Differential Equations, Exercises of Differential Equations

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SCHAUM'S OUTLINE OF

THEORY AND PROBLEMS

OF

COLLEGE PHYSICS

Ninth Edition

.

FREDERICK J. BUECHE, Ph.D.

Distinguished Professor at Large

University of Dayton

EUGENE HECHT, Ph.D.

Professor of Physics

Adelphi University

.

SCHAUM'S OUTLINE SERIES

McGRAW-HILL

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Preface

The introductory physics course, variously known as ``general physics'' or

``college physics,'' is usually a two-semester in-depth survey of classical topics

capped o€ with some selected material from modern physics. Indeed the name

``college physics'' has become a euphemism for introductory physics without

calculus. Schaum's Outline of College Physics was designed to uniquely

complement just such a course, whether given in high school or in college. The

needed mathematical knowledge includes basic algebra, some trigonometry, and a

tiny bit of vector analysis. It is assumed that the reader already has a modest

understanding of algebra. Appendix B is a general review of trigonometry that

serves nicely. Even so, the necessary ideas are developed in place, as needed. And

the same is true of the rudimentary vector analysis that's requiredÐit too is taught as

the situation requires.

In some ways learning physics is unlike learning most other disciplines. Physics

has a special vocabulary that constitutes a language of its own, a language

immediately transcribed into a symbolic form that is analyzed and extended with

mathematical logic and precision. Words like energy, momentum, current, ¯ux,

interference, capacitance, and so forth, have very speci®c scienti®c meanings.

These must be learned promptly and accurately because the discipline builds layer

upon layer; unless you know exactly what velocity is, you cannot know what

acceleration or momentum are, and without them you cannot know what force is,

and on and on. Each chapter in this book begins with a concise summary of the

important ideas, de®nitions, relationships, laws, rules, and equations that are

associated with the topic under discussion. All of this material constitutes the

conceptual framework of the discourse, and its mastery is certainly challenging in

and of itself, but there's more to physics than the mere recitation of its principles.

Every physicist who has ever tried to teach this marvelous subject has heard the

universal student lament, ``I understand everything; I just can't do the problems.''

Nonetheless most teachers believe that the ``doing'' of problems is the crucial

culmination of the entire experience, it's the ultimate proof of understanding and

competence. The conceptual machinery of de®nitions and rules and laws all come

together in the process of problem solving as nowhere else. Moreover, insofar as the

problems re¯ect the realities of our world, the student learns a skill of immense

practical value. This is no easy task; carrying out the analysis of even a

moderately complex problem requires extraordinary intellectual vigilance and

un¯agging attention to detail above and beyond just ``knowing how to do it.''

Like playing a musical instrument, the student must learn the basics and then

practice, practice, practice. A single missed note in a sonata is overlookable; a

single error in a calculation, however, can propagate through the entire e€ort

producing an answer that's completely wrong. Getting it right is what this book is

all about.

Although a selection of new problems has been added, the 9th-edition revision

of this venerable text has concentrated on modernizing the work, and improving the

pedagogy. To that end, the notation has been simpli®ed and made consistent

throughout. For example, force is now symbolized by F and only F; thus

centripetal force is FC, weight is F W, tension is F T, normal force is F N, friction is

Ff , and so on. Work (W ) will never again be confused with weight (FW), and period

iii

Copyright 1997, 1989, 1979, 1961, 1942, 1940, 1939, 1936 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

(T ) will never be mistaken for tension (F T). To better match what's usually written in

the classroom, a vector is now indicated by a boldface symbol with a tiny arrow

above it. The idea of signi®cant ®gures is introduced (see Appendix A) and

scrupulously adhered to in every problem. Almost all the de®nitions have been

revised to make them more precise or to re¯ect a more modern perspective. Every

drawing has been redrawn so that they are now more accurate, realistic, and

readable.

If you have any comments about this edition, suggestions for the next edition, or

favorite problems you'd like to share, send them to E. Hecht, Adelphi University,

Physics Department, Garden City, NY 11530.

Freeport, NY E UGENE H ECHT

iv SIGNIFICANT FIGURES

Chapter 8 IMPULSE AND MOMENTUM............................. 87

Linear momentum. Impulse. Impulse causes change in momentum. Conservation of linear momentum. Collisions and explosions. Perfectly elastic collision. Coecient of restitution. Center of mass.

Chapter 9 ANGULAR MOTION IN A PLANE.......................... 99

Angular displacement. Angular speed. Angular acceleration. Equations for uniformly accelerated motion. Relations between angular and tangential quantities. Centripetal acceleration. Centripetal force.

Chapter 10 RIGID-BODY ROTATION................................. 111

Torque (or moment). Moment of inertia. Torque and angular acceleration. Kinetic energy of rotation. Combined rotation and translation. Work. Power. Angular momentum. Angular impulse. Parallel-axis theorem. Analogous linear and angular quantities.

Chapter 11 SIMPLE HARMONIC MOTION AND SPRINGS................ 126

Period. Frequency. Graph of a vibratory motion. Displacement. Restoring force. Simple harmonic motion. Hookean system. Elastic potential energy. Energy interchange. Speed in SHM. Acceleration in SHM. Reference circle. Period in SHM. Acceleration in terms of T. Simple pendulum. SHM.

Chapter 12 DENSITY; ELASTICITY.................................. 138

Mass density. Speci®c gravity. Elasticity. Stress. Strain. Elastic limit. Young's modulus. Bulk modulus. Shear modulus.

Chapter 13 FLUIDS AT REST....................................... 146

Average pressure. Standard atmospheric pressure. Hydrostatic pressure. Pascal's principle. Archimedes' principle.

Chapter 14 FLUIDS IN MOTION.................................... 157

Fluid ¯ow or discharge. Equation of continuity. Shear rate. Viscosity. Poiseuille's Law. Work done by a piston. Work done by a pressure. Bernoulli's equation. Torricelli's theorem. Reynolds number.

Chapter 15 THERMAL EXPANSION.................................. 166

Temperature. Linear expansion of solids. Area expansion. Volume expansion.

vi PREFACE

Chapter 16 IDEAL GASES......................................... 171

Ideal (or perfect) gas. One mole of a substance. Ideal Gas Law. Special cases. Absolute zero. Standard conditions or standard temperature and pressure (S.T.P.). Dalton's Law of partial pressures. Gas-law problems.

Chapter 17 KINETIC THEORY...................................... 179

Kinetic theory. Avogadro's number. Mass of a molecule. Average translational kinetic energy. Root mean square speed. Absolute temperature. Pressure. Mean free path.

Chapter 18 HEAT QUANTITIES..................................... 185

Thermal energy. Heat. Speci®c heat. Heat gained (or lost). Heat of fusion. Heat of vaporization. Heat of sublimation. Calorimetry problems. Absolute humidity. Relative humidity. Dew point.

Chapter 19 TRANSFER OF HEAT ENERGY............................ 193

Energy can be transferred. Conduction. Thermal resistance. Convection. Radiation.

Chapter 20 FIRST LAW OF THERMODYNAMICS....................... 198

Heat. Internal energy. Work done by a system. First Law of Thermodynamics. Isobaric process. Isovolumic process. Isothermal process. Adiabatic process. Speci®c heats of gases. Speci®c heat ratio. Work related to area. Eciency of a heat engine.

Chapter 21 ENTROPY AND THE SECOND LAW........................ 209

Second Law of Thermodynamics. Entropy. Entropy is a measure of disorder. Most probable state.

Chapter 22 WAVE MOTION........................................ 213

Propagating wave. Wave terminology. In-phase vibrations. Speed of a transverse wave. Standing waves. Conditions for resonance. Longitudinal (compressional) waves.

Chapter 23 SOUND............................................... 223

Sound waves. Equations for sound speed. Speed of sound in air. Intensity. Loudness. Intensity (or loudness) level. Beats. Doppler e€ect. Interference e€ects.

SIGNIFICANT FIGURES vii

Chapter 33 ELECTRIC GENERATORS AND MOTORS................... 315

Electric generators. Electric motors.

Chapter 34 INDUCTANCE; R-C AND R-L TIME CONSTANTS............. 321

Self-inductance. Mutual inductance. Energy stored in an inductor. R-C time constant. R-L time constant. Exponential functions.

Chapter 35 ALTERNATING CURRENT............................... 329

Emf generated by a rotating coil. Meters. Thermal energy generated or power lost. Forms of Ohm's Law. Phase. Impedance. Phasors. Resonance. Power loss. Transformer.

Chapter 36 REFLECTION OF LIGHT................................. 338

Nature of light. Law of re¯ection. Plane mirrors. Spherical mirrors. Mirror equation. Size of the image.

Chapter 37 REFRACTION OF LIGHT................................. 346

Speed of light. Index of refraction. Refraction. Snell's Law. Critical angle for total internal re¯ection. Prism.

Chapter 38 THIN LENSES......................................... 353

Type of lenses. Object and image relation. Lensmaker's equation. Lens power. Lenses in contact.

Chapter 39 OPTICAL INSTRUMENTS................................ 359

Combination of thin lenses. The eye. Magnifying glass. Microscope. Telescope.

Chapter 40 INTERFERENCE AND DIFFRACTION OF LIGHT.............. 366

Coherent waves. Relative phase. Interference e€ects. Di€raction. Single-slit di€raction. Limit of resolution. Di€raction grating equation. Di€raction of X-rays. Optical path length.

Chapter 41 RELATIVITY.......................................... 374

Reference frame. Special theory of relativity. Relativistic linear momentum. Limiting speed. Relativistic energy. Time dilation. Simultaneity. Length contraction. Velocity addition formula.

SIGNIFICANT FIGURES ix

Chapter 42 QUANTUM PHYSICS AND WAVE MECHANICS............... 382

Quanta of radiation. Photoelectric e€ect. Momentum of a photon. Compton e€ect. De Broglie waves. Resonance of de Broglie waves. Quantized energies.

Chapter 43 THE HYDROGEN ATOM................................. 390

Hydrogen atom. Electron orbits. Energy-level diagrams. Emission of light. Spectral lines. Origin of spectral series. Absorption of light.

Chapter 44 MULTIELECTRON ATOMS............................... 396

Neutral atom. Quantum numbers. Pauli exclusion principle.

Chapter 45 NUCLEI AND RADIOACTIVITY............................ 399

Nucleus. Nuclear charge and atomic number. Atomic mass unit. Mass number. Isotopes. Binding energies. Radioactivity. Nuclear equations.

Chapter 46 APPLIED NUCLEAR PHYSICS............................. 409

Nuclear binding energies. Fission reaction. Fusion reaction. Radiation dose. Radiation damage potential. E€ective radiation dose. High-energy accelerators. Momentum of a particle.

Appendix A SIGNIFICANT FIGURES.................................. 417

Appendix B TRIGONOMETRY NEEDED FOR COLLEGE PHYSICS.......... 419

Appendix C EXPONENTS........................................... 422

Appendix D LOGARITHMS.......................................... 424

Appendix E PREFIXES FOR MULTIPLES OF SI UNITS; THE GREEK

ALPHABET............................................ 427

Appendix F FACTORS FOR CONVERSIONS TO SI UNITS................. 428

Appendix G PHYSICAL CONSTANTS.................................. 429

Appendix H TABLE OF THE ELEMENTS............................... 430

INDEX...................................................... 433

x CONTENTS

The resultant is represented by an arrow with its tail end at the starting point and its tip end at the

tipof the last vector added. If ~R is the resultant, R ˆ j~Rj is the size or magnitude of the resultant.

PARALLELOGRAM METHOD for adding two vectors: The resultant of two vectors acting at

any angle may be represented by the diagonal of a parallelogram. The two vectors are drawn as

the sides of the parallelogram and the resultant is its diagonal, as shown in Fig. 1-2. The direc-

tion of the resultant is away from the origin of the two vectors.

SUBTRACTION OF VECTORS: To subtract a vector ~B from a vector ~A, reverse the direction

of ~B and add individually to vector ~A, that is, ~A  ~B ˆ ~A ‡

 ~B†:

THE TRIGONOMETRIC FUNCTIONS are de®ned in relation to a right angle. For the right tri-

angle shown in Fig. 1-3, by de®nition

sin  ˆ

opposite

hypotenuse

B

C

; cos  ˆ

adjacent

hypotenuse

A

C

; tan  ˆ

opposite

adjacent

B

A

We often use these in the forms

B ˆ C sin  A ˆ C cos  B ˆ A tan 

A COMPONENT OF A VECTOR is its e€ective value in a given direction. For example, the x-

component of a displacement is the displacement parallel to the x-axis caused by the given displa-

cement. A vector in three dimensions may be considered as the resultant of its component vectors

resolved along any three mutually perpendicular directions. Similarly, a vector in two dimensions

2 INTRODUCTION TO VECTORS [CHAP. 1

Fig. 1-

Fig. 1-

may be resolved into two component vectors acting along any two mutually perpendicular direc-

tions. Figure 1-4 shows the vector ~R and its x and y vector components, ~Rx and ~Ry, which have

magnitudes

j~Rxj ˆ j~Rj cos  and j~Ryj ˆ j~Rj sin 

or equivalently

Rx ˆ R cos  and R y ˆ R sin 

COMPONENT METHOD FOR ADDING VECTORS: Each vector is resolved into its x-, y-,

and z-components, with negatively directed components taken as negative. The scalar x-component

R x of the resultant ~R is the algebraic sum of all the scalar x-components. The scalar y- and z-

components of the resultant are found in a similar way. With the components known, the magni-

tude of the resultant is given by

R ˆ

R^2 x ‡ R^2 y ‡ R^2 z

q

In two dimensions, the angle of the resultant with the x-axis can be found from the relation

tan  ˆ

R y

R x

UNIT VECTORS have a magnitude of one and are represented by a boldface symbol topped

with a caret. The special unit vectors ^i, ^j, and ^k are assigned to the x-, y-, and z-axes, respec-

tively. A vector 3^i represents a three-unit vector in the ‡x-direction, while  5 ^k represents a ®ve-

unit vector in the z-direction. A vector ~R that has scalar x-, y-, and z-components Rx, R y, and

R z, respectively, can be written as ~R ˆ Rx^i ‡ Ry^j ‡ R z ^k.

THE DISPLACEMENT: When an object moves from one point in space to another the displa-

cement is the vector from the initial location to the ®nal location. It is independent of the actual

distance traveled.

CHAP. 1] INTRODUCTION TO VECTORS 3

Fig. 1-

1.4 Add the following two force vectors by use of the parallelogram method: 30 N at 30 8 and 20 N at

1408. Remember that numbers like 30 N and 20 N have two signi®cant ®gures.

The force vectors are shown in Fig. 1-8(a). We construct a parallelogram using them as sides, as shown in Fig. 1-8(b). The resultant ~R is then represented by the diagonal. By measurement, we ®nd that ~R is 30 N at 728 :

1.5 Four coplanar forces act on a body at point O as shown in Fig. 1-9(a). Find their resultant

graphically.

Starting from O, the four vectors are plotted in turn as shown in Fig. 1-9(b). We place the tail end of each vector at the tipend of the preceding one. The arrow from O to the tipof the last vector represents the resultant of the vectors.

CHAP. 1] INTRODUCTION TO VECTORS 5

Fig. 1-

Fig. 1-

Fig. 1-

We measure R from the scale drawing in Fig. 1-9(b) and ®nd it to be 119 N. Angle is measured by protractor and is found to be 37 8. Hence the resultant makes an angle  ˆ 1808  378 ˆ 1438 with the positive x-axis. The resultant is 119 N at 143 8 :

1.6 The ®ve coplanar forces shown in Fig. 1-10(a) act on an object. Find their resultant.

(1) First we ®nd the x- and y-components of each force. These components are as follows:

Notice the ‡ and  signs to indicate direction.

(2) The resultant ~R has components Rx ˆ  Fx and R (^) y ˆ  Fy, where we read  F (^) x as ``the sum of all the x- force components.'' We then have R (^) x ˆ 19 :0 N ‡ 7 :50 N  11 :3 N  9 :53 N ‡ 0 N ˆ ‡ 5 :7 N Ry ˆ 0 N ‡ 13 :0 N ‡ 11 :3 N  5 :50 N  22 :0 N ˆ  3 :2 N

(3) The magnitude of the resultant is

R ˆ

R^2 x ‡ R^2 y

q ˆ 6 :5 N

(4) Finally, we sketch the resultant as shown in Fig. 1-10(b) and ®nd its angle. We see that

tan  ˆ

3 :2 N

5 :7 N

from which  ˆ 298. Then  ˆ 3608  298 ˆ 3318. The resultant is 6.5 N at 331 8 (or  298 ) or ~R ˆ 6 :5 N Ð 3318 FROM ‡X-AXIS.

6 INTRODUCTION TO VECTORS [CHAP. 1

Force x-Component y-Component

19.0 N 19.0 N 0 N 15.0 N 15 :0 N) cos 60: 08 ˆ 7 :50 N 15 :0 N) sin 60: 08 ˆ 13 :0 N 16.0 N  16 :0 N) cos 45: 08 ˆ  11 :3 N 16 :0 N) sin 45: 08 ˆ 11 :3 N 11.0 N  11 :0 N) cos 30: 08 ˆ  9 :53 N  11 :0 N) sin 30: 08 ˆ  5 :50 N 22.0 N 0 N  22 :0 N

Fig. 1-

As shown in Fig. 1-13, the components of the 60 N force are 39 N and 46 N. (a) The pull along the ground is the horizontal component, 46 N. (b) The lifting force is the vertical component, 39 N.

1.10 A car whose weight is FW is on a rampwhich makes an angle  to the horizontal. How large a

perpendicular force must the ramp withstand if it is not to break under the car's weight?

As shown in Fig. 1-14, the car's weight is a force ~FW that pulls straight down on the car. We take components of ~F along the incline and perpendicular to it. The ramp must balance the force component FW cos  if the car is not to crash through the ramp.

1.11 Express the forces shown in Figs. 1-7(c), 1-10(b), 1-11, and 1-13 in the form ~R ˆ R x^i ‡ R y^j ‡ Rz ^k

(leave out the units).

Remembering that plus and minus signs must be used to show direction along an axis, we can write

For Fig. 1-7(c): ~R ˆ  0 : 88 ^i ‡ 4 : 48 ^j For Fig. 1-10(b): ~R ˆ 5 : 7 ^i  3 : 2 ^j For Fig. 1-11: ~R ˆ  94 ^i ‡ 71 ^j For Fig. 1-13: ~R ˆ 46 ^i ‡ 39 ^j

1.12 Three forces that act on a particle are given by ~F 1 ˆ

20 ^i  36 ^j ‡ 73 k^† N,

~F 2 ˆ

 17 ^i ‡ 21 ^j  46 k^† N, and ~F 3 ˆ  12 ^k† N. Find their resultant vector. Also ®nd the mag-

nitude of the resultant to two signi®cant ®gures.

We know that

Rx ˆ  F (^) x ˆ 20 N  17 N ‡ 0 N ˆ 3 N Ry ˆ  F (^) y ˆ 36 N ‡ 21 N ‡ 0 N ˆ 15 N R (^) z ˆ  F (^) z ˆ 73 N  46 N  12 N ˆ 15 N

Since ~R ˆ R (^) x^i ‡ R (^) y^j ‡ Rz k^, we ®nd ~R ˆ 3 ^i  15 ^j ‡ 15 ^k

To two signi®cant ®gures, the three-dimensional pythagorean theorem then gives

R ˆ

R^2 x ‡ R^2 y ‡ R^2 z

q ˆ

p ˆ 21 N

8 INTRODUCTION TO VECTORS [CHAP. 1

Fig. 1-13 Fig. 1-

1.13 Perform graphically the following vector additions and subtractions, where ~A, ~B, and ~C are the

vectors shown in Fig. 1-15: (a) ~A ‡ ~B; (b) ~A ‡ ~B ‡ ~C; (c) ~A  ~B; (d ) ~A ‡ ~B  ~C:

See Fig. 1-15(a) through (d ). In (c), ~A  ~B ˆ ~A ‡ ~B†; that is, to subtract ~B from ~A, reverse the direction of ~B and add it vectorially to ~A. Similarly, in (d ), ~A ‡ ~B  ~C ˆ ~A ‡ ~B ‡ ~C†, where ~C is equal in magnitude but opposite in direction to ~C:

1.14 If ~A ˆ  12 ^i ‡ 25 ^j ‡ 13 ^k and ~B ˆ  3 ^j ‡ 7 ^k, ®nd the resultant when ~A is subtracted from ~B:

From a purely mathematical approach, we have ~B  ~A ˆ  3 ^j ‡ 7 ^k†   12 ^i ‡ 25 ^j ‡ 13 ^k†

ˆ  3 ^j ‡ 7 k^ ‡ 12 ^i  25 ^j  13 ^k ˆ 12 ^i  28 ^j  6 ^k

Notice that 12^i  25 ^j  13 ^k is simply ~A reversed in direction. Therefore we have, in essence, reversed ~A and added it to ~B.

1.15 A boat can travel at a speed of 8 km/h in still water on a lake. In the ¯owing water of a stream, it

can move at 8 km/h relative to the water in the stream. If the stream speed is 3 km/h, how fast can

the boat move past a tree on the shore when it is traveling (a) upstream and (b) downstream?

(a) If the water was standing still, the boat's speed past the tree would be 8 km/h. But the stream is carrying it in the opposite direction at 3 km/h. Therefore the boat's speed relative to the tree is 8 km=h  3 km=h ˆ 5 km=h: (b) In this case, the stream is carrying the boat in the same direction the boat is trying to move. Hence its speed past the tree is 8 km=h ‡ 3 km=h ˆ 11 km=h:

1.16 A plane is traveling eastward at an airspeed of 500 km/h. But a 90 km/h wind is blowing south-

ward. What are the direction and speed of the plane relative to the ground?

The plane's resultant velocity is the sum of two velocities, 500 km/h Ð EAST and 90 km/h Ð SOUTH. These component velocities are shown in Fig. 1-16. The plane's resultant velocity is then

R ˆ

500 km=h†^2 ‡ 90 km=h†^2

q ˆ 508 km=h

CHAP. 1] INTRODUCTION TO VECTORS 9

Fig. 1-