Essential Trig-based Physics Study Guide Workbook: Waves, Fluids, Sound, Heat, and Light

422 Pages·2017·10.898 MB·English

by  Chris McMullen

Essential Trig-based Physics Study Guide Workbook
Volume 3: Waves, Fluids, Sound, Heat, and Light
Learn Physics Step-by-Step

Chris McMullen, Ph.D.
Physics Instructor
Northwestern State University of Louisiana
Copyright © 2017 Chris McMullen, Ph.D.
www.monkeyphysicsblog.wordpress.com
www.improveyourmathfluency.com
www.chrismcmullen.wordpress.com
Zishka Publishing
All rights reserved.

ISBN: 978-1-941691-18-2

Textbooks > Science > Physics
Study Guides > Workbooks> Science
CONTENTS
Introduction 5
Chapter 1 – Sine Waves 7
Chapter 2 – Simple Harmonic Motion 25
Chapter 3 – Oscillating Spring 49
Chapter 4 – Oscillating Pendulum 57
Chapter 5 – Wave Motion 69
Chapter 6 – Doppler Effect and Shock Waves 81
Chapter 7 – Standing Waves 91
Chapter 8 – Density 105
Chapter 9 – Pressure 111
Chapter 10 – Archimedes’ Principle 117
Chapter 11 – Fluid Dynamics 129
Chapter 12 – Pascal’s Law 137
Chapter 13 – Temperature Conversions 141
Chapter 14 – Thermal Expansion 147
Chapter 15 – Heat Transfer 153
Chapter 16 – Ideal Gases 173
Chapter 17 – The Laws of Thermodynamics 179
Chapter 18 – Heat Engines 199
Chapter 19 – Light Waves 217
Chapter 20 – Reflection and Refraction 223
Chapter 21 – Snell’s Law 231
Chapter 22 – Total Internal Reflection 239
Chapter 23 – Dispersion and Scattering 245
Chapter 24 – Thin Lenses 255
Chapter 25 – Spherical Mirrors 279
Chapter 26 – Single-slit Diffraction 297
Chapter 27 – Diffraction Grating 303
Chapter 28 – Double-slit Interference 307
Chapter 29 – Polarization 313
Hints, Intermediate Answers, and Explanations 321
EXPECTATIONS
•
Prerequisites:
As this is the third volume of the series, the student should already have studied the
material from first-semester physics, including motion, applications of Newton’s
•
second law, conservation of energy, Hooke’s law, and rotation.
The student should know some basic algebra skills, including how to combine like
terms, how to isolate an unknown, how to solve the quadratic equation, and how to
•
apply the method of substitution. Needed algebra skills were reviewed in Volume 1.
The student should have prior exposure to trigonometry. Essential trigonometry
skills were reviewed in Volume 1.
•
Use:
This book is intended to serve as a supplement for students who are attending
•
physics lectures, reading a physics textbook, or reviewing physics fundamentals.
The goal is to help students quickly find the most essential material.
•
Concepts:
Each chapter reviews relevant definitions, concepts, laws, or equations needed to
•
understand how to solve the problems.
This book does not provide a comprehensive review of every concept from physics,
but does cover most physics concepts that are involved in solving problems.
•
Strategies:
Each chapter describes the problem-solving strategy needed to solve the problems
•
at the end of the chapter.
This book covers the kinds of fundamental problems which are commonly found in
standard physics textbooks.
•
Help:
Every chapter includes representative examples with step-by-step solutions and
explanations. These examples should serve as a guide to help students solve similar
•
problems at the end of each chapter.
Each problem includes the main answer(s) on the same page as the question. At the
back of the book, you can find hints, intermediate answers, directions to help walk
you through the steps of each solution, and explanations regarding common issues
that students encounter when solving the problems. It’s very much like having your
own physics tutor at the back of the book to help you solve each problem.
INTRODUCTION
The goal of this study guide workbook is to provide practice and help carrying out essential
problem-solving strategies that are standard in waves, fluids, sound, heat, and light. The
aim here is not to overwhelm the student with comprehensive coverage of every type of
problem, but to focus on the main strategies and techniques with which most physics
students struggle.
This workbook is not intended to serve as a substitute for lectures or for a textbook,
but is rather intended to serve as a valuable supplement. Each chapter includes a concise
review of the essential information, a handy outline of the problem-solving strategies, and
examples which show step-by-step how to carry out the procedure. This is not intended to
teach the material, but is designed to serve as a time-saving review for students who have
already been exposed to the material in class or in a textbook. Students who would like
more examples or a more thorough introduction to the material should review their lecture
notes or read their textbooks.
Every exercise in this study guide workbook applies the same strategy which is solved
step-by-step in at least one example within the chapter. Study the examples and then
follow them closely in order to complete the exercises. Many of the exercises are broken
down into parts to help guide the student through the exercises. Each exercise tabulates
the corresponding answers on the same page. Students can find additional help in the hints
section at the back of the book, which provides hints, answers to intermediate steps,
directions to walk students through every solution, and explanations regarding issues that
students commonly ask about.
Every problem in this book can be solved without the aid of a calculator. You may use
a calculator if you wish, though it is a valuable skill to be able to perform basic math
without relying on a calculator.
The better you truly understand the underlying concepts,
the easier it becomes to solve the physics problems.

— Chris McMullen, Ph.D.
Essential Trig-based Physics Study Guide Workbook
1 SINE WAVES

Relevant Terminology
Crest – a point on a wave where there is a maximum (the top of a hill).
Trough – a point on a wave where there is a minimum (the bottom of a valley).
Equilibrium – a horizontal line about which a sine wave oscillates.
Amplitude – the vertical height between a crest and the equilibrium position (or one-half of
the vertical height between a crest and trough).
Period – the horizontal time between two consecutive crests. Period is time it takes for the
wave to complete exactly one oscillation.
Frequency – the number of oscillations completed per second.
Angular frequency – the number of radians completed per second. The angular frequency
equals 2𝜋𝜋 times the frequency.
Phase angle – an angle representing how much the wave is shifted horizontally. A positive
phase angle represents a wave that is shifted to the left.
Symbols and SI Units

Symbol Name SI Units
𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 the maximum value of 𝑥𝑥 (at a crest) m
𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 the minimum value of 𝑥𝑥 (at a trough) m
𝑥𝑥𝑒𝑒 the equilibrium position m
𝑥𝑥0 the initial position m
𝑥𝑥 the instantaneous position m
𝐴𝐴 the amplitude of oscillation m
𝑡𝑡 time s
𝑇𝑇 period s
𝛿𝛿𝑡𝑡 the phase shift in seconds s
𝜑𝜑 phase angle rad
𝑓𝑓 frequency Hz
𝜔𝜔0 angular frequency rad/s
7
Chapter 1 – Sine Waves
Sine Wave Equations

In physics problems, a sine wave can generally be expressed as
𝑥𝑥 = 𝐴𝐴sin(𝜔𝜔0𝑡𝑡+𝜑𝜑)+𝑥𝑥𝑒𝑒
•
where the symbols represent the following:
•
𝑥𝑥 is the position of the oscillating object.
•
𝑥𝑥𝑒𝑒 is the equilibrium position (how much the graph is shifted vertically).
•
𝐴𝐴 is the amplitude of the sine wave.
•
𝑡𝑡 is time.
•
𝜔𝜔0 is the angular frequency in radians per second.
𝜑𝜑 is the phase angle (how much the graph is shifted horizontally).
The amplitude (𝐴𝐴) is the vertical distance between the maximum value (𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚) and the
equilibrium position (𝑥𝑥𝑒𝑒), or one-half of the vertical distance between the maximum value
(𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚) and the minimum value (𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚).
𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 −𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚
𝐴𝐴 = 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 −𝑥𝑥𝑒𝑒 =
2
The angular frequency (𝜔𝜔0) equals 2𝜋𝜋 times the frequency (𝑓𝑓).
𝜔𝜔0 = 2𝜋𝜋𝑓𝑓
Frequency (𝑓𝑓) and period (𝑇𝑇) share a reciprocal relationship.
1 1
𝑓𝑓 = , 𝑇𝑇 =
𝑇𝑇 𝑓𝑓
•
There are three ways to determine the phase angle (𝜑𝜑) for a graph of a sine wave:
Set 𝑡𝑡 equal to zero and algebraically solve for 𝜑𝜑. You will get an inverse sine:
−1 𝑥𝑥0 −𝑥𝑥𝑒𝑒
𝜑𝜑 = sin � �
𝐴𝐴
It’s important to realize that the answer may lie in any of the four Quadrants. (It
may help to review the trigonometry essentials from Chapter 7 of Volume 1.) The
method of the following bullet point will help you determine the correct Quadrant.
•
Note that 𝑥𝑥0 is the initial position: It’s the value of 𝑥𝑥 for which 𝑡𝑡 = 0.
You can estimate phase angle visually by drawing 0°, 90°, 180°, 270°, and 360° in
the usual positions and extrapolating to 𝑡𝑡 = 0, as discussed on pages 10-11 and as
shown in one of the examples. It’s wise to do this in addition to the inverse sine
•
from the first bullet point.
When there is a graph, you could measure how much the sine wave is shifted
horizontally in seconds, 𝛿𝛿𝑡𝑡, and use the following equation.
𝜑𝜑 𝛿𝛿𝑡𝑡
=
2𝜋𝜋 𝑇𝑇

o
It’s important to note that 𝛿𝛿𝑡𝑡 may be negative.

o
If you measured a shift to the right, 𝛿𝛿𝑡𝑡 is negative.
If you measured a shift to the left, 𝛿𝛿𝑡𝑡 is positive.
8
Essential Trig-based Physics Study Guide Workbook
Essential Concepts

Consider the general sine wave plotted below. It corresponds to the following equation.
𝑥𝑥 = 𝐴𝐴sin(𝜔𝜔0𝑡𝑡+𝜑𝜑)+𝑥𝑥𝑒𝑒
period (𝑇𝑇)
𝑥𝑥 (m)
maximum (𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚)
amplitude (𝐴𝐴)
equilibrium (𝑥𝑥𝑒𝑒)
initial position (𝑥𝑥0)
minimum (𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚)
𝑡𝑡 (s)

•
The diagram above visually demonstrates the following quantities:
The period (𝑇𝑇) is the horizontal time (not distance! – note that the horizontal axis
has 𝑡𝑡, not 𝑥𝑥) between two crests. Note: Remember to look at the crests (the peaks
at the top) and not to look along the equilibrium line. (Students who measure
points along the equilibrium line instead of between two crests often make a
•
mistake and measure one-half of the period.)
The equilibrium position (𝑥𝑥𝑒𝑒) is the vertical value on the graph, which corresponds
•
to a horizontal line that bisects the sine wave between its crests and troughs.
•
The crests correspond to 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚, which designates the highest points.
•
The troughs correspond to 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚, which designates the lowest points.
The amplitude (𝐴𝐴) is the vertical distance between each crest and equilibrium (or
one-half of the vertical distance from crest to trough).
𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 −𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚
𝐴𝐴 = 𝑥𝑥𝑚𝑚𝑚𝑚𝑚𝑚 −𝑥𝑥𝑒𝑒 =
2
•
The following quantities can also be determined from the graph:
1
•
Find the frequency (𝑓𝑓) from the period (𝑇𝑇): 𝑓𝑓 = 𝑇𝑇.
•
Find the angular frequency (𝜔𝜔0) from the frequency (𝑓𝑓): 𝜔𝜔0 = 2𝜋𝜋𝑓𝑓.
Find the phase angle (𝜑𝜑) from an inverse sine:
−1 𝑥𝑥0 −𝑥𝑥𝑒𝑒
𝜑𝜑 = sin � �
𝐴𝐴
Be sure to pick the correct Quadrant, as discussed on pages 10-11. Note that 𝑥𝑥0 is
the initial position: It’s the vertical intercept of the graph.
9
Chapter 1 – Sine Waves
Phase Angle
The phase angle (𝜑𝜑), represented by a lowercase Greek phi, provides a measure of how
•
much a wave is shifted horizontally compared to a standard sine wave.
•
A positive phase angle represents a wave that is shifted to the left.
A negative phase angle represents a wave that is shifted to the right.
𝑥𝑥 (m) 𝑥𝑥 (m) 𝑥𝑥 (m)
positive 𝜑𝜑 𝜑𝜑 = 0 negative 𝜑𝜑
shifts left no shift shifts right
𝑡𝑡 (s) 𝑡𝑡 (s) 𝑡𝑡 (s)

Why does a positive phase angle shift the graph to the left? You can see this by considering
2
a more familiar function, the basic parabola. The equation 𝑦𝑦 = 𝑥𝑥 is an unshifted parabola.
2
If you graph the equation 𝑦𝑦 = (𝑥𝑥+2) , you will get the same shape, but the positive 2 shifts
the entire graph to the left. Why? Because 𝑦𝑦 = 0 when 𝑥𝑥 = −2 (since that’s the value of 𝑥𝑥
2
that makes 𝑥𝑥+2 equal zero). If you graph the equation 𝑦𝑦 = (𝑥𝑥−2) , you again get the
same shape, but the negative 2 shifts the entire graph to the right. It works the same way
with the phase angle and sine waves.
𝑦𝑦 𝑦𝑦 𝑦𝑦
𝑥𝑥 𝑥𝑥 𝑥𝑥
2 positive 2 2 2 negative 2
𝑦𝑦 = (𝑥𝑥+2) 𝑦𝑦 = 𝑥𝑥 no shift 𝑦𝑦 = (𝑥𝑥−2)
shifts left shifts right
In order to figure out which Quadrant the phase angle lies in, label the angles 0°, 90°, 180°,
270°, and 360° on the sine wave. These are labeled below for a basic sine wave.
90°
I II
180°
0° 360°
III IV
270°

10