Summary Physical science study guide

Worked Examples from Introductory Physics
Vol. II: Rotation, Vibrations and Waves

David Murdock
Tenn. Tech. Univ.

December 27, 2008

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,Contents

To the Student. Yeah, You. i

1 Rotation of an Object About a Fixed Axis 1
1.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Rigid Bodies; Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Angular Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Angular Acceleration; Constant Angular Acceleration . . . . . . . . . 3
1.1.5 Relationship Between Angular and Linear Quantities . . . . . . . . . 3
1.1.6 Rotational Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.7 The Moment of Inertia; The Parallel Axis Theorem . . . . . . . . . . 5
1.1.8 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.9 Torque and Angular Acceleration (Newton’s Second Law for Rotations) 8
1.1.10 Work, Energy and Power in Rotational Motion . . . . . . . . . . . . 9
1.1.11 The New Equations Look Like the Old Equations . . . . . . . . . . . 9
1.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Angular Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.3 Angular Acceleration; Constant Angular Acceleration . . . . . . . . . 11
1.2.4 Relationship Between Angular and Linear Quantities . . . . . . . . . 15
1.2.5 Rotational Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.6 The Moment of Inertia (and More Rotational Kinetic Energy) . . . . 17
1.2.7 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.8 Torque and Angular Acceleration (Newton’s Second Law for Rotations) 21
1.2.9 Work, Energy and Power in Rotational Motion . . . . . . . . . . . . 27

2 Rolling Motion; Angular Momentum 35
2.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Rolling Without Slipping . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.2 Torque as a Vector (A Cross Product) . . . . . . . . . . . . . . . . . 36
2.1.3 Angular Momentum of a Particle and of Systems of Particles . . . . . 36
2.1.4 Angular Momentum for Rotation About a Fixed Axis . . . . . . . . . 37
2.1.5 The Conservation of Angular Momentum . . . . . . . . . . . . . . . . 37
2.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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2.2.1 Rolling Without Slipping . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 Torque as a Vector (A Cross Product) . . . . . . . . . . . . . . . . . 43
2.2.3 Angular Momentum of a Particle and of Systems of Particles . . . . . 44
2.2.4 Angular Momentum for Rotation About a Fixed Axis . . . . . . . . . 45
2.2.5 The Conservation of Angular Momentum . . . . . . . . . . . . . . . . 46

3 Static Equilibrium 55
3.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Conditions for Equilibrium of a Rigid Object . . . . . . . . . . . . . . 55
3.1.2 Two Important Facts for Working Statics Problems . . . . . . . . . . 56
3.1.3 Examples of Rigid Objects in Static Equilibrium . . . . . . . . . . . . 56
3.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Examples of Rigid Objects in Static Equilibrium . . . . . . . . . . . . 56

4 Oscillatory Motion 69
4.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 Mass Attached to a Spring . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.3 Energy and the Simple Harmonic Oscillator . . . . . . . . . . . . . . 72
4.1.4 Relation to Uniform Circular Motion . . . . . . . . . . . . . . . . . . 73
4.1.5 The Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Mass Attached to a Spring . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 Energy and the Simple Harmonic Oscillator . . . . . . . . . . . . . . 81
4.2.4 The Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.5 Physical Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Waves I: Generalities, Superposition & Standing Waves 87
5.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.1 Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.2 Types of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.3 Mathematical Description of a Wave; Wavelength, Frequency and Wave
Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.4 Waves on a Stretched String . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.5 The Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . 89
5.1.6 Interference of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.7 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.8 Standing Waves on Strings Under Tension . . . . . . . . . . . . . . . 91
5.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1 Wavelength, Frequency and Speed . . . . . . . . . . . . . . . . . . . . 91
5.2.2 Waves on a Stretched String . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.3 Superposition; Interference of Waves . . . . . . . . . . . . . . . . . . 93
5.2.4 Standing Waves on a Stretched String . . . . . . . . . . . . . . . . . 94