Table Of ContentAn Advanced Introduction
to Calculus-Based Physics
(Mechanics)
Chris McMullen, Ph.D.
Northwestern State University of Louisiana
An Advanced Introduction
to Calculus-Based Physics (Mechanics)
Copyright © 2012, 2014, 2017 Chris McMullen, Ph.D.
All rights are reserved. This includes the right to
reproduce any portion of this book in any form.
Chris McMullen, Ph.D.
Physics Instructor
Department of Mathematics and Physical Sciences
Northwestern State University of Louisiana
http://mathematics.nsula.edu/faculty
[email protected]
FIRST EDITION, May, 2012
FIRST E-BOOK EDITION, July, 2014
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Contents
Introduction 6
How to Succeed in Physics 11
Suggestions for Reading This Book 16
Which Material Is Practical, Which Is
19
Abstract?
Units, Dimensions, and Significant Figures 24
What Is Physics? 37
1 Single-Component Motion 40
1.1 Basic 1D Motion Quantities 42
1.2 Net and Average Values in 1D 49
1.3 Instantaneous Values in 1D 59
1.4 Calculus Applied to 1D Motion 66
1.5 Single-Component Motion Graphs 71
1.6 Uniform Acceleration in 1D 97
1.7 Vertical Free Fall 116
1.8 Non-Uniform Acceleration in 1D 130
1.9 Multiple Moving Objects in 1D 139
2 Multi-Component Motion 169
2.1 Scalars and Vectors 171
2.2 Vector Addition and Subtraction 195
2.3 Vector Motion Quantities 215
2.4 Polar and Cylindrical Coordinates 240
2.5 Spherical Coordinates 249
2.6 Projectile Motion 257
2.7 Multi-Dimensional Motion 282
3 Newton’s Laws of Motion 304
3.1 Newton’s Laws 309
3.2 Applying Newton’s Second Law 328
3.3 Mass and Weight 338
3.4 Tension Forces 348
3.5 Normal and Friction Forces 361
3.6 Hooke’s Law 375
3.7 Mutual Surface Forces 393
3.8 Air Resistance 403
3.9 Gravitational Forces 414
4 Uniform Circular Motion 466
4.1 Basic Rotational Quantities 467
4.2 Centripetal Acceleration 475
4.3 Centripetal Forces 484
4.4 Swinging in a Circle 491
4.5 Rounding a Turn 503
4.6 Satellite Motion 513
5 Work and Energy 536
5.1 The Scalar Product 539
5.2 Work and Power 545
5.3 Potential and Kinetic Energy 571
5.4 Conservation of Energy 588
5.5 Conservative Force Fields 614
5.6 Nonconservative Work 630
6 Systems of Objects 667
6.1 Center of Mass 668
6.2 Momentum and Impulse 699
6.3 Conservation of Momentum 709
6.4 Motion of the Center of Mass 718
6.5 Collisions in 1D 730
6.6 Collisions in 2D 751
6.7 The Ballistic Pendulum 769
6.8 Scattering 774
iv
6.9 Rocket Motion 784
7 Rotation 822
7.1 Angular Acceleration 826
7.2 Moment of Inertia 843
7.3 The Vector Product 874
7.4 Torque 907
7.5 Summing the Torques 917
7.6 Static Equilibrium 939
7.7 Rotational Kinetic Energy 953
7.8 Angular Momentum 974
7.9 Conservation of Angular Momentum 981
Appendix A: Propagation of Errors 1032
Appendix B: Linear Regression 1043
Hints 1053
Selected Answers 1195
About the Author 1236
v
Introduction
I was fortunate to teach a unique calculus-based physics
course for several years. Two features made this course
very unique: The audience included many advanced
students, and the first semester of the course had a
Calculus II corequisite. The norm for calculus-based
physics at colleges and universities is a Calculus I
corequisite. This means that my students were fluent in
differentiation and basic integration at the beginning of
the course, instead of just learning these skills during the
course.
The unique corequisite of Calculus II proved to be
meaningful in several ways. For one, the students were
eager for me to incorporate a high level of calculus in my
lectures from Day 1. They thrived on the higher-level
math skills, both in the lectures and on the homework.
Once I realized this, I applied calculus as often as possible
in my lectures and examples, and also embedded it as a
standard part of the homework, quiz, and exam problems.
Standard physics textbooks, however, expect the course
to have a Calculus I corequisite, which means that I had to
create much of my own material in order to fully utilize
their calculus skills during the first semester.
One of the major advantages of having a Calculus II
corequisite on the first-semester calculus-based physics
course has to do with the transition to the second
semester. Traditionally, calculus plays a minor role in the
first semester of calculus-based physics, where students
An Advanced Introduction to Calculus-Based Physics (Mechanics)
are just learning calculus along with the course, but
suddenly becomes a major component of the second
semester of calculus-based physics – especially, in the
mathematical techniques of electricity and magnetism.
The sudden change in the level of mathematics makes
this abrupt transition very challenging. When students
have a Calculus II corequisite during the first semester,
however, and get the opportunity to apply their calculus
skills as a regular part of the lectures and coursework,
most of the students are already comfortable with this
level of mathematics when the second semester begins.
This makes for a much smoother transition. For example,
if the students perform rigorous center of mass and
moment of inertia integrals in cylindrical and spherical
coordinates, using the methods of trig substitution
learned in Calculus II, these skills carry right into the
electric field integrals that are standard in electricity and
magnetism.
I find, also, that students have a better opportunity to
fully grasp the precise meaning of the concepts when
there is a higher math prerequisite. One of my top
students called my approach “mathematically-motivated
physics concepts” – and she was right on the button of
exactly what I strive to do. I emphasize that students are
most likely to make mistakes – or be totally in the dark –
trying to reason concepts out using their intuition, but can
usually use a precise definition, a mathematical equation,
or a physical law to reason concepts out correctly. After a
student answers a conceptual question incorrectly during
7
Introduction
a class discussion, I often tell the class to let the equations
be their guide. The equations convey the concepts most
precisely when we express them with more eloquent
mathematics. Therefore, by requiring the students to
have higher mathematics skills to get into the course and
by expressing the mathematics at as high a level as
possible, they have the best opportunity to learn the
concepts most precisely, and to understand more subtle
points.
Having taught such a unique calculus-based physics
course, I saw how it could potentially be beneficial to
students learning calculus-based physics to have a text
that incorporated a higher level of calculus in the
beginning and which emphasized the connections
between the mathematics and the concepts. This could
be useful for advanced students who have a strong
background in mathematics. It could also serve as a useful
reference for physics majors and engineers who have
gone beyond the first year of physics, but who would like
to review the fundamentals as they explore more
advanced fields of physics.
Since there are very few first-year physics courses that
place a Calculus II corequisite on first-semester physics, I
do not expect this text to be adopted as a required
textbook. There are, however, many very bright students
taking or reviewing physics who may benefit from a text
that incorporates a higher level of calculus, emphasizes
connections between formal mathematics and precise
concepts, and was written not for a general audience, but
8
An Advanced Introduction to Calculus-Based Physics (Mechanics)
specifically for advanced students. For these reasons, I
have prepared this text for independent use by bright,
motivated students who have already completed Calculus
I, who may be learning physics for the first time or who
are reviewing the fundamentals. No prerequisite physics
knowledge is assumed of the reader.
The following features of this book are intended to
help students who are using this book independently to
improve their understanding of physics. There are
separate sections of hints and answers at the back of the
book for the end-of-chapter exercises. It is highly
recommended that you try to solve a variety of problems
to assess your understanding of the material. The hints
section, which you won’t find in standard textbooks, is
specifically designed to help independent learners who
might get stuck on a problem. I hope that you appreciate
that 100% of the conceptual questions and problems have
hints. Also, 100% of the conceptual questions have
answers, since it’s very important to understand the
concepts well if you want to become adept at solving the
problems. Some of the problems have answers so that
you can gain some confidence by reproducing the same
answers, but even those problems that don’t have
answers do have hints to help you see if you are solving
them correctly. The examples were selected based on
their instructiveness in elucidating important concepts or
illustrating how to carry out important problem-solving
strategies; quality was favored over quantity. Simple plug-
and-chug examples and problems are scarce, since the
9
Introduction
audience for this book is advanced, independent
students. Several notes highlight common mistakes to
help you learn from mistakes that many other students
have made without having to suffer through them
yourself.
May you develop a passion for physics – whether you
love it or hate it (or love to hate it or hate to love it), may
you be passionate toward it (after all, there’s just a simple
minus sign difference between the two). But please don’t
be indifferent about the subject.
Chris McMullen, Ph.D.
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