Physics: Calculus

1336 Pages·1996·107.123 MB·English

by  Eugene Hecht

PHYSICS
C a lc u lu s
H E C H T
UNIT CONVERSIONS
Length Force
1 in. = 2.540 cm 1 N = 0.224 8 lb
1 cm = 0.393 7 in. 1 lb = 4.448 N
1 ft = 30.48 cm
Energy
1 m = 3.281 ft
1 mi = 5280 ft = 1.609 km 1 J = 0.7376 ft lb
1 km = 0.621 4 mi 1 ft·lb = 1.36 J = 1.29 X 10~3 Btu = 3.24 X 10“4 kcal
1 A = 10'10 m = 0.10 nm 1 kcal = 4.186 kJ
1 light-year (ly) = 9.461 X 1015 m 1 Btu = 1055 J
1 kWh = 3.600 MJ = 860 kcal
Time
1 eV = 1.60218 X 10~19 J
1 d = 1.44 X 103 min 1 MeV = 1.60218 X ΚΓ13 J
1 y = 365.24 d = 3.156 X 107 s 1 megaton = 4 X 1015 J
Speed Power
1 mi/h = 1.467 ft/s = 1.609 km/h = 0.447 0 m/s 1 W = 1 J/s = 0.7376 ft-lb/s = 1.341 X ΚΓ3 hp
1 km/h = 0.277 8 m/s = 0.6214 mi/h = 0.9113 ft/s 1 hp = 550 ft-lb/s = 745.7 W
1 ft/s = 0.304 8 m/s = 0.6818 mi/h
Pressure
1 m/s = 3.281 ft/s = 3.60 km/h
1 Pa = 1 N/m2 = 1.450 X 10"4 lb/in.2
Volume
1 atm = 0.101325 MPa = 14.7 lb/in.2
1 cm3 = 0.061 in.3 1 lb/in.2 = 6.90 X 103 N/m2
1 liter = 1000 cm3 = 10”3 m3 = 0.035 3 ft3 1 mm Hg = 1.333 X 102 Pa
1 in.3 = 16.4 cm3 1 in. Hg = 3.386 X 103 Pa
1 ft3 = 28.3 X 103 cm3
Angle
1 m3 = 1 X 106 cm3 = 61 X 103 in.3
1 rad = 57.30°
Mass
1° = 0.01745 rad
1 kg = 1000 g 360° = 2π rad
1 atomic mass unit (u) = 1.660 540 X 10-27 kg 180° = 77 rad
[1 kg weighs 2.20 lb where g — 9.80 m/s2]
SI DERIVED UNITS SI PREFIXES
Quantity Unit Equivalents Power Prefix Symbol
Force newton N J/m kg-m/s2 1012 tera T
Energy joule J N-m kg-m2/s2 109 giga G
Power watt W J/s kg-m2/s3 106 mega M
Pressure pascal Pa N/m2 kg/m-s2 103 kilo k
Frequency hertz Hz cycle/s s-‘ 102 hecto h
Electric charge coulomb C A-s 10“‘ deci d
Electric potential volt V J/C kg-m2/A-s3 10^2 centi c
Electric resistance ohm Ω V/A kg-m2/A2-s3 10~3 milli m
Capacitance farad F C/V AV/kg-m2 10"6 micro μ­
Magnetic field tesla T N-s/C-m kg/A-s2 10“9 nano η
Magnetic flux weber Wb Tm2 kg-m2/A-s2 10~12 pico P
Inductance henry H V-s/A kg-m2/A2-s2 10“15 femto f
MATHEMATICAL REVIEW
Area of a circle of radius R A = 7tR2
Circumference of a circle C = 2ttR
Surface area of a sphere A = 47tR1
Volume of a sphere V = f trtf3
Area of a triangle A = \bh
Volume of a circular
cylinder of length l V = 77 R2l
Pythagorean Theorem C2 =-- A2 + B2
sin Θ = A/C
sin Θ
cos Θ = B/C tan Θ =
cos t)
tan θ — A/B
Quadratic Equation: Where ax2 + bx + IIo
—b ± Vb2 — 4ac
2 a
PHYSICAL CONSTANTS
Quantity Symbol Value
Gravitation constant G 6.672 59 X 10"11 N-m2/kg:
Speed of light in vacuum c 2.997 924 58 X 10s m/s
Electron charge e 1.60218 X 10~19C
Planck’s Constant h 6.626076 X 10"34 J-s
4.135 669 X 10"15 eV-s
Universal gas constant R 8.314510 J/mol-K
Avogadro’s number na 6.022 137 X 1023 mol-1
Boltzmann Constant 1.38066 X 10'23 J/K
8.617 39 X 10'5 eV/K
Coulomb force constant k0 8.987 55 X 109 N-m2/C2
Permittivity of free space (l/u0c2) 8.854 19 X 10"12 C2/N-m2
to
Permeability of free space 1.25664 X 10"6 T-m/A
Mo
Permeability constant Mo/4tt 10-7 T-m/A
Electron mass me 9.109 39 x 10'31 kg
Electron rest energy me c2 0.510 999 MeV
Electron magnetic moment 9.284 77 X 10"24J/T
M=
Electron charge/mass ratio e/me 1.758 82 X 1011 C/kg
Electron Compton wavelength 2.426 31 X 10"12 m
Ac
Proton mass mp 1.672 623 X 10“27 kg
1.007 276 u
Proton rest energy m„c2 938.272 MeV
Proton magnetic moment 1.410 608 X 10"26 J/T
R-p
Neutron mass m„ 1.674929 X 10"27 kg
1.008 66 u
Neutron rest energy mnc2 939.566 MeV
Neutron magnetic moment 9.662 37 X 10"27J/T
Bohr magneton Ms 9.274015 X 10"24J/T
Stefan-Boltzmann Constant σ 5.670 51 X 10-8 W/m2-K4
Rydberg constant R 1.097 373 X 107 m"1
Bohr radius U 5.29177 X 10“u m
Faraday constant F 9.648 53 X 104 C/mol

PHYSICS
Calculus
EUGENE HECHT
Adelphi University
Brooks/Cole Publishing Company
l(T)P
An International Thomson Publishing Company
Pacific Grove · Albany · Bonn · Boston · Cincinnati · Detroit ·
London · Madrid · Melbourne · Mexico City · New York · Paris ·
San Francisco · Singapore · Tokyo · Toronto · Washington
Brooks/Cole Publishing Company
A Division of International Thomson Publishing Inc.
Sponsoring Editor: Harvey Pantzis Permissions Editor: May Clark &
Marketing: Connie Jirovsky & Margaret Jennifer Burke
Parks Interior & Cover Design: E. Kelly
Editorial Associate: Beth Wilbur Shoemaker & Vernon T. Boes
Production Editor: Ellen Brownstein Photo Coordination & Digital Photo
Production Service: HRS Electronic Text Design: Larry Molmud & HRS Electronic
Management Text Mgmt.
Manuscript Editor: Monique Condon Photo Researcher: Stuart Renter &
Art Coordinator: HRS Electronic Text Mgmt. Carolyn Hecht
Interior Illustration: Precision Graphics, Typesetting: Beacon Graphics
Carl Brown, LM Graphics, Matrix Cover Printing: Lehigh Press
Communications, and HRS Electronic Lithographers
Text Management Printing and Binding: Quebecor/Hawkins
Cover Photo: Tom Skrivan
COPYRIGHT © 1996 by Eugene Hecht in any and all countries.
For more information, contact:
BROOKS/COLE PUBLISHING COMPANY International Thomson Editores
511 Forest Lodge Road Seneca 53
Pacific Grove, CA 93950 Col. Polanco
USA 11560 Mexico, D. F., Mexico
International Thomson Publishing Europe International Thomson Publishing GmbH
Berkshire House 168-173 Konigswinterer Strasse 418
High Holborn 53227 Bonn
London WC1V 7AA Germany
England
International Thomson Publishing Asia
Thomas Nelson Australia 221 Henderson Road
102 Dodds Street #05-10 Henderson Building
South Melbourne, 3205 Singapore 0315
Victoria, Australia
International Thomson Publishing Japan
Nelson Canada Hirakawacho Kyowa Building, 3F
1120 Birchmount Road 2-2-1 Hirakawacho
Scarborough, Ontario Chiyoda-ku, Tokyo 102
Canada M1K 5G4 Japan
All rights reserved. No part of this work may be reproduced, stored in a retrieval
system, or transcribed, in any form or by any means—electronic, mechanical,
photocopying, recording, or otherwise—without the prior written permission of the
publisher, Brooks/Cole Publishing Company, Pacific Grove, California 93950.
Printed in the United States of America
10 9 8 7 6 5 4 3 2
Library of Congress Cataloging-In-Publication Data
Hecht, Eugene.
PHYSICS: calculus / Eugene Hecht.
p. cm.
Includes index.
ISBN 0-534-33985-9
L Physics. 2. Calculus. 3. Mathematical physics. I. Title.
QC21.2.H43 1996 95-37110
530-dc20 CIP
To Ca, b. w. I.

Preface
Physics is the study of the material Universe—all there is. And that’s a bold and
wonderful agenda. The Universe is incredibly awesome and tantalizingly mysteri­
ous, and we, after all, are just beginning to understand it. Almost 3000 years in the
making, physics—incomplete as it is—stands as one of the great creations of the
human intellect. It has been a privilege and an unending joy to have spent much of
my life studying physics, and it is out of gratitude and admiration that this book
takes its form. If this work, while insightfully teaching basic physics, transmits a
sense of the grandeur, unity, and vitality of the subject, it will have met my primary
objectives.
TO THE STUDENT
This book has been designed for the calculus-based Introductory Physics course,
and it contains the standard range of material from kinematics to quantum mechan­
ics. It is predicated, however, on the belief that it’s time to return to fundamentals;
today’s texts have become too mathematical and too advanced. By contrast, this
work limits the associated math to basic calculus and very basic vector analysis.
It omits a variety of obscure high-level topics and instead uses its facilities to trans­
mit a deeper understanding of the fundamental concepts of modern-day physics. It
covers all the grand insights, but with a self-restraint that stops short of examining
every possible side issue. As a result the book can progress at a slower pace and pro­
vide much more support in the process.
The text assumes that the student comes with only a modest knowledge of alge­
bra, geometry, and trigonometry. It presumes that the reader is now studying or has
already studied calculus but retains little more than a cloudy memory of that experi­
ence. Of course, the derivative of x2 equals 2x, even though one may not remember
exactly what a derivative is. In short, whatever mathematics is required will be
retaught, in place, as the need arises and is further elaborated in an extensive
tutorial appendix. It is strongly suggested that the math appendix be reviewed
before starting the course.
TO THE PROFESSOR
Back to Basics
Over the last five decades the Introductory Physics course and the texts that support
it have undergone a dramatic transformation. Prior to the Second World War a typi­
cal text dealt almost entirely with the discussion of concepts and principles; there
were comparatively few equations, little analysis, and a small selection of straight­
forward problems. The idea of a vector quantity was introduced, but only in a
qualitative way that allowed for the addition of forces. The mathematics was algebra
with a touch of trigonometry—no calculus, no vector analysis. The war brought
physics into a new social prominence, and even before it ended, calculus was al­
ready finding its way, if only tentatively, into the introductory course. By the mid-
1950s the distinction had been established between so-called College Physics
(algebra/trig. based) and University Physics (calculus based). Standard treatises
(e.g., University Physics by Sears and Zemansky, 2nd edition, 1955) fully embraced
calculus but contained no rigorous discussion of vectors. The rudiments of vector
analysis entered the introductory discourse only in the early 1960s.
Driven by a seemingly endless supply of high-quality students of engineering
and science, University Physics became increasingly more sophisticated mathemati­
cally, even as it remained philosophically naive and minimally developed conceptu­
ally. The tension between competing texts (Physics for Students of Science and
Engineering by Halliday and Resnick, 1960) and a boundless optimism in the teach­
ing community reshaped the next generation of books and the courses that used
them. By 1970 serious difficulties were already cropping up and there was a token
reduction in the “level of sophistication” (Fundamentals of Physics by Halliday and
Resnick, 1st edition), but even the revised works were doing vector calculus (with i,
j, k vectors) by Chapter 3. Texts ranged well beyond the once-traditional bounds of
the curriculum, treating numerous subtleties (e.g., the Poynting vector, the
Maxwell-Boltzmann distribution, and the power associated with a wave on a
string) that had previously always been dealt with in higher level courses.
Today’s established texts are the end product of that rush toward analytic and
computational prowess that rolled on, untempered and unchallenged, across the
1970s and 1980s. A quick thumb through any of these works will reveal a splendid
but dauntingly unintegrated minicompendium of contemporary physics. Scattered
among the foundational concepts (and undifferentiated from them) one can find a
plethora of such unlikely items as the gradient operator in all its glory, a derivation
of the partial differential wave equation, the relativistic Doppler effect, an analysis
of forced oscillations, and the quantum mechanical treatment of a particle in a well.
Any first-year graduate student should be pleased to have mastered the scope and
depth of the material now proffered as Introductory Physics.
This state of affairs is only acceptable for a core cadre of highly motivated
gifted students of the physical sciences. The clientele of the 1990s is much broader
than that and these people are not at all well served by the standard texts of the day.
Indeed, for many students these books are simply overwhelming, and that puts inor­
dinate demands on the professor. From their earliest chapters these texts require an
extensive familiarity with vector analysis (based on i, j, k vectors); something few
undergraduates have at the time. Thus, at the very beginning of the experience,
when students are most vulnerable, and when they should be concentrating on
learning how to learn physics, they are made to deal with the unnecessary burden
of mastering mathematical machinery, albeit elegant machinery, that they can
easily do without. Moreover, the standard texts seem to make the assumption
that someone who is taking, or has taken, a course in calculus has learned cal­
culus. With little or no additional preparation, they swiftly go on to incorporate
vector calculus, something quite unheard of, even for physics majors, thirty or forty
years ago.
There should be a realistic alternative, especially one that is thoroughly mod­
ern, philosophically mature, and pedagogically effective. And that’s what this book
is all about. Where appropriate, it restrains the mathematics and limits the range
of the discourse so that it can focus on, elaborate, and teach the fundamentals of
physics. Calculus and vector analysis are both painstakingly developed as tools and