Classical Mechanics

[Pages:297]Classical Mechanics

An introductory course

Richard Fitzpatrick Associate Professor of Physics The University of Texas at Austin

Contents

1 Introduction

7

1.1 Major sources: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 What is classical mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 mks units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Standard prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Other units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Precision and significant figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Motion in 1 dimension

18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Motion with constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Free-fall under gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Motion in 3 dimensions

32

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Vector displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Vector magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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3.6 Scalar multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.7 Diagonals of a parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.8 Vector velocity and vector acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.9 Motion with constant velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.10 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.11 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.12 Relative velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Newton's laws of motion

53

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Newton's first law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Newton's second law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Hooke's law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Newton's third law of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Mass and weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Strings, pulleys, and inclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.8 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.9 Frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Conservation of energy

78

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Energy conservation during free-fall . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 Conservative and non-conservative force-fields . . . . . . . . . . . . . . . . . . . . . 88

5.5 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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5.6 Hooke's law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.7 Motion in a general 1-dimensional potential . . . . . . . . . . . . . . . . . . . . . . 96 5.8 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Conservation of momentum

107

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2 Two-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Multi-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4 Rocket science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.6 Collisions in 1-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.7 Collisions in 2-dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7 Circular motion

136

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3 Centripetal acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4 The conical pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.5 Non-uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.6 The vertical pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.7 Motion on curved surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8 Rotational motion

160

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.2 Rigid body rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.3 Is rotation a vector? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

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8.4 The vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.5 Centre of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.6 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.7 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.8 Power and work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.9 Translational motion versus rotational motion . . . . . . . . . . . . . . . . . . . . . 186 8.10 The physics of baseball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.11 Combined translational and rotational motion . . . . . . . . . . . . . . . . . . . . . 190

9 Angular momentum

204

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.2 Angular momentum of a point particle . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.3 Angular momentum of an extended object . . . . . . . . . . . . . . . . . . . . . . . 206

9.4 Angular momentum of a multi-component system . . . . . . . . . . . . . . . . . . . 209

10 Statics

217

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.2 The principles of statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10.3 Equilibrium of a laminar object in a gravitational field . . . . . . . . . . . . . . . . 220

10.4 Rods and cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.5 Ladders and walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

10.6 Jointed rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

11 Oscillatory motion

237

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

11.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

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11.3 The torsion pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 11.4 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.5 The compound pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 11.6 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

12 Orbital motion

253

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

12.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

12.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

12.4 Gravitational potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

12.5 Satellite orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

12.6 Planetary orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

13 Wave motion

279

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

13.2 Waves on a stretched string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

13.3 General waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

13.4 Wave-pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

13.5 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

13.6 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

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1 INTRODUCTION

1 Introduction

1.1 Major sources:

The sources which I consulted most frequently whilst developing this course are:

Analytical Mechanics: G.R. Fowles, Third edition (Holt, Rinehart, & Winston, New York NY, 1977).

Physics: R. Resnick, D. Halliday, and K.S. Krane, Fourth edition, Vol. 1 (John Wiley & Sons, New York NY, 1992).

Encyclop?dia Brittanica: Fifteenth edition (Encyclop?dia Brittanica, Chicago IL, 1994).

Physics for scientists and engineers: R.A. Serway, and R.J. Beichner, Fifth edition, Vol. 1 (Saunders College Publishing, Orlando FL, 2000).

1.2 What is classical mechanics?

Classical mechanics is the study of the motion of bodies (including the special case in which bodies remain at rest) in accordance with the general principles first enunciated by Sir Isaac Newton in his Philosophiae Naturalis Principia Mathematica (1687), commonly known as the Principia. Classical mechanics was the first branch of Physics to be discovered, and is the foundation upon which all other branches of Physics are built. Moreover, classical mechanics has many important applications in other areas of science, such as Astronomy (e.g., celestial mechanics), Chemistry (e.g., the dynamics of molecular collisions), Geology (e.g., the propagation of seismic waves, generated by earthquakes, through the Earth's crust), and Engineering (e.g., the equilibrium and stability of structures). Classical mechanics is also of great significance outside the realm of science. After all, the sequence of events leading to the discovery of classical mechanics--starting with the ground-breaking work of Copernicus, continuing with the researches of Galileo, Kepler, and Descartes, and culminating in the monumental achievements

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1 INTRODUCTION

1.2 What is classical mechanics?

of Newton--involved the complete overthrow of the Aristotelian picture of the Universe, which had previously prevailed for more than a millennium, and its replacement by a recognizably modern picture in which humankind no longer played a privileged role.

In our investigation of classical mechanics we shall study many different types of motion, including:

Translational motion--motion by which a body shifts from one point in space to another (e.g., the motion of a bullet fired from a gun).

Rotational motion--motion by which an extended body changes orientation, with respect to other bodies in space, without changing position (e.g., the motion of a spinning top).

Oscillatory motion--motion which continually repeats in time with a fixed period (e.g., the motion of a pendulum in a grandfather clock).

Circular motion--motion by which a body executes a circular orbit about another fixed body [e.g., the (approximate) motion of the Earth about the Sun].

Of course, these different types of motion can be combined: for instance, the motion of a properly bowled bowling ball consists of a combination of translational and rotational motion, whereas wave propagation is a combination of translational and oscillatory motion. Furthermore, the above mentioned types of motion are not entirely distinct: e.g., circular motion contains elements of both rotational and oscillatory motion. We shall also study statics: i.e., the subdivision of mechanics which is concerned with the forces that act on bodies at rest and in equilibrium. Statics is obviously of great importance in civil engineering: for instance, the principles of statics were used to design the building in which this lecture is taking place, so as to ensure that it does not collapse.

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