By Linda Huetinck, PhD, and Scott Adams

Physics

By Linda Huetinck, PhD, and Scott Adams

An International Data Group Company New York, NY • Cleveland, OH • Indianapolis, IN

About the Author Publisher’s Acknowledgments

Linda Huetinck, an award-winning educator, has

Editorial

taught physics for 23 years. A former editor of the Project Editor: Tracy Barr Journal of the California Science Teachers’ organ-

Acquisitions Editor: Sherry Gomoll ization, Ms. Huetinck is currently a professor of

Technical Editor: David A. Herzog computer, mathematics, and science education in

Editorial Assistant: Michelle Hacker the Department of Secondary Education at Cali-

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fornia State University. Indexer: TECHBOOKS Production Services Proofreader: Joel Showalter

Scott V. Adams is earning his PhD in physics at Hungry Minds Indianapolis Production Services Vanderbilt University. His main interest is in

molecular biophysics, especially electrophysiology.

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Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Visit Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Chapter 1: Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Kinematics in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Definition of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Displacement and velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Average acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Graphical interpretations of displacement, velocity,

and acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Definitions of instantaneous velocity and instantaneous acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .9 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . .9 Kinematics in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 Addition and subtraction of vectors: geometric method . . . . . . . .11 Addition and subtraction of vectors: Component method . . . . . . .12 Multiplication of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Velocity and acceleration vectors in two dimensions . . . . . . . . . . .15 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 Uniform circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Mass and weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 Force diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 Centripetal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 Universal gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 Momentum and impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 The conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Elastic and inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 Center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Rotational Motion of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . .37 Angular velocity and angular acceleration . . . . . . . . . . . . . . . . . . .37 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 Rotational Motion of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . .37 Angular velocity and angular acceleration . . . . . . . . . . . . . . . . . . .37 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

Elasticity and Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . .42 Elastic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 The relation of SHM to circular motion . . . . . . . . . . . . . . . . . . . .44 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 SHM energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Density and pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Pascal’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Archimedes’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 Bernoulli’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Chapter 2: Waves and Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 Wave Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49

Transverse and longitudinal waves . . . . . . . . . . . . . . . . . . . . . . . . .50 Wave characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 Superposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 Standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52

Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53 Intensity and pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 Forced vibrations and resonance . . . . . . . . . . . . . . . . . . . . . . . . . .55 Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

Chapter 3: Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

Thermometers and temperature scales . . . . . . . . . . . . . . . . . . . . . .58 Thermal expansion of solids and liquids . . . . . . . . . . . . . . . . . . . .60

Development of the Ideal Gas Law . . . . . . . . . . . . . . . . . . . . . . . . . .61 Boyle’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 Charles/Gay-Lussac law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 Definition of a mole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 The ideal gas law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62 Avogadro’s number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 The kinetic theory of gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 Heat capacity and specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . .64 Mechanical equivalent of heat . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 Latent heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 The heat of fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 The heat of vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

Table of Contents

The Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 The first law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . .69 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 Definitions of thermodynamical processes . . . . . . . . . . . . . . . . . . .70 Carnot cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 The second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . .73 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

Chapter 4: Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . .76 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77

Electric charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 Electric fields and lines of force . . . . . . . . . . . . . . . . . . . . . . . . . . .80 Electric flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 Gauss’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83 Potential difference and equipotential surfaces . . . . . . . . . . . . . . .84 Electrostatic potential and equipotential surfaces . . . . . . . . . . . . . .86

Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 Parallel plate capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 Parallel and series capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90

Current and Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 Resistance and resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 Electrical power and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94

Direct Current Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 Series and parallel resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 Kirchhoff’s rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

Electromagnetic Forces and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . .97 Magnetic fields and lines of force . . . . . . . . . . . . . . . . . . . . . . . . .98 Force on a moving charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98 Force on a current-carrying conductor . . . . . . . . . . . . . . . . . . . .100 Torque on a current loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100 Galvanometers, ammeters, and voltmeters . . . . . . . . . . . . . . . . . .101 Magnetic field of a long, straight wire . . . . . . . . . . . . . . . . . . . . .101 Ampere’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102 Magnetic fields of the loop, solenoid, and toroid . . . . . . . . . . . . .103

Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 Magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Lenz’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 Generators and motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 Mutual inductance and self-inductance . . . . . . . . . . . . . . . . . . . .106 Maxwell’s equations and electromagnetic waves . . . . . . . . . . . . . .107 Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 Magnetic flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Lenz’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 Generators and motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 Mutual inductance and self-inductance . . . . . . . . . . . . . . . . . . . .106 Maxwell’s equations and electromagnetic waves . . . . . . . . . . . . . .107

Alternating Current Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 Alternating currents and voltages . . . . . . . . . . . . . . . . . . . . . . . .107 Resistor-capacitor circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 Resistor-inductance circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109 Reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109 Resistor-inductor-capacitor circuit . . . . . . . . . . . . . . . . . . . . . . .111

Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Chapter 5: Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 Characteristics of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114

Electromagnetic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 Speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116

Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 The law of reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 Plane mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 Concave mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 Convex mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 The law of refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 Brewster’s angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 Total internal reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 Optical lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 The compound microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Dispersion and prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132

Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 Huygens’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 Young’s experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136 Chapter 6: Modern Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140

Frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141 Michelson-Morley experiment . . . . . . . . . . . . . . . . . . . . . . . . . .141 The special theory of relativity . . . . . . . . . . . . . . . . . . . . . . . . . .142 Addition of velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 Time dilation and the Lorentz contraction . . . . . . . . . . . . . . . . .143 The twin paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 Relativistic momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 Relativistic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145

Table of Contents

vii

Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 Blackbody radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 Photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 Particle-wave duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 De Broglie waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 The Heisenberg uncertainty principle . . . . . . . . . . . . . . . . . . . . .150

Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150 Atomic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151 The Bohr atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151 Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153 De Broglie waves and the hydrogen atom . . . . . . . . . . . . . . . . . .154

Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154 Nucleus structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154 Binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 Half-life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156 Nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157

CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159 CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179

Introduction

P hysics is a branch of physical science that deals with physical changes

of objects. The mental, idealized models on which it is based are most frequently expressed in mathematical equations that simplify the condi- tion of the real world for ease of analysis. Even though the equations are derived from ideal conditions, they approximate real situations closely enough to allow accurate prediction of the behaviors of complex systems.

The primary task in studying physics is to understand its basic principles. Understanding these formal principles enables better understanding of the phenomena observed in the universe.

The system of units used throughout this book is the International Sys- tem of Units (SI). This is the metric system, with which you may be familiar. The basic units are length (meter, m), mass (kilogram, kg), time (second, s), temperature (degrees celsius,°C, or Kelvin, K), electric current (amperes, A), and amount (mole). Standard prefixes are often used; for example, millimeter (10 -3 meter) is abbreviated mm. A list of the most commonly used prefixes is included in the Pocket Guide. Conversions between the fundamental units of the SI and the common American units (feet, pounds, and so on) are given on the Pocket Guide.

This book is written with a broad audience in mind. Therefore, concepts are presented at varying levels of mathematical sophistication. Each topic is generally first presented in a manner that requires only basic geometry and trigonometry. In some cases, formulae requiring knowledge of calculus are given for those readers familiar with it. However, calculus is not required to understand the concepts in this book. In addition, some of the vector algebra and trigonometry used are presented in the first chapter and on the Pocket Guide.

Why You Need This Book

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2 CliffsQuickReview Physics

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Look for areas of interest in the book’s Table of Contents, or use the index to find specific topics.

Flip through the book, looking for subject areas at the top of each page.

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Use the “Chapter Checkout” at the end of each chapter to gauge your grasp of the important information you need to know.

Test your knowledge more completely in the CQR Review and look for additional sources of information in the CQR Resource Center.

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Introduction

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Chapter 1 CLASSICAL MECHANICS

Chapter Check-In

❑ Understanding motion (kinematics) in one and two dimensions, and rotational motion

❑ Applying Newton’s Laws and analyzing force diagrams ❑ Using the concepts of energy and momentum ❑ Learning about periodic motion and elasticity ❑ Applying classical mechanics to fluids

M echanics is the study of the motion of material objects. Classical or

Newtonian mechanics deals with objects and motions familiar in our everyday world. Most people possess some intuition about classical mechanics; we all have watched a ball fly through the air or a bicycle tire spin. You should not be afraid to connect the formalism in this book with your intuition. Indeed, this is often the easiest way to see the answer to a difficult problem. Allow the formal physics and math to illuminate what you already know.

Although textbook examples usually deal with blocks, springs, and other mundane devices, keep in mind that classical mechanics describes phenomena on a vast range of scales, from large molecules such as DNA to the planets of our solar system and beyond. In this chapter, you learn about the basic laws that these diverse systems all obey.

Kinematics in One Dimension

Kinematics analyzes the positions and motions of objects as a function of time, without regard to the causes of motion. It involves the relationships between the quantities displacement (d), velocity (v), acceleration (a),

6 CliffsQuickReview Physics

Definition of a vector

A vector is a physical quantity with direction as well as magnitude, for example, velocity or force. In contrast, a quantity that has only magnitude and no direction, such as temperature or time, is called a scalar. A vector is commonly denoted by an arrow drawn with a length proportional to the given magnitude of the physical quantity and with direction shown by the orientation of the head of the arrow.

Displacement and velocity

Imagine that a car begins traveling along a road after starting from a specific sign post. To know the exact position of the car after it has traveled a given distance, it is necessary to know not only the miles it traveled but also its heading. The displacement, defined as the change in position of the object, is a vector with the magnitude as a distance, such as 10 miles, and a direction, such as east. Velocity is a vector expression with a magnitude equal to the speed traveled and with an indicated direction of motion. For motion defined on a number line, the direction is specified by a positive or negative sign. Average velocity is mathematically defined as

total displacement average velocity = time elapsed

Note that displacement (distance from starting position) is not the same as distance traveled. If a car travels one mile east and then returns one mile west, to the same position, the total displacement is zero and so is the average velocity over this time period. Displacement is measured in units of length, such as meters or kilometers, and velocity is measured in units of length per time, such as meters/second (meters per second).

Average acceleration

Acceleration, defined as the rate of change of velocity, is given by the following equation:

final velocity initial velocity - average acceleration =

time elapsed Acceleration units are expressed as length per time divided by time such

as meters/second/second or in abbreviated form as 2 ms / .

Chapter 1: Classical Mechanics

Graphical interpretations of displacement, velocity, and acceleration

The distance versus time graph in Figure 1-1 shows the progress of a person: (I) standing still, (II) walking with a constant velocity, and (III) walking with a slower constant velocity. The slope of the line yields the speed. For example, the speed in segment II is

( 40 - ) m 4 m . 8 ms / ( - 10 5 ) s = 5 s =

Figure 1-1 Motion of a walking person.

8 III

4 II

Distance (meters)

5 10 15 Time (seconds)

Each segment in the velocity versus time graph in Figure 1-2 depicts a different motion of a bicycle: (I) increasing velocity, (II) constant velocity, (III) decreasing velocity, and (IV) velocity in a direction opposite the initial direction (negative). The area between the curve and the time axis represents the distance traveled. For example, the distance traveled during segment I is equal to the area of the triangle with height 15 and base 10. Because the area of a triangle is (1/2)(base)(height), then (1/2)(15 m/s) (10 s) = 75 m. The magnitude of acceleration equals the calculated slope. The acceleration calculation for segment III is (–15 m/s)/(10 s) = –1.5 m/s/s or –1.5 2 ms / .

The more realistic distance-versus-time curve in Figure 1-3(a) illustrates gradual changes in the motion of a moving car. The speed is nearly constant in the first 2 seconds, as can be seen by the nearly constant slope of the line; however, between 2 and 4 seconds, the speed is steadily decreasing and the instantaneous velocity describes how fast the object is moving at a given instant.

8 CliffsQuickReview Physics

Figure 1-2 Accelerating motion of a bicycle.

10 20 30 40 V elocity (meters/seconds) −15

Time (seconds)

Figure 1-3 Motion of a car: (a) distance, (b) velocity, and (c) acceleration change in time.

Distance (meters) 0 2 4 6 8 10 12 14

Time (seconds) (a)

V elocity (meters/second)

Time (seconds) (b)

0 2 4 6 8 10 12 14 −2.5 ation (meters/second

Acceler

Time (seconds) (c)

Chapter 1: Classical Mechanics

Instantaneous velocity can be read on an odometer in the car. It is calculated from a graph as the slope of a tangent to the curve at the specified time. The slope of the line sketched at 4 seconds is 6 ms. Figure 1-3(b) is

a sketch of the velocity-versus-time graph constructed from the slopes of the distance-versus-time curve. In like fashion, the instantaneous acceleration is found from the slope of a tangent to the velocity-versus-time curve at a given time. The instantaneous acceleration-versus-time graph in Figure 1-3(c) is the sketch of the slopes of the velocity-versus-time graph of Figure 1-3(b). With the vertical arrangement shown, it is easy to com- pute the displacement, velocity, and acceleration of a moving object at the same time.

For example, at time t = 10 s, the displacement is 47 m, the velocity is –5

m/s, and the acceleration is 2 - 5 ms / .

Definitions of instantaneous velocity and instantaneous acceleration

The instantaneous velocity, by definition, is the limit of the average velocity as the measured time interval is made smaller and smaller. In formal

terms, v = lim ∆ t " 0 ∆ d ∆ t . The notation lim ∆ t " 0 means the ratio ∆ d ∆ t is evaluated as the time interval approaches zero. Similarly, instantaneous acceleration is defined as the limit of the average acceleration as the time

interval becomes infinitesimally short. That is, a = lim ∆ t " 0 ∆ v ∆ t .

Motion with constant acceleration

When an object moves with constant acceleration, the velocity increases or decreases at the same rate throughout the motion. The average acceleration equals the instantaneous acceleration when the acceleration is constant. A negative acceleration can indicate either of two conditions:

Case 1: The object has a decreasing velocity in the positive direction.

Case 2: The object has an increasing velocity in the negative direction.

For example, a ball tossed up will be under the influence of a negative (downward) acceleration due to gravity. Its velocity will decrease while it travels upward (case 1); then, after reaching its highest point, the velocity will increase downward as the object returns to earth (case 2).

10 CliffsQuickReview Physics

Using v o (velocity at the beginning of time elapsed), v f (velocity at the end of the time elapsed), and t for time, the constant acceleration is v f - v o

[Equation 1] Substituting the average velocity as the arithmetic average of the original

or v f = v o + at

and final velocities v avg = ( v o + v f )/ 2 into the relationship between distance and average velocity d = ( v avg )( ) t yields

[Equation 2] Substitute v f from Equation 1 into Equation 2 to obtain dvt 2 =

d = 1 2 ( v o + v f )t

[Equation 3] Finally, substitute the value of t from Equation 1 into Equation 2 for 2 v 2

+ 1 2 at

f = v o + 2 ad [Equation 4] These four equations relate vv o , f , t, a, and d. Note that each equation has

a different set of four of these five quantities. Table 1-1 summarizes the equations for motion in a straight line under constant acceleration.

A special case of constant acceleration occurs for an object under the influence of gravity. If an object is thrown vertically upward or dropped, the acceleration due to gravity of 2 - 98 . ms / is substituted in the above equations to find the relationships among velocity, distance, and time.

Table 1-1 Equations and Variables of Kinematics in One Dimension

Information Given Variables Equation

by Equation v o v f t a d v f = v o + at

Velocity as a function of time ✓ ✓ ✓ ✓ X d = 1 ( v o + vt f ) Displacement varying with

✓ ✓ ✓ X ✓ 2 velocity and time

1 2 Displacement as a function ✓ X ✓ ✓ dvt ✓ =

+ 2 at of time

2 v 2 f = v o + 2 ad Velocity as a function ✓ ✓ X ✓ ✓ of displacement

Kinematics in Two Dimensions

Up to this time, only forward and backward motion along a number line

Chapter 1: Classical Mechanics 11

analysis, many motions can be simplified to two dimensions. For example, an object fired into the air moves in a vertical, two-dimensional plane; also, horizontal motion over the earth’s surface is two-dimensional for short distances. Elementary vector algebra is required to examine the relationships between vector quantities in two dimensions.

Addition and subtraction of vectors: geometric method

The vector A shown in Figure 1-4(a) represents a velocity of 10 m/s north- east, and vector B represents a velocity of 20 m/s at 30 degrees north of east. (A vector is named with a letter in boldface, nonitalic type, and its magnitude is named with the same letter in regular, italic type. You will often see vectors in the figures of the book that are represented by their magnitudes in the mathematical expressions.) Vectors may be moved over the plane if the represented length and direction are preserved.

Figure 1-4 Graphical addition of vectors, A + B = C.

In Figure 1-4(b), the same vectors are positioned to be geometrically added. The tail of one vector, in this case A, is moved to the head of the other vector (B). The vector sum (C) is the vector that extends from the tail of one vector to the head of the other. To find the magnitude of C, measure along its length and use the given scale to determine the velocity represented. To find the direction θ of C, measure the angle to the horizontal axis at the tail of C.

Figure 1-5(a) shows that A + B = B + A. The sum of the vectors is called the resultant and is the diagonal of a parallelogram with sides A and B. Figure 1-5(b) illustrates the construction for adding four vectors. The result-

12 CliffsQuickReview Physics

Figure 1-5 (a) A + B = B + A. (b) Graphical addition of several vectors.

(a) (b)

To subtract vectors, place the tails together. The difference of the two vectors (D) is the vector that begins at the head of the subtracted vector (B) and goes to the head of the other vector (A). An alternate method is to add the negative of a vector, which is a vector with the same length but pointing in the opposite direction. The second method is demonstrated in Figure 1-6.

Figure 1-6 Graphical subtraction of vectors, A – B = D.

−B

Addition and subtraction of vectors: Component method

For precision in adding vectors, an analytical method using basic trigonometry is required because scale drawings do not give accurate values.

Consider vector A in the rectangular coordinate system of Figure 1-7. The vector A can be expressed as the sum of two vectors along the x and y axes,

A = A x + A y , where A x and A y are called the components of A. The direction of A x is parallel to the x axis, and that of A y is parallel to the y axis.

Chapter 1: Classical Mechanics 13

The magnitudes of the components are obtained from the definitions of the sine and cosine of an angle: cos θ = AA x / and sin θ = AA y / , or

A x = A cos θ A y = A sin θ

Figure 1-7 Components of a vector. y

tanθ = A y x

To add vectors numerically, first find the components of all the vectors. The signs of the components are the same as the signs of the cosine and sine in the given quadrant. Then, sum the components in the x direction, and sum the components in the y direction. As shown in Figure 1-8, the sum of the x components and the sum of the y components of the given vectors (A and

B) comprise the x and y components of the resultant vector (C). These resultant components form the two sides of a right angle with a

hypotenuse of the magnitude of C; thus, the magnitude of the resultant is

Figure 1-8 Component method of vector addition, A + B = C.

14 CliffsQuickReview Physics

The direction of the resultant (C) is calculated from the tangent because - tan 1 θ= CC

x / y . To solve for the angle θ, use θ = tan ( CC y / x .) The procedure can be summarized as follows:

1. Sketch the vectors on a coordinate system.

2. Find the x and y components of all the vectors, with the appro- priate signs.

3. Sum the components in both the x and y directions.

4. Find the magnitude of the resultant vector from the Pythagorean theorem.

5. Find the direction of the resultant vector using the tangent function.

Follow the same procedure to subtract vectors by calculating the appro- priate algebraic sum of the components in Step 3.

Multiplication of vectors

The dot product: There are two different ways in which two vectors may

be multiplied together. The first is the dot product, also called the scalar product, which is written A ⋅ B. This can be evaluated in two ways:

A⋅B = A x B x +A y B y

A⋅B = AB cos θ, where θ is the angle between the vectors when they are set tail to tail, and A and B are the lengths of the vectors.

Note that the order of the vectors does not matter and that the result of the dot product is a scalar rather than a vector. Note that if two vectors are per- pendicular, their dot product is zero according to the second rule above.

Cross product: The second way to multiply vectors is called the cross

product or the vector product. It is written A ⋅ B. It can be evaluated in two ways:

A⋅B=( A x B y –A y B x )z, when the vectors A and B both are in x-y plane. The z indicates that the result is a vector that points along the z axis. In general, the vector resulting from a cross product is always per- pendicular to both of the vectors being multiplied together.

A⋅B= ABz sin θ, where θ is the angle between the vectors A and B when they are placed tail to tail. Again, the result is a vector perpen- dicular to A and B (and therefore points along the z axis if A and B

Chapter 1: Classical Mechanics 15

The result of a cross product does depend on the order of the vectors. Note from the first rule that A ⋅ B = –B ⋅ A. Also, if A and B are parallel, the second rule implies that their cross product is zero.

Finally, the cross product give rise to the “right hand rule,” which allows you to easily determine the direction of the resulting vector. For the general expression A × B = C, point your thumb in the direction of A. Now point your index finger in the direction of B; if necessary, flip over your hand. The vector C points outward from your palm. For an illustration of this procedure, flip ahead to Chapter 4 (Figure 4-19), where the rule is applied to the equation F = qv × B.

Velocity and acceleration vectors in two dimensions

For motion in two dimensions, the earlier kinematics equations must be expressed in vector form. For example, the average velocity vector is

v = ( d f - d o )t , where d o and d f are the initial and final displacement vectors and t is the time elapsed. As noted earlier, the velocity and displacement vectors are shown in bold type, whereas the scalar (t) is not. In similar

fashion, the average acceleration vector is a = ( v f - v o )t , where v o and v f

are the initial and final velocity vectors. An important point is that the acceleration can arise from a change in the

magnitude of the velocity (speed) as well as from a change in the direction of the velocity. If an object travels around a circle at a constant speed, there is an acceleration due to the change in the direction of the velocity, even though the magnitude of the velocity does not change. A mass moves in a horizontal circle with a constant speed in Figure 1-9. The velocity vectors at positions 1 and 2 are subtracted to find the average acceleration, which is directed toward the center of the circle. (Note that the average acceleration vector is placed at the midpoint of the path in the given time interval.)

Figure 1-9 Velocity and acceleration vectors of an object moving in a circle.

(a) (b)

16 CliffsQuickReview Physics

The following discussion summarizes the four different cases for acceleration in a plane:

Case 1: Zero acceleration

Case 2: Acceleration due to changing direction but not speed

Case 3: Acceleration due to changing speed but not direction

Case 4: Acceleration due to changing both speed and direction. Imagine a ball rolling on a horizontal surface that is illuminated by a stro-

boscopic light. Figure 1-10(a) shows the position of the ball at even inter- vals of time along a dotted path. Case 1 is illustrated in positions 1 through 3; the magnitude and direction of the velocity do not change (the pictures are evenly spaced and in a straight line), and therefore, there is no acceleration. Case 2 is indicated for positions 3 through 5; the ball has constant speed but changing direction, and therefore, an acceleration exists. Figure

1-10(b) illustrates the subtraction of v 3 and v 4 and the resulting acceleration toward the center of the arc. Case 3 occurs from positions 5 to 7; the direction of the velocity is constant, but the magnitude changes. The acceleration for this portion of the path is along the direction of motion. The ball curves from position 7 to 9, showing case 4; the velocity changes both direction and magnitude. In this case, the acceleration is directed nearly upward between 7 and 8 and has a component toward the center of the arc due to the change in direction of the velocity and a component along the path due to the change in the magnitude of the velocity.

Figure 1-10 (a) Path of a ball on a table. (b) Acceleration between points 3 and 4.

3 v 4 2 1 (a)

(b)

Chapter 1: Classical Mechanics 17

Projectile motion

Anyone who has observed a tossed object—for example, a baseball in flight—has observed projectile motion. To analyze this common type of motion, three basic assumptions are made: (1) acceleration due to gravity is constant and directed downward, (2) the effect of air resistance is negligible, and (3) the surface of the earth is a stationary plane (that is, the cur- vature of the earth’s surface and the rotation of the earth are negligible).

To analyze the motion, separate the two-dimensional motion into vertical and horizontal components. Vertically, the object undergoes constant acceleration due to gravity. Horizontally, the object experiences no acceleration and, therefore, maintains a constant velocity. This velocity is illustrated in Figure 1-11 where the velocity components change in the y direction; however, they are all of the same length in the x direction (constant). Note that the velocity vector changes with time due to the fact that the vertical component is changing.

Figure 1-11 Projectile motion. y

v y =0 v

v y =− v y 0 v

In this example, the particle leaves the origin with an initial velocity () v o , up at an angle of θ o . The original x and y components of the velocity are given by v xo = v o cos θ o and v yo = v o sin θ o .

18 CliffsQuickReview Physics

With the motions separated into components, the quantities in the x and y directions can be analyzed with the one-dimensional motion equations subscripted for each direction: for the horizontal direction, v x = v xo and xvt 2 =

xo ; for vertical direction, v y = v yo - gt and yvt = yo - ` 12 j gt , where x and y represent distances in the horizontal and vertical directions, respec- tively, and the acceleration due to gravity (g) is 2 98 . ms / . (The negative sign is already incorporated into the equations.) If the object is fired down at an angle, the y component of the initial velocity is negative. The speed of the projectile at any instant can be calculated from the components at that time from the Pythagorean theorem, and the direction can be found from the inverse tangent on the ratios of the components:

- = 1 tan d vv y x n

By Linda Huetinck, PhD, and Scott Adams

Physics