University Physics. Arfken Griffing Kelly Priest

328 Pages·1984·17.68 MB·English

by  T. William Houk

Study Guide
to accompany
University Physics
Arfken Griffing Kelly Priest
T William Houk
James Poth
John W Snider
Miami University Oxford, Ohio
Academic Press, Inc.
(Harcourt Brace Jovanovich, Publishers)
Orlando San Diego San Francisco New York
London Toronto Montreal Sydney Tokyo Säo Paulo
Copyright © 1984 by Academic Press, Ine
All rights reserved
No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher
Academic Press, Ine
Orlando, Florida 32887
United Kingdom edition published by Academic Press, Ine (London) Ltd
24/28 Oval Road, London NWl 7DX
ISBN: 0-12-059868-X
Printed in the United States of America
Preface to the Student
OBJECTIVES OF THE STUDY GUIDE
This study guide is a companion volume to UNIVERSITY PHYSICS by Arfken, Griffing, Kelly, and
Priest. We have adhered both to the style and format of the text. Extensive reference to the text
as well as use of the same titles, section headings, and equation numbers should make the study
guide simple to use as an auxiliary to the text. This study guide has been created as a supplement
to the text and not as a replacement for it. We have relied on our experience to provide concise
summaries of the textual material, added information, example problems, and additional exercises.
Our coverage of the material has been selective rather than exhaustive. We have stressed
topics that we judge to be those that are the most difficult either conceptually or operationally
for you to understand. We have provided examples which we believe illustrate the principles of the
physics involved and we have worked them out fully with comments which should clarify the
principles as well as the analytical and synthetic methods employed to solve problems. We have
also included a number of supplementary problems which should help you to solidify your grasp of
the material covered in the text and the study guide. Answers to these problems are included in an
appendix at the end of the study guide so that you may check your work.
Problem solving is an important feature of the text. Accordingly, this study guide is strongly
problem oriented. Perhaps the most important aspect of your introduction to physics is the
development of the ability to analyze problems and synthesize their solutions. Students often feel
that the answer justifies the means. Nothing is further from the truth. Although obtaining the
correct answer to a problem should not be minimized, the primary goal of your efforts should be to
understand the material well enough that you can be successful at developing methods to solve the
problems presented to you. The outlines of various methods for attacking problems are presented in
the text as well as in the study guide. These methods are standard but they cannot replace the
ability to think analytically. Much factual material that you learn in this course will be
repeated, albeit at a more sophisticated level, in coursework which you will take as you progress
through your college careers. However the ability to use this material effectively must begin to
be developed at this level.
ORGANIZATION OF THE STUDY GUIDE
Chapter titles in this study guide coincide with those in the text. Each chapter of the study
guide consists of four major parts:
PREVIEW - This brief section describes the major ideas to be covered in the
chapter as well as the goals of the chapter.
SUMMARY - This section presents a brief synopsis of the material presented
in the text. It is broken into subsections that are numbered and
vii
titled identically to those in the text. Equations in this section
are also numbered as they are in the text. Although the summaries
are primarily synoptic in character, we have added material, defi­
nitions, special notes, and hints to clarify, elaborate, or emphasize
the textual material.
EXAMPLE PROBLEMS - This section contains problems chosen to illustrate both
the physical principles presented in the chapter and the methods used
to solve these problems. We have provided complete solutions to these
problems and have included extensive commentary in order that you may
understand clearly why we have done what is done to solve the problem.
PROBLEMS - We have presented in this section a number of problems for you to
work independently in addition to those which you may find in the text.
We have tried to choose problems that reinforce and extend the concepts
and methods in the Example Problems section.
SUGGESTIONS FOR USING THE STUDY GUIDE.
It is important that you realize that the primary source for the material which you will be
studying and learning in this course is your text. Ancillaries, such as this study guide, are
meant to aid you in your use of the text and in understanding what is in it. We would like to make
a number of suggestions regarding your study of Physics and the appropriate use of this book.
The lecture-demonstration method is probably the most common teaching method used in physics
courses. In order that you may derive the maximum benefit from the lectures which you will attend,
it is imperative that you read ahead. Students often make the mistake of thinking that sitting
and listening to a lecture over material they have not encountered before will transfer
sufficient information for them to understand the material. In Physics, this is rarely the case.
Lectures are meant to clarify the material you have studied and to answer questions that you may
have about it. So expend every effort to keep up with your assigned reading.
The second important activity you should undertake is to carefully study the examples that are
worked in your text and in the study guide. It is very important that you study them with the
objective of understanding not only what operations have been done to solve the problem but why
they are done.
Next, you must try to apply your new knowledge to solving problems. We suggest that you first
perform the problems assigned to you from the text and then turn your attention to those in the
Study Guide.
Finally, the study guide can serve as an excellent review and practice to prepare you for
examinations. The chapter summaries serve as good reviews of the material covered, and the
unworked problems can provide you with practice in problem solving.
Good Luck.
ACKNOWLEDGEMENTS
We express our sincere appreciation to Jane Kelly for her outstanding work in the preparation
of this manuscript and thank Jeff Holtmeier of Academic Press for his help throughout the project.
We are especially grateful to our families for their understanding and encouragement.
viii
Chapter 1
General Introduction
PREVIEW
This chapter introduces the concepts upon which the quantitative nature of physics as a science
depends. The types of quantities with which physics deals are defined and their nature is
discussed. The concepts of units and dimensions are introduced and discussed.
SUMMARY
1.1 THE DEVELOPMENT OF SCIENCE
The primary characteristic of a science is that it seeks to discover the interrelationships
which exist between quantifiable properties of systems. In some sciences the properties of the
relevant systems cannot be quantified numerically, but must be quantified descriptively. Much of
the social sciences as well as some branches of the natural sciences are in this category. The
science of physics, which seeks to discover the relationships between physically observable
quantities, is one in which numerical values can and are assigned to observed quantities. Many of
the areas which physics addresses are listed in Table 1.1 of the text.
1.2 SCIENCE AND MEASUREMENT
In physics the quantities with which we most often deal are operationally defined. This means
that there must be an explicit definition for each quantity which specifies how this quantity is to
be measured. The stated measurement process provides an operational definition of the desired
quantity. j
In physics as in most of the natural sciences and mathematics there are two types of
quantities.
Definition. Fundamental quantities: those which form the primary set of quantities
on which all others are based or in terms of which all others can be defined.
In order to not to have to resort to circular definitions, as does a dictionary, the actual
meanings of fundamental quantities cannot be expressed in terms of other quantities but must rely
on a mutually agreed upon understanding of what these quantities are.
As an example of this process we turn to plane geometry. In plane geometry one often accepts
the "point" as a mutually understood fundamental quantity. From this beginning one can then
construct other quantities.
Definition. Derived quantities: those which are defined in terms of
fundamental quantities.
In geometry the line defined as a collection of points is a derived quantity, assuming our
choice of the point as the fundamental quantity.
1
2 Chapter 1
The choice of a set of fundamental quantities is not unique. In our geometry we could have
chosen the line as our fundamental "undefinable" quantity and then defined a point (now a derived
quantity) as the intersection of two lines.
Since fundamental quantities must be agreed upon by all using them and to some extent
understood without formal definition, the minimum number which can suffice to provide a complete
description of an area of science is chosen. We also try to choose for that set a group which have
yery clear perceptual meanings.
Currently there are seven fundamental quantities employed in physics. These quantities are
those defined in the System International d'Unites (SI units) for mass, length, time, temperature,
number of particles in one mole of a substance, current, and luminous intensity. We consider only
the first three of these at this time since they suffice for a complete set for mechanics which is
the first area of physics which we will study.
1.3 LENGTH
The original metric standard of length is a platinum-iridium bar housed at the International
Bureau of Weights and Measures in France. Since this standard for the meter is difficult to use
and reproduce, the current definition of the SI meter is that 1 meter is the distance traveled by
light in vacuum during 1/299792458 of a second (this definition was adopted on October 30, 1983).
The extremely large range of many physical measurements requires the use of scientific notation and
a conventional set of prefixes has been developed to be used with it. These prefixes are listed in
Table 1.3 and should be committed to memory.
1.4 TIME
The SI unit of time (as well as the unit in most other systems of measurement) is the second.
Currently the second is defined as the duration of 9,191,631,770 periods of the microwave radiation
emitted by cesium-133.
1.5 MASS
Of the three quantities which we address in this chapter, mass is the most difficult to
establish conceptually. Although it is proper to say that the mass of an object reflects the
amount of matter which it contains, this is a circular definition since at this point in our
development we do not have a definition of matter. A truly satisfactory development of this
concept must wait until we have developed some of the tools of mechanics which will allow us to
describe inertial mass and gravitational mass. Inertial mass reflects the resistance of an object
to acceleration. Gravitational mass is related to the gravitational interaction between objects.
Currently the standard for mass is a platinum-iridium cylinder housed at the International
Bureau of Weights and Measures. This standard kilogram is currently the only artificial non-atomic
based standard in use. Although it would be preferable to have an easily accessible atomic
standard for mass, the current state of technology makes such a standard difficult to achieve.
1.6 DIMENSIONS AND UNITS
Dimensions are a way of describing the qualitative nature of a physical quantity.
Chapter 1 3
Definition. Dimension or dimensions of a quantity impart the relationship
between that quantity and the concept of the fundamental quantities from which
they are derived.
As an example all quantities which measure length, such as a meter, a yard, a mile, etc. are
said to have dimensions of length. Hours, seconds, years, days, on the other hand, have dimensions
of time.
Examples of derived quantities' dimensions are those for velocity or speed [length/time],
3 2
density [mass/length ], force in pounds or Newtons [mass x length/time ]. Dimensional
statements are usually expressed using L, T, and M to denote length, time, and mass respectively.
Dimensions:
Length = L
Time = T
Mass = M
Thus the dimensions of force can be written as ML/T and those of speed as L/T.
All relationships in physics must be stated between quantities which have the same dimensions.
This is reminiscent of the statement made to you when you were learning arithmetic that you cannot
add apples and pears. One way of checking all of your work in problems is to make sure that your
expressions are dimensionally correct.
Definition. Units: a concrete expression of a quantity's measure based
upon a set of standards which define the system of units employed.
Examples of the units employed to measure the quantities we have discussed above are for
speed: meters/sec, miles/hr, kilometers/hr; for length: meters, feet, miles, kilometers, light
years; for force: pounds, newtons; and for density: kilograms/cubic meter, slugs/cubic foot.
One of the most useful techniques you should learn is the conversion of units from one set to
another. This is accomplished by multiplying the original quantity by appropriate sets of
conversion factors each of which is dimensionless and has magnitude one. Two examples are provided
in the text and others are given below.
EXAMPLE PROBLEMS
Example 1
One of the most important quantities in physics is energy. The kinetic energy (or energy of
2
motion) of an object is defined as (l/2)mv , where m stands for mass and v for speed.
a) Is energy a fundamental or derived quantity? b) What are its dimensions? c) What are its
units in the SI system?
Solution
a) Since the definition of energy involves the derived quantity of speed, it is also a derived
quantity. b) Using M for mass, L for length, and T for time and denoting the operation of
determining the dimensions of a quantity by square brackets [ ], we see:
[d/2)mv2] = MU/T)2 = ML2/T2
c) In the SI system the units are:
M = kg L=m T = sec. (or s)
4 Chapter 1
2 2
therefore the units of energy are kg m /s . Note that the dimensionless quantity 1/2 is
ignored in determining the dimensions of energy.
Example 2
Poiseuille's equation for the change in pressure of a viscous fluid flowing through a pipe of
3
radius R and length ι at a rate of Q ft /s is:
P = (8Q TI*,)/ R4
where η is the viscosity. What are the dimensions of viscosity?
Solution
2
In the United States we normally measure pressure in units of lbs/in . The left hand side of
2
Poiseuille's equation has the same dimensions as Force/Length . From our discussion in the
2
Summary the dimensions of force are ML/T .
[P] = (ML/T2)/L2 = M/LT2
The right hand side of this equation must also have these dimensions.
[Q] = L3/T [1] = L [R] = L
Thus the dimensions of can be calculated as:
[(8Q ηΑ)/ R4] = [η] X [Ql/R4]
= [η] x (L 3/T)(L)/L4
M/LT2 = [ η] x (1/T)
and therefore the dimensions of η are:
[ η] = M/LT
Example 3
International track and field events use SI units. What is the distance of the Olympic
marathon which is 40 km in miles?
Solution / / / / i ll I
40 km = (40 m (1000j//U) (100 J\M) (1 i//2.54 ψ) (1 fit/12 i/) (1 mi/5280 Jt) = 24.85 mi.
PROBLEMS
1. The force on a certain object obeys the equation F = -kx where x is the position of the object.
What are the dimensions of k?
2. What are the dimensions of area and volume?
4 2
3. In most of the world land area is measured in hectares (1 hectare =10 m ). How many
2
acres (1 acre = 43,560 ft ) are there in a hectare?
3 3
4. A litre is 1000 cm and a gallon is 231 in . How many litres are there in one quart?
5. The speed of light is now defined to be 299,792,458 m/s. How fast is this in mi/s and mi/hr?
Chapter 2
Vector Algebra
PREVIEW
In this chapter the concepts of scalars and vectors are introduced and the rules for
performing mathematical operations such as addition, subtraction, and multiplication on vector
quantities are explained. These rules are compared with the more familiar rules for operating on
scalar quantities. Two different but equivalent techniques are described for adding and
subtracting vectors, and situations in which each is useful are exemplified.
SUMMARY
2.1 SCALARS AND VECTORS
Of the concepts which have been developed to describe natural phenomena, some have
intrinsically associated with them a direction while others do not. The former are defined as
vector concepts or quantities and require a number (magnitude), a unit, and a direction in order
to be completely described. The latter are defined as scalar quantities and require only a
number and a unit for a complete description.
Definition. Vector: a quantity which has associated with it both a magnitude
and a direction.
Definition. Scalar: a quantity which has associated with it only a magnitude.
Examples of vector quantities are velocity, force, and magnetic field, while examples of scalar
quantities are mass, time, and energy. Vector quantities are identified in print by using an arrow
above the symbol, or by using a wavy underline, or by BOLDFACE type. Thus, the equation,
A + B = C
means that the vector A is to be added according to the rules explained in the next paragraph,
to the vector B to obtain the vector C.
2.2 ADDITION AND SUBTRACTION OF VECTORS
Vector quantities can best be visualized by
representing them with arrows, the length of
the arrow representing the magnitude of the
quantity and the orientation of the arrow
representing the direction associated with the
quantity. Two like quantities (i.e., both
forces or both velocities) may be added together
FIG. 1
by first drawing the vectors representing the
two quantities with the tail end of the second
5
6 Chapter 2
arrow beginning at the arrowhead end of the first one. The sum or resultant of the two vectors
is then represented by the arrow drawn from the tail of the first arrow to the head of the second.
This rule may be extended to any number of vectors, each succeeding arrow being laid off from the
head of the preceding one; the resultant of the several vectors is then represented by the arrow
drawn from the tail of the first arrow to the head of the last. Fig. 1 illustrates this method of
adding two vectors.
Subtraction of two or more vectors may be accomplished by simply reversing the direction of the
vector(s) to be subtracted and applying the rule outlined on the previous page for addition.
A - B = A + (-B)
WARNING: It must be remembered that, just as in the case of scalar quantities, two or more
vectors representing different concepts cannot be added or subtracted.
Any vector may be multipled or divided by a scalar. The only change resulting from such
multiplication or division is a change in the magnitude of the vector, the direction of the vector
being unaffected. As in the case of the multiplication of two scalar quantities, the vector and
the scalar need not represent the same concept, and the dimensions of the resulting vector will be
the same as the product of the dimensions of the two quantities which were multiplied together.
2.3 COMPONENTS
A procedure which is essentially the reverse of that outlined above for the addition of two or
more vectors is called the resolution of a vector into its components. The components of a
vector are those two or more vectors which, when added to each other, will yield the original
vector as a resultant. While it is not a part of the above definition, it is usually convenient to
require that these components be mutually perpendicular to each other so that they may be
described in terms of a coordinate system with each component lying parallel to one of the axes of
the coordinate system. Such a coordinate system with the three axes mutually perpendicular is
called an orthogonal coordinate system.
Definition. Component: the projection of a vector on one of the coordinate
axes, there being one component corresponding to each of the three axes of the
coordinate system. These components when added together vectorially yield as a
resultant the original vector.
The components of a vector can be found by use of the trigonometric functions. Thus, as shown
in equations (2.9a) and (2.9b) in the text, the components of a vector corresponding to any one of
the axes of a coordinate system may be found by
multiplying the magnitude of the vector by the
cosine of the angle between the vector and that
axis. This same procedure may be used to find
the component of a vector along any direction
whether that direction corresponds to one of
the axes of a coordinate system or not.
Note that any of the components of a vector
may be either positive or negative; the
FIG.
direction along a coordinate axis which is