233
52.1 FirstPartialDerivatives 233
52.2 Higher-OrderPartialDerivatives 234
53 Lagrangian Mechanics 235
53.1 Examples 236
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54 Hamiltonian Mechanics 238
54.1 Examples 238
55 Special Relativity 241
55.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
55.2 Postulates 241
55.3 TimeDilation 241
55.4 LengthContraction 242
55.5 AnExample 242
55.6 Momentum 242
55.7 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
55.8 Energy 243
56 Quantum Mechanics 245
56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
56.2 ReviewofNewtonianMechanics 245
56.3 QuantumMechanics 245
56.4 Example: Simple Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
56.5 TheHeisenbergUncertaintyPrinciple 248
AFurtherReading 250
B Greek Alphabet 254
C Trigonometry 255
D Useful Series 258
ESIUnits 259
F Gaussian Units 262
G Units of Physical Quantities 264
H Physical Constants 267
IAstronomicalData 268
J Unit Conversion Tables 269
K Angular Measure 272
K.1 PlaneAngle 272
K.2 SolidAngle 272
L Vector Arithmetic 274
M Matrix Properties 277
N Newton’s Laws of Motion (Original) 279
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O The Simple Plane Pendulum: Exact Solution 280
O.1 EquationofMotion 280
O.2 Solution, Â.t/ 281
O.3 Period 281
P Motion of a Falling Body 285
Q Tabl e of Viscosities 287
R TI-83+ / TI-84+ Calculator Programs 289
R.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
R.2 Kepler’sEquation 290
R.3 HyperbolicKepler’sEquation 290
R.4 Barker’sEquation 291
R.5 ReductionofanAngle 291
R.6 Helmert’sEquation 292
R.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
R.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
S TI-89 / TI-92 / Voyage 200 Calculator Programs 295
S.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
S.2 Kepler’sEquation 296
S.3 HyperbolicKepler’sEquation 296
S.4 Barker’sEquation 297
S.5 ReductionofanAngle 298
S.6 Helmert’sEquation 298
S.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
S.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
T HP 35s / HP 15C Calculator Programs 301
T.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
T.2 Kepler’sEquation 303
T.3 HyperbolicKepler’sEquation 304
T.4 Barker’sEquation 306
T.5 ReductionofanAngle 307
T.6 Helmert’sEquation 308
T.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
T.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
U HP 48 / HP 50g Calculator Programs 313
U.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
U.2 Kepler’sEquation 314
U.3 HyperbolicKepler’sEquation 314
U.4 Barker’sEquation 315
U.5 ReductionofanAngle 315
U.6 Helmert’sEquation 316
U.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
U.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
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