A

Student's

Pocket Companion

FUNDAMENTALS

OF PHYSICS
FIFTH EDITION
J.

Richard Christman

Halli day

NEDS BOOKSTORE
No

Return w/o labe


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A STUDENT'S POCKET COMPANION
J.

Richard Christman

U.S.

Coast

to

Guard Academy

accompany

FUNDAMENTALS OF

PHYSICS
FEFTH EDITION

David Haluday
University of Pittsburgh

Robert Resnick
Rensselaer Polytechnic Institute

Jearl Walker
Cleveland State University

John Wiley

New York

Chichester


&

Sons, Inc.

Brisbane

Toronto

Singapore


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Copyright

©

1997 by John Wiley

& Sons, Inc.

All rights reserved.

Reproduction or translation of any part of this work
beyond that permitted by Sections 107 and 108 of
the 1976 United States Copyright Act without the
permission of the copyright owner is unlawfiil.
Requests for permission or further information
should be addressed to the Permissions Department,


John Wiley

&

Sons, Inc.

ISBN 0-47 1-09675-X
Printed in the United States of America

1098765432
Printed and bound by Courier Westford


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PREFACE
A Student's Pocket Companion io Fundamentals of Physics, fifth ediby Halliday, Resnick, and Walker lists the important ideas covered
each section of the text, with a few sentences about each, and gives the
basic equations. It serves three purposes. First, it can be taken to class as
a substitute for the text. You might want to check off the topics covered
wide left
and make short notes to remind yourself of important points.
margin and a notes section at the end of each chapter are provided for
this purpose. Second, it can be used as a handy reference for ideas and
equations while working problem assignments. Third, it can be used to
review text material before an exam or when you need to recall an idea
from a previous chapter.
Readingyl Student's Pocket Companion is NOT a substitute for reading the text. Some derivations and applications are outlined in^ Pocket
Student's Companion but they are necessarily shortened. The text contains much more detail and much fuller explanations. Study the text well,
preferably before class, then use>4 Student's Pocket Companion to remind

yourself of the material you have studied. If it fails to jog your memory,
restudy the appropriate portion of the text.
short vocabulary list is provided at the beginning of each chapter. In order to understand the material of the chapter you should know the meanings of these words and
phrases. Some definitions are given in^ Student's Pocket Companion-, for
other definitions you should refer to the text.
Full understanding of the ideas outlined in^ Student's Pocket Companion will help you greatly in solving problems. However, problem solving techniques are not explicitly covered. For help in solving problems
refer to the Sample Problems of the text, the full size Student's Companion, and the Solutions Manual. Also read the Problem Solving Tactics
tion,

in

A

A

sections of the text.

Preface

iii


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&

Many good people at John Wiley Sons helped put
together y4 Student's Pocket Companion. Among them, Cliff Mills, Joan
Kalkut, Erica Liu, and Rita Kerrigan were instrumental in the conception, design, and production. Catherine Donovan, in her usual efficient


Acknowledgements

manner, carried out a myriad of essential tasks. Stuart Johnson, the current Physics Editor, has supported the latest edition. Monica Stipanov
and Jennifer Bruer have each contributed in a great many ways. I am
grateful to them all. I am also grateful to Karen Christman, who carefully proofread the manuscript. Very special thanks goes to Mary Ellen
Christman, whose support and encouragement seem to know no bound.
J.

Richard Christman

U.S. Coast

Guard Academy

New London, CT

iv

06320

Preface


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TABLE OF CONTENTS
Measurement

1


Chapter!

Motion Along a Straight Line

5

Chapters

Vectors

Chapter 4

Motion

Chapter 5

Force and Motion

Chapter

1

13
in T\vo

and Three Dimensions

21

Chapter 6


—
Force and Motion —

Chapter 7

Kinetic Energy and

Chapter 8

Potential

Chapter 9

Systems of Particles

59

Chapter 10

Collisions

67

Chapter 11

Rotation

73


Chapter 12

Rolling, Tbrque,

Chapter 13

Equilibrium and Elasticity
Gravitation

Chapter 15

Fluids

and Angular

43

Momentum

81

89

Ill

Chapter 18
Chapter 19

Tbmperature, Heat,


Chapter 20

The

Chapter 21

Entropy and the Second

Chapter 22

Electric

Chapter 23

Electric Fields

Chapter 24

Gauss*

and the

I

119

II

127


First

Law of Thermodynamics

Kinetic Theory of Gases

Thermodynamics

Table of Contents

49

95

Oscillations

Waves

of

...

103

—
—
Waves

Chapter 17


37

II

Work

Energy and Conservation of Energy

Chapter 14

Chapter 16

29

I

Charge

Law

135
145

Law
153
161

165
171



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Chapter 25

Electric Potential

179

Chapter 26

Capacitance

187

Chapter 27

Current and Resistance

193

Chapter 28

Circuits

201

Chapter 29

Magnetic Fields


Chapter 30

Magnetic Fields

Chapter 31

Induction and Inductance

Chapter 32

Magnetism of Matter; Maxwell's Equations

Chapter 33

Electromagnetic Oscillations
and Alternating Current

245

Chapter 34

Electromagnetic Waves

255

Chapter 35

Images


265

Chapter 36

Interference

275

Chapter 37

Diffraction

283

Chapter 38

Relativity

291

Chapter 39

Photons and Matter Waves

301

Chapter 40

More About Matter Waves


311

Chapter 41

All

Chapter 42

Conduction of Electricity in Solids

Chapter 43

Nuclear Physics

337

Chapter 44

Energy from the Nucleus

343

Chapter 45

Quarks, Leptons, and the Big Bang

349

vi


207

Due

to Currents

About Atoms

215
221
.

.

.

233

319
329

Table of Contents


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Chapter

1


MEASUREMENT
Physics is an experimental science and relies strongly on accurate measurements of physical quantities. All measurements are comparisons, either direct or indirect, with standards. This

means

that for every quantity

you must not only have a qualitative understanding of what the quantity
represents but also an understanding of how it is measured. A length
measurement is a familiar example. You should know that the length
of an object represents its extent in space and also that length might be
measured by comparison with a meter stick, say, whose length is accurately known in terms of the SI standard for the meter. Make a point of
understanding both aspects of each new quantity as it is introduced.

Important Concepts

n
n
n

unit

standard

base quantity
(base unit, base standard)

n

International System of Units


1-1

n
D
D
D
D

conversion factor

meter
second
kilogram

atomic mass unit

Measuring Things

n A unit

is

a well-defined quantity with which other quantities

are compared in a measurement.
length is the meter, the unit of time
of mass is the kilogram.

n


Some

Examples: the unit of
is

the second, the unit

units are defined in terms of others.

the unit for speed

is

For example,

the meter per second. Others are base

units and are defined in terms of standards. Ideally, a standard should be accessible and invariable.

n A system of units consists of a unit for each physical quantity, organized so that all can be derived from a small number of independent base units.

Chapter 1:

Measurement

1


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1-2

The International System of Units

n

This system

is

called the SI system (previously, the metric

system).

n

The three International System base

units used in

mechan-

ics are:

length:

time:

mass:


D

meter (abbreviation: m)
second (abbreviation: s)
kilogram (abbreviation: kg)

SI prefixes are used to represent powers of ten.
lowing are used the most:

Power of Tbn

Svmbol

kilo:

10^

k

mega:

10^

M

centi:

10-2


c

milli:

10-3

m

micro:

10-6

nano:

10-9

pico:

10-12

Prefix

The

fol-

f^

n
P


Memorize them. When evaluating an algebraic expression,
substitute the value using the appropriate

That

is,

for example,

if

a length

is

power of

ten.

given as 25 /im, sub-

25 X 10-6 m. Qj^g catch: the SI unit for mass is
the kilogram Thus, a mass of 25 kg is substituted directly,

stitute

.

while a mass of 25 g


1-3

substituted as 25 x 10" ^ kg.

Changing Units

n
2

is

There are usually several common units for each physical
quantity. For example, length can be measured in meters,
feet, yards, miles, light years, and other units.
Chapter

1:

Measurement


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n

A quantity given in one unit

is converted to another by mulby a conversion factor. Some conversion factors
are listed in Appendix D of the text.


tiplying

n

Carefully study Section 1-3 to see how a quantity given
one unit is converted to another. Cultivate the good

in

habit of saying the words associated with a conversion. Sup-

pose you want to convert 50 ft to meters. Appendix D tells
that 1ft is equivalent to 0.3048 m. Say "Since 1ft is
equivalent to 0.3048 m, then 50 ft must be equivalent to
(50 ft) X (0.3048 m/ft) = 15 m".

you

1-4

Length

n
n
n

The

SI standard for the meter


light

during a time interval of 1/299, 792, 458s.

the distance traveled by

is

This makes the speed of light exactly 299, 792, 458 m/s.
T^ble 1-3 gives

some

lengths.

Note the wide range of val-

ues.

1-5

Time

n

The

SI standard for the second is the time taken for ex770 vibrations of a certain light emitted by
cesium- 133 atoms.

actly 9, 192, 631,

n

'Rible

1-4

lists

some time

intervals.

Note the wide range of

values.

1-6

Mass

n

The SI standard

for the kilogram

is


the mass of a platinum-

iridium cylinder carefully stored at the International Bureau of Weights and Measures near Paris, France.

n A second mass unit

is the atomic mass unit, abbreviated u
and defined so the mass of a carbon-12 atom is exactly 12 u.

1

n
Chapter 1:

u = 1.6605402 X lO'^^

'Rible

1-5

lists

Measurement

kg.

some masses. Note

the wide range of values.



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NOTES:

Chapter

1:

Measurement


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Chapter 2

MOTION ALONG A STRAIGHT LINE
This chapter introduces you to some of the concepts used to describe motion; most important are those of position, velocity, and acceleration. Pay
particular attention to their definitions and to the relationships between

them.

Important Concepts

n (instantaneous) speed
D average acceleration
n (instantaneous) acceleration
n motion with constant

n particle

n coordinate axis
n origin
n coordinate
D displacement
n average velocity
n average speed
n (instantaneous) velocity
2-1

acceleration

n
n

free-fall acceleration
free-fall

motion

Motion

n

In this section of the

text, objects

are treated as particles.

A particle has no extent in space and has no internal parts

that can

move

relative to

each other.

It

may have

other

properties, such as mass.

An extended object can be treated as a particle if all points
it move along parallel lines. It cannot rotate and it cannot deform. If an extended object can be treated as a par-

in

we may pick one point on the object and follow its
motion. The position of a crate, for example, means the
position of the point on the crate we have chosen to follow,
perhaps one of its corners.

ticle,

Chapter


2:

Motion Along a Straight Line


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2-2

Position and Displacement

n

The motion of a particle in one dimension can be described
by giving its coordinate x as a function of time t. Draw a
coordinate axis along the line of motion of the particle and
select one point on the axis to be the origin. The distance
from the origin to the particle is the magnitude of the coordinate. The coordinate is positive if it is on the side of the
origin designated positive and negative if it is on the side
designated negative.

n

You must

carefully distinguish between an instant of time
and an interval of time. The symbol t represents an instant
and has no extension. Thus, t might be exactly 12 min, 2.43 s
after noon on a certain day. At any other time, no matter
how close, t has a different value. On the other hand, an

interval extends from some initial time to some final time:
two instants of time are required to describe it. Note that a
value of the time may be positive or negative, depending on
whether the instant is after or before the instant designated

as

n

^

=

0.

Similarly, a value of the coordinate x specifies a point

the X

axis. It

on

has no extension in space.

n A displacement

is

a difference in two coordinates. If a par-


goes from xi to X2 during some interval of time, its
displacement during that interval is Ax = X2 - xi. Notice
that the initial coordinate is subtracted from the final coordinate. This definition is valid no matter what the signs of
ticle

XI and X2'

n

particles that start at xj and end at X2 have the same
displacement no matter what their motions. The magnitude of the displacement during a time interval may be different from the distance traveled during the interval. The
difference is pronounced if, for example, the particle moves

TWo

back and forth several times

Chapter

2:

in the interval.

Motion Along a Straight Line


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2-3


Average Velocity and Average Speed

n

If a particle goes from xj at time ^i to X2 at time ^2» its
average velocity v in the interval from
_ _ X2 — XI _ Ax
" A/"'
ti -ti
where

n

If

Ax =

X2

-

xi and

At =

you are given the function

t2

-t-

I.

and are asked for the avfrom ti to /2» first evaluate

x(^)

erage velocity in some interval
the function for < = into the defining equation.

n

On a graph of x vs.
from

^1 to ^2 is

t the average velocity over the interval
given by the slope of the line from ^i, xj to

^2' On the graph below ti = 2.0s and ti = 8.0 s. The
average velocity in this interval is the slope of the dotted

hy

line.

n

A downward sloping line (from left to right) has a negative
slope and indicates a negative average velocity.

An upward

sloping line has a positive slope and indicates a positive velocity.

n

Carefully distinguish between average velocity and average
speed.

The average speed over a time interval At

by
s

=

d

A^
Chapter

2:

Motion Abng a Straight Line

is

defined


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where d

is distance traveled in the interval. This may be
quite different from the displacement if the particle moves

back and forth during the

2-4

interval.

Instantaneous Velocity and Speed

n

The instantaneous

velocity is the velocity at an instant of
time (not over an interval). It is defined as the limiting
value of the average velocity in an interval as the interval
shrinks to zero. The term "velocity" means instantaneous
velocity. The adjective "instantaneous" is implied.

n A positive velocity means that the particle
positive X direction.

that the particle

n

is

is

traveling in the

Similarly, a negative velocity

means

traveling in the negative x direction.

the function x(t) is known, the velocity at any time ti is
found by finding the derivative ofx(t) with respect to t and
evaluating the result for / = ti. The velocity at any time t
is given by
If

On a graph of x vs. t, the velocity at any time ti is the slope
of the straight line that is tangent to the curve at the point
corresponding to ^ = ^i. On the graph below the slope of
the slanted dotted line gives the velocity at time = 7.0 s.

2

4

6

8

10

t(s)

Chapter

2:

Motion Along a Straight Line


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n

The instantaneous speed

(or just plain speed) of a particle
the magnitude of its velocity. If the velocity is -5.0 m/s,
for example, then the speed is +5.0 m/s.

is

2-5

Acceleration

D

If

the velocity of a particle changes from vi at time ^i to V2
its average acceleration a over the

at a later time /2> then

from

interval

ti

tot 2

is

given by

t2-ti
where

At;

=

t;2

-

vi

" a7'

and At = ii-^i- Notice that the veof the interval is subtracted from the

locity at the beginning

velocity at the end. Also notice that instantaneous velocities

appear

n

in the definition.

function x(/) is given, first differentiate it with respect
to / to find the velocity as a function of time, then evaluate
the result for times ti and ^2 to find values for v^ and t;2Substitute the values into the defining equation for the avIf the

erage acceleration.

n

On a
terval

graph of v vs t, the average acceleration over the infrom tiioti'^ the slope of the line from t\yVi ioti,

V2.

n

The instantaneous

acceleration gives the acceleration of
the particle at an instant of time, rather than over an inter-

val. It is

the limiting value of the average acceleration as the

interval

becomes

acceleration"

n

zero. "Acceleration"

mean

the

same

is known, the instantaneous accelerafound by differentiating it with respect to i. If the
function x{t) is known, the instantaneous acceleration is
found by differentiating it twice with respect to i:

If

the function v{i)

tion

is

dv

Chapter

2:

and "instantaneous

thing.

Motion Along a Straight Line

d^x

9


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n

On a graph of v vs. t, the instantaneous acceleration at any
time ti is the slope of the straight line that is tangent to the
curve at/ = ti.

n

Note that a

positive acceleration does not necessarily

the particle speed

mean

increasing and a negative acceleration
does not necessarily mean the particle speed is decreasing.

The speed
the

same

increases

sign

At the
ation

if

the velocity and acceleration have

and decreases

matter what the

n

is

instant a particle

not necessarily

is

if

they have opposite signs, no

signs.

is

0.

momentarily at rest its accelerIf, at an instant, the velocity is

zero but the acceleration is not, then in the next instant the
is not zero and the particle is moving.

velocity

2-6

Constant Acceleration:

n

A Special

Case

moving along the x axis with constant accelcoordinate and velocity are given as functions

If a particle is

eration

a, its

of time

t

=

x(t)

by
XQ

+

VQt

+

and

jat^

v(t)

=

vq

-\-

at

is its coordinate at / =
and vq is its velocity at
Notice that the second equation is the derivative of

where xq
t

=

0.

the first

n

Eq. 2-16 of the text
V
It

n

is

also extremely useful.

=

Vq

-{-

2a{x

—

Xq)

It is

.

gives the velocity as a function of the coordinate.

Problems involving motion with constant acceleration are
worked by solving these equations for the unknowns. Usually two events are described in the problem statement. Select the time to be zero at one of the events, xq is the coordinate of the particle and vq is its velocity then. The other
event occurs at some time t. x is the coordinate and v is the
velocity then. The other quantity that enters is the accela. Of the six quantities that occur in the equations,
four are usually given or implied and two are unknown.

eration

10

Chapter

2:

Motion Along a Straight Line

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2-7

Another Look

Constant Acceleration

n

Integration can be used to derive the equations for motion
with constant acceleration. If a is the acceleration, then the
velocity is given by f (^) = J adt + C = at + Cy where C is a
constant of integration. Its physical meaning is the velocity
in the equation above.
at / = 0. lb see this, just set / =
Thus, C = VQ and v(t) = vq -k- at.

n

The coordinate

C

= / v(t) d< +
= f(vQ +
+ C\ where C' is a constant of
integration. It is the coordinate when ^ = 0. Thus, C' = xq
and x(t) = Xq + 1^0^ + jat^.
at)(lt

2-8

at

+

C

=

given by x(t)

is

VQt

+

^at^

Free-Fail Acceleration

n

Every object near the surface of the Earth, if acted on only
by the gravitational force of the Earth, has the same acceleration, called the free-fall acceleration and denoted by g.
It is downward, toward the Earth. The value of g is independent of the mass or shape of the object. It varies slightly

from place

n

to place

on the Earth.

In the absence of air resistance, a ball (or any other object)

thrown upward has the same acceleration
ing

its

upward

the very top of

n

flight,
its

during

its

at all times: dur-

downward

fall,

and even

at

trajectory.

The constant acceleration equations are
to deal with free-fall motion.

somewhat
drawn with

revised

A vertical y axis

is

the upward direction being positive and the equations are
written
2/(0

and

=

yo

+

^0^

-

v'^

'^(O

J9^^y

= v^-

2g{y

-

= ^0-

9iy

y^).

They can be obtained from the equations given previously
for

motion along an x

axis

by substituting y for x and

-g

for a.

Chapter

2:

Motion Along a Straight Line

11


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n

Problems dealing with free fall are solved in exactly the
as other problems dealing with constant accel-

same way
eration.

2-9

The

Particles of Physics

n

Quantum physics deals with phenomena at the atomic and
subatomic levels. Matter ultimately is not continuous but is
made up of quanta of matter, called particles. Other quantities, such as energy, are also "lumpy" (quantized) at the
atomic level and below.

n

An atom consists of a central, highly compact nucleus, surrounded by one or more electrons and

is

held together by

electrical forces.

n

An atomic nucleus is composed
ticles,

called neutrons,

called protons.

and

The protons

of electrically neutral par-

electrically

charged particles,

repel each other electrically,

neutrons and protons in a nucleus attract each other
The neutrons provide the glue
that holds a nucleus together.

but

all

via a strong nuclear force.

n

Protons, neutrons, and many other particles (but not electrons) are not fundamental but rather are composed of particles known as quarks.

n

Quarks and leptons (of which the electron is one) seem
be the fundamental building blocks of nature.

to

NOTES:

12

Chapter

2:

Motion Along a Straight Line


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Chapter 3

VECTORS
You
ter,

will deal with

you

will learn

mathematically.
dividends later.

A

vector quantities throughout the course. In this chapabout their properties and how they are manipulated
solid understanding of this material will pay handsome

Important Concepts

n vector
n scalar
n component of a vector
n unit vector
n vector addition
n negative of a vector
5-1

n vector subtraction
D multiplication of a vector
by a scalar

D scalar product of two vectors
D vector product of two vectors

Vectors and Scalars

n A vector quantity has a direction as well as a magnitude and
obeys the rules of vector addition (discussed later in this
chapter). In contrast, a scalar quantity has only a magnitude and obeys the rules of ordinary arithmetic. Displacement, velocity, acceleration, and force are vector quantities; mass, speed, charge, and temperature are scalar quantities.

n A vector

is

represented graphically by an arrow in the di-

rection of the vector, with length proportional to the

mag-

nitude of the vector (according to some scale). As an algebraic symbol, a vector is written in boldface (a, for example) or with an arrow over the symbol (a). The magnitude
of a is written a, in italics (or not bold and without an arrow) or as |a| (or \a\). Be sure you follow this convention.

Chapter

3:

Vectors

13


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helps you distinguish vectors from scalars and components of vectors. It helps you communicate properly with
It

your instructors and exam graders.

you mean

a

+

6,

for example.

Do not write a + 6 when

They have

entirely different

meanings!

n

Displacement vectors are used
this chapter.

as

A displacement vector

examples of vectors in
is a vector from the po-

sition of a particle at the beginning of a time interval to

position at the end of the interval.

vector

tells

Note

its

that a displacement

us nothing about the path of the object, only the

relationship between the initial and final positions.

you need an example

When

to illustrate addition or subtraction

of vectors, think of displacement vectors.

3-2

Adding Vectors: Graphical Method

n

If

di

is

and di

.4

to point

the displacement vector from point

B

to point C,

then the

from

n

sum

(written d^

-I-

d2)

is

the displacement vector

A to C:

To add two vectors, place the tail of the second vector at
the head of the first, then draw the vector from the tail of
the first to the head of the second. The order in which you
draw the vectors is not important: a-)-b = b + a. Itis
important that the tail of one be at the head of the other
and that the resultant vector be from the "free" tail to the
"free" head, as in the diagram above.
Except

magnitude of the reof the magnitudes of the vec-

in special circumstances, the

sultant vector

14

B

the displacement vector from point
is

is

not the

sum

Chapter

3:

Vectors


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sum and the direction of the resultant vecnot in the direction of any of the vectors entering the

tors entering the

tor

is

sum.

n

Remember you can reposition a vector as long as you do not
change its magnitude and direction. Thus, if two vectors
you wish to add graphically do not happen to be placed with
the tail of one at the head of the other, simply move one

into the proper position.

n

The

idea of the negative of a vector is used to define vecThe negative of a vector is a vector that is

tor subtraction.

parallel to the original vector but in the opposite direction:

D

Vector subtraction

is

defined by a

-

b

=

a

+

(-b). That

is,

you add a and -b.

n

Notice that vector subtraction is defined so that if a + b = 0,
then a = -b and if c = a + b, then a = c - b. Just subtract
b from both sides of each equation. Vector subtraction is
clearly useful for solving vector equations.

3-3

Components
lb find the x component of a vector, draw lines from the
head and tail to the x axis, both perpendicular to the axis.
The X component of the vector is the projection of the vector on the axis and is indicated by the separation of the
points where the lines meet the axis. Similarly, to find the y
component draw lines from the head and tail to the y axis.
The diagram below shows the components of a vector.

Vectors and Their

n

y
a

Chapter

3:

Vectors

15


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n

The components of a vector are not vectors themselves but
they can be either positive or negative. The vector in the
diagram above has positive x and y components. The vector in the diagram below has a positive x component and a
negative y component.

n

Given the magnitude a of a vector and the angle 6 it makes
axis, you can find the components using

with the X

ax

= acos^

and

= asm

For these expressions to be valid, 6 must be measured counterclockwise from the positive x direction:

and 90°, both the x and y components are
between 90° and 180°, the x component is
negative and the y component is positive; if ^ is between
180° and 270°, both the x and y components are negative;
if 9 is between 270° and 360°, the x component is positive
and the y component is negative.
If 9 is

between

positive; if ^

D

You must

is

be able to find the magnitude and orientawhen you are given its components. Suppose a vector a lies in the xy plane and its components a^
also

tion of a vector

16

Chapter 3:

Vectors


www.pdfgrip.com

and Qy are

given. In terms of the

tude of a

given by

is

a

=

+a^

yja 2

and the angle a makes with the
by
tan^

=

components, the magni

positive x direction

is

given

^.
Clx

The magnitude

always positive. There are two possible

is

solutions to the equation for 9, the one given by your calculator and that angle plus 180°. You must look at the ori-

entation of the vector to see which

makes

physical sense.

Oy/ox = 0.50, then 6 = 26.6° or 206.6°.
In the first case, both components are positive, while in the
second, both are negative.

For example,

3-4

if

Unit Vectors

n

The unit vectors i, j, and k are used when a vector is written
in
1

n

terms of its components. These vectors have magnitude

and are

Ox i
Oy j

is

and

z directions respectively.

a vector parallel to the x axis with x

component

a^,

a vector parallel to the y axis with y component ay,
a^ k is a vector parallel to the z axis with z component

The vector a

is

a

where the

n

t/,

is

and
Gz.

in the positive x,

given by

= Oxi +

fly j

+

flz

k

rules of vector addition apply.

The symbols

t,

j,

and k are used

to

hand write the unit vec-

tors.

n

Chapter

3:

Units are associated with the components Ox, Oy, and a^
of a vector but not with the unit vectors i, j, and k. Thus,
the same unit vectors can be used to write any vector, no
matter what its units.
Vectors

17