Worked Examples from Introductory Physics (Algebra–Based ...
Worked Examples from Introductory Physics (Algebra?Based)
Vol. I: Basic Mechanics
David Murdock, TTU
October 3, 2012
2
Contents
Preface
i
1 Mathematical Concepts
1
1.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Measurement and Units in Physics . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Metric System; Converting Units . . . . . . . . . . . . . . . . . . 2
1.1.3 Math: You Had This In High School. Oh, Yes You Did. . . . . . . . . 3
1.1.4 Math: Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.5 Vectors and Vector Addition . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.6 Components of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Measurement and Units . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Vectors and Vector Addition . . . . . . . . . . . . . . . . . . . . . . . 14
2 Motion in One Dimension
19
2.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Speed and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 Motion With Constant Velocity . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.5 Motion Where the Acceleration is Constant . . . . . . . . . . . . . . 21
2.1.6 Free-Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Motion Where the Acceleration is Constant . . . . . . . . . . . . . . 23
2.2.2 Free-Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Motion in Two Dimensions
33
3.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Motion in Two Dimensions, Coordinates and Displacement . . . . . . 33
3
4
CONTENTS
3.1.2 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.3 Motion When the Acceleration Is Constant . . . . . . . . . . . . . . . 35 3.1.4 Free Fall; Projectile Problems . . . . . . . . . . . . . . . . . . . . . . 36 3.1.5 Ground?To?Ground Projectile: A Long Example . . . . . . . . . . . 36 3.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 Motion for Constant Acceleration . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Free?Fall; Projectile Problems . . . . . . . . . . . . . . . . . . . . . . 41
4 Forces I
49
4.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2 Newton's 1st Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.3 Newton's 2nd Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.4 Units and Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.5 Newton's 3rd Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.6 The Force of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.7 Other Forces Which Appear In Our Problems . . . . . . . . . . . . . 54
4.1.8 The Free?Body Diagram: Draw the Damn Picture! . . . . . . . . . . 56
4.1.9 Simple Example: What Does the Scale Read? . . . . . . . . . . . . . 56
4.1.10 An Important Example: Mass Sliding On a Smooth Inclined Plane . 58
4.1.11 Another Important Example: The Attwood Machine . . . . . . . . . 61
4.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Newton's Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 The Force of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Applying Newton's Laws of Motion . . . . . . . . . . . . . . . . . . . 65
5 Forces II
69
5.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.2 Friction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.3 An Important Example: Block Sliding Down Rough Inclined Plane . 70
5.1.4 Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.5 Circular Motion and Force . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.6 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 Friction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.2 Uniform Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.3 Circular Motion and Force . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.4 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
CONTENTS
5
6 Energy
87
6.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.3 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.1.4 The Work?Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.5 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.6 The Spring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.7 The Principle of Energy Conservation . . . . . . . . . . . . . . . . . . 91
6.1.8 Solving Problems With Energy Conservation . . . . . . . . . . . . . . 92
6.1.9 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.1 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.2 The Spring Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.3 Solving Problems With Energy Conservation . . . . . . . . . . . . . . 94
7 Momentum
99
7.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.1.1 Momentum; Systems of Particles . . . . . . . . . . . . . . . . . . . . 99
7.1.2 Relation to Force; Impulse . . . . . . . . . . . . . . . . . . . . . . . . 99
7.1.3 The Principle of Momentum Conservation . . . . . . . . . . . . . . . 100
7.1.4 Collisions; Problems Using the Conservation of Momentum . . . . . . 102
7.1.5 Systems of Particles; The Center of Mass . . . . . . . . . . . . . . . . 104
7.1.6 Finding the Center of Mass . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Rotational Kinematics
107
8.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.1.1 Rigid Bodies; Rotating Objects . . . . . . . . . . . . . . . . . . . . . 107
8.1.2 Angular Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.1.3 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.1.4 Angular Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.1.5 The Case of Constant Angular Acceleration . . . . . . . . . . . . . . 111
8.1.6 Relation Between Angular and Linear Quantities . . . . . . . . . . . 112
8.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.2.1 Angular Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.2.2 Angular Velocity and Acceleration . . . . . . . . . . . . . . . . . . . 113
8.2.3 Rotational Motion with Constant Angular Acceleration . . . . . . . . 114
8.2.4 Relation Between Angular and Linear Quantities . . . . . . . . . . . 114
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