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General Physics I:

Classical Mechanics

David G. Simpson

Dept. of Natural Sciences,Prince George"sCommunity College, Largo,Maryland

Larry L. Simpson

Union Carbide Corporation (ret.), South Charleston, West Virginia

Fall 2020

Last updated: October 8, 2020

Acknowledgments11

1 What is Physics?12

2 Units14

2.1 Systems of Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 CGS Systems of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 BritishEngineering Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Units as an Error-Checking Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Unit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Currency Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Odds and Ends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Problem-Solving Strategies 24

4 Density26

4.1 Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Density Trivia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Kinematics in One Dimension 29

5.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.4 Higher Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.5 Dot Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.6 Inverse Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.7 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.9 Geometric Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Vectors37

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Vector Arithmetic: Graphical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3 Vector Arithmetic: Algebraic Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.4 The Zero Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.5 Derivatives.......................................... 43

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6.6 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.7 Other Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7 The Dot Product45

7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2 Component Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.4 Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8 Kinematics in Two or Three Dimensions 49

8.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.2 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.4 Inverse Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.5 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.6 Vertical vs. Horizontal Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

9 Projectile Motion54

9.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9.2 Maximum Altitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9.3 Shape of the Projectile Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9.4 Hittinga Target on the Ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9.5 Hittinga Target on a Hill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

9.6 ExplodingProjectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.7 Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.8 The Monkey and the Hunter Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

10 Newton"s Method63

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

10.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

10.3 Example: Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

10.4 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

11 Mass66

12 Force67

12.1 The Four Forces of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.2 Hooke's Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.3 Weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.4 Normal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.5 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

13 Newton"s Laws of Motion 70

13.1 First Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

13.2 Second Law of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

13.3 Third Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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14 The Inclined Plane73

15 Atwood"s Machine75

16 Statics79

16.1 Mass Suspended by Two Ropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

16.2 The Elevator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

16.3 The Catenary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

17 Friction84

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17.2 Static Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

17.3 Kinetic Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17.4 RollingFriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

17.5 The Coefficient of Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

18 Blocks and Pulleys87

18.1 Horizontal Block and Vertical Block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

18.2 Inclined Block and Vertical Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

19 Resistive Forces in Fluids 91

19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

19.2 Model I:F

R /v....................................... 91

19.3 Model II:F

R /v 2 ...................................... 93

20 Circular Motion96

20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

20.2 Centripetal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

20.3 CentrifugalForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

20.4 Relations between Circular and Linear Motion. . . . . . . . . . . . . . . . . . . . . . . . 99

20.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

21 Work100

21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

21.2 Case I: ConstantFkr....................................100

21.3 Case II: ConstantF¬r...................................101

21.4 Case III: VariableFkr...................................101

21.5 Case IV (General Case): VariableF¬r...........................102

21.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

22 Simple Machines103

22.1 Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

22.2 Wheel and Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

22.3 Pulley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

22.4 Lever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

22.5 Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

22.6 Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

22.7 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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23 Energy110

23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

23.2 Kinetic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

23.3 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

23.4 Other Forms of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

23.5 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

23.6 The Work-Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

23.7 The VirialTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

24 Conservative Forces 117

25 Power118

25.1 Energy Conversion of a Falling Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

25.2 Rate of Change of Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

25.3 Vector Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

26 Linear Momentum121

26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

26.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

26.3 Newton's Second Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

27 Impulse123

28 Collisions125

28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

28.2 The Coefficient of Restitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

28.3 Perfectly Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

28.4 Perfectly Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

28.5 Newton's Cradle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

28.6 Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

28.7 Collisionsin Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

29 The BallisticPendulum 131

30 Rockets133

30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

30.2 The Rocket Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

30.3 Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

30.4 Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

31 Center of Mass136

31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

31.2 Discrete Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

31.3 ContinuousBodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

32 The Cross Product140

32.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

32.2 Component Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

32.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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32.4 Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

32.5 Inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

33 RotationalMotion145

33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

33.2 Translational vs. Rotational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

33.3 Example Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

34 Moment of Inertia149

34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

34.2 Radius of Gyration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

34.3 Parallel Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

34.4 Plane Figure Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

34.5 Routh'sRule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

34.6 Lees' Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

35 Torque157

35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

35.2 RotationalVersion of Hooke's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

35.3 Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

36 Measuring the Moment of Inertia 159

36.1 Torque Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

36.2 Pendulum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

37 Newton"s Laws of Motion: RotationalVersions 162

37.1 First Law of RotationalMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

37.2 Second Law of RotationalMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

37.3 Third Law of RotationalMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

38 The Pendulum164

38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

38.2 The Simple Plane Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

38.3 The Spherical Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

38.4 The Conical Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

38.5 The Torsional Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

38.6 The Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

38.7 Other Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

39 Simple Harmonic Motion 170

39.1 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

39.2 Frequency and Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

39.3 The Vertical Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

39.4 Frequency and Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

39.5 Mass on a Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

39.6 More on the Spring Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

40 Rocking Bodies178

40.1 The Half-Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

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41 Rolling Bodies181

41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

41.2 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

41.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

41.4 Kinetic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

41.5 The Wheel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

41.6 Ball Rollingin a Bowl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

42 Galileo"s Law187

42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

42.2 Modern Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

43 The Coriolis Force189

43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

43.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

44 Angular Momentum 191

44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

44.2 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

45 Conservation Laws193

46 The Gyroscope194

46.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

46.2 Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

46.3 Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

47 Elasticity196

47.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

47.2 Longitudinal(Normal) Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

47.3 Transverse (Shear) Stress - Translational . . . . . . . . . . . . . . . . . . . . . . . . . . 197

47.4 Transverse (Shear) Stress - Torsional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

47.5 Volume Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

47.6 Elastic Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

47.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

48 Fluid Statics200

48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

48.2 Archimedes' Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

48.3 Floating Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

48.4 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

48.5 Change in Fluid Pressure with Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

48.6 Pascal's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

49 Fluid Dynamics204

49.1 The ContinuityEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

49.2 Bernoulli'sEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

49.3 Torricelli'sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

49.4 The Siphon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6 Prince George's Community College General PhysicsI Simpson & Simpson

49.5 Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

49.6 The Reynolds Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

49.7 StokesÕs Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

49.8 Fluid Flow througha Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

49.9 Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

49.10 Superßuids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

50 Hydraulics and Pneumatics 217

50.1 Hydraulics: The Hydraulic Press. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

50.2 Pneumatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

51 Gravity220

51.1 Newton's Law of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

51.2 GravitationalPotential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

51.3 The Cavendish Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

51.4 Helmert's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

51.5 Earth Density Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

51.6 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

51.7 Gauss's Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

51.8 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

51.9 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

52 Earth Rotation230

52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

52.2 Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

52.3 Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

52.4 Polar Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

52.5 RotationRate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

53 Geodesy234

53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

53.2 Radius of the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

53.3 The Cosine Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

53.4 Vincenty's Formulae: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

53.5 Vincenty's Formulae: Direct Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

53.6 Vincenty's Formulae: Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

54 Celestial Mechanics 241

54.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

54.2 Kepler's Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

54.3 Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

54.4 Orbit Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

54.5 Orbital Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

54.6 RightAscension and Declination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

54.7 Computinga Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

54.8 The Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

54.9 Corrections to the Two-BodyCalculation . . . . . . . . . . . . . . . . . . . . . . . . . . 246

54.10 Bound and Unbound Orbits.................................247

54.11 TheVis VivaEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

7 Prince George's Community College General PhysicsI Simpson & Simpson

54.12 BertrandÕs Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

54.13 Differential Equation for an Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

54.14 Lagrange Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

54.15 The Rings of Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

54.16 Hyperbolic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

54.17 Parabolic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

55 Astrodynamics254

55.1 Circular Orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

55.2 Geosynchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

55.3 EllipticalOrbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

55.4 The Hohmann Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

55.5 Gravity Assist Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

55.6 The InternationalCometary Explorer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

56 Partial Derivatives265

56.1 First Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

56.2 Higher-Order Partial Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

57 Lagrangian Mechanics 267

57.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

58 HamiltonianMechanics 270

58.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

59 Special Relativity273

59.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

59.2 Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

59.3 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

59.4 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

59.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

59.6 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

59.7 Additionof Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

59.8 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

60 Quantum Mechanics 277

60.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

60.2 Review of Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

60.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

60.4 Example: Simple Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

60.5 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

61 The Standard Model 281

61.1 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

61.2 Antimatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

61.3 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

61.4 The Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Appendices287

A Greek Alphabet289

B Trigonometry290

C Hyperbolic Trigonometry 296

D Useful Series298

E Table of Derivatives 299

F Table of Integrals301

G Mathematical Subtleties 303

H SI Units305

I Gaussian Units308

J British Engineering Units 310

K Units of Physical Quantities 312

L Physical Constants315

M Astronomical Data316

N Unit Conversion Tables 317

O Angular Measure321

P Vector Arithmetic323

Q Matrix Properties326

R Newton"s Laws of Motion (Original) 328

S The Simple Plane Pendulum: Exact Solution 329

T Motion of a Falling Body 334

U Table of Viscosities336

V Calculator Programs 338

W Round-Number Handbook of Physics 339

X Short Glossary of Particle Physics 341

9 Prince George's Community College General PhysicsI Simpson & Simpson Y Fundamental Physical Constants Extensive Listing 342

Z Periodic Table of the Elements 349

References349

Index353

Acknowledgments

The author (DGS) wishes to express his deepest thanks to his father, L.L. Simpson, for valuable comments

and contributions to material throughout the text, especially in the areas of resistive forces in fluids, fluid

dynamics, geodesy. 11

Chapter 1 What is Physics?

Physicsis the most fundamental of the sciences. Its goal is to learn how the Universe works at the most

fundamental level - and to discover the basic laws by which it operates.Theoretical physicsconcentrates

on developing the theory and mathematics of these laws, whileapplied physicsfocuses attention on the

application of the principles of physics to practical problems.Experimental physicslies at the intersection

of physics and engineering; experimental physicists have the theoretical knowledge of theoretical physicists,

andthey know how to build and work with scientific equipment.

Physics is divided into a number of sub-fields, and physicists are trained to have some expertise in all of

them. This variety is what makes physics one of the most interesting of the sciences - and it makes people

with physics trainingvery versatile in their ability to do work in many different technical fields.

The major fields of physics are:

•Classical mechanicsis the study the motion of bodies according to Newton's laws of motion, and is

the subject of this course.

•Electricity andmagnetismare twoclosely related phenomena that are together considereda singlefield

of physics.

•Quantum mechanicsdescribes the peculiar motion of very small bodies (atomic sizes and smaller).

•Opticsis the study of light. •Acousticsis the study of sound. •Thermodynamicsandstatisticalmechanicsare closely related fields that study the nature of heat. •Solid-statephysicsis the study of solids - mostoften crystalline metals. •Plasma physicsis the study of plasmas (ionized gases).

•Atomic, nuclear, and particlephysicsstudyof the atom, the atomic nucleus, and the particles that make

up the atom.

•Relativityincludes Albert Einstein's theories of special and general relativity.Special relativityde-

scribes the motion of bodies moving at very high speeds (near the speed of light), whilegeneral rela-

tivityis Einstein's theory of gravity.

The fields ofcross-disciplinary physicscombine physics with other sciences. These includeastrophysics

(physics of astronomy),geophysics(physics of geology),biophysics(physics of biology),chemical physics

(physics of chemistry), andmathematicalphysics(mathematical theories related to physics). 12 Prince George's Community College General PhysicsI Simpson & Simpson Besides acquiringa knowledgeofphysicsforitsownsake, thestudyofphysicswillgiveyoua broadtech-

nical background and set of problem-solving skills that you can apply to wide variety of other Þelds. Some

students of physics go on to study more advanced physics, while others Þnd ways to apply their knowledge

of physics to such diverse subjects as mathematics, engineering, biology, medicine, and Þnance.

Another beneÞt of learning physics is that, unlike courses in technology, everything you learn in this

course will never be obsolete. Although theories at the cutting edge of physics research may change, the

basic physics youÕll learn in these courses will not. You will be able to use what you learn in this course

throughoutyour life.

Deductive Logic

Solvingphysicsproblemsmakes extensive use ofdeductive logic. One beginswitha setofknownfacts (given intheproblem)anda setofrelevantequationsanddefinitions(whichyouselect, based ontheproblem). Using logic and mathematics, you then deduce the conclusion (the solutionto the problem). As a simple example, suppose youare given thata bodytravels 700meters in10 seconds, andare asked to

find itsaverage speed. You must search yourknowledge ofphysics to decide what additionalfacts are needed

to solve this problem. In this case, you decide to use the definition of "average speed": the total distance

divided by the total time. Puttingthe given information together with this definition, you find the solution to

be 700 meters divided by 10 seconds, or 70 meter per second.

If you enjoy solving logic problems, cryptograms, and similar puzzles, then you'llenjoy solving physics

problems. Solving physics problems is the primary skill you'll be developing in this course. Professional

physicists solve similar types of problems - often more complex problems. They also do experiments to try

to deduce the correct laws of Nature. In this course we'll present some of the laws of Nature that have been

deduced so far, along with some of the importantresults and consequences of those laws. 13

Chapter 2

Units

The phenomena of Nature have been foundto obey certain physicallaws; one of the primary goals of physics

research is to discover those laws. It has been known for several centuries that the laws of physics are

appropriately expressed in the language ofmathematics, so physics and mathematics have enjoyed a close

connection for quite a long time. In order to connect the physical world to the mathematical world, we need to makemeasurementsof the

real world. In making a measurement, we compare a physical quantity with some agreed-upon standard, and

determine how many such standard units are present. For example, we have a precise definition of a unit of

length called amile, and have determined that there are about 92,000,000 such miles between the Earth and

the Sun.

Itis importantthatwe have very precise definitionsofphysicalunits - notonlyforscientific use, butalso

for trade and commerce. In practice, we define a fewbase units, and derive other units from combinationsof

those base units. For example, if we define units for length and time, then we can define a unit for speed as

the length divided by time (e.g. miles/hour). How many base units do we need to define? There is no magic number; in fact it is possible to define

a system of units using onlyonebase unit (and this is in fact done for so-callednatural units). For most

quotesdbs_dbs21.pdfusesText_27

[PDF] [PDF] General Physics I - Prince Georges Community College