Introductory Physics I
Elementary Mechanics
by
Robert G. Brown
Duke University Physics Department
Durham, NC 27708-0305
rgb@phy.duke.edu
Copyright Notice
Copyright Robert G. Brown 1993, 2007, 2013
Notice
This physics textbook is designed to support my personal teaching activities at Duke University, in particular teaching its Physics 141/142, 151/152, or 161/162 series (Introductory Physics for life science majors, engineers, or potential physics majors, respectively). It is freely available in its entirety in a downloadable PDF form or to be read online at:
http://www.phy.duke.edu/ rgb/Class/intro physics 1.php
It is also available in an inexpensive (really!) print version via Lulu press here:
http://www.lulu.com/shop/product-21186588.html
where readers/users can voluntarily help support or reward the author by purchasing either this paper copy or one of the even more inexpensive electronic copies.
By making the book available in these various media at a cost ranging from free to cheap, I enable the text can be used by students all over the world where each student can pay (or not) according to their means.
Nevertheless, I am hoping that students who truly find this work useful will purchase a copy through Lulu or a bookseller (when the latter option becomes available), if only to help subsidize me while I continue to write inexpensive textbooks in physics or other subjects.
This textbook is organized for ease of presentation and ease of learning. In particular, they are hierarchically organized in a way that directly supports e cient learning. They are also remarkably complete in their presentation and contain moderately detailed derivations of many of the important equations and relations from first principles while not skimping on simpler heuristic or conceptual explanations as well.
As a “live” document (one I actively use and frequently change, adding or deleting material or altering the presentation in some way), this textbook may have errors great and small, “stub” sections where I intend to add content at some later time but haven’t yet finished it, and they cover and omit topics according to my own view of what is or isn’t important to cover in a one-semester course. Expect them to change with little warning or announcement as I add content or correct errors.
Purchasers of the paper version should be aware of its probable imperfection and be prepared to either live with it or mark up their copy with corrections or additions as need be. The latest (and hopefully most complete and correct) version is always available for free online anyway, and people who have paid for a paper copy are especially welcome to access and retrieve it.
I cherish good-hearted communication from students or other instructors pointing out errors or suggesting new content (and have in the past done my best to implement many such corrections or suggestions).
Books by Robert G. Brown
Physics Textbooks
•Introductory Physics I and II
A lecture note style textbook series intended to support the teaching of introductory physics, with calculus, at a level suitable for Duke undergraduates.
•Classical Electrodynamics
A lecture note style textbook intended to support the second semester (primarily the dynamical portion, little statics covered) of a two semester course of graduate Classical Electrodynamics.
Computing Books
•How to Engineer a Beowulf Cluster
An online classic for years, this is the print version of the famous free online book on cluster engineering. It too is being actively rewritten and developed, no guarantees, but it is probably still useful in its current incarnation.
Fiction
•The Book of Lilith
ISBN: 978-1-4303-2245-0
Web: http://www.phy.duke.edu/ rgb/Lilith/Lilith.php
Lilith is the first person to be given a soul by God, and is given the job of giving all the things in the world souls by loving them, beginning with Adam. Adam is given the job of making up rules and the definitions of sin so that humans may one day live in an ethical society. Unfortunately Adam is weak, jealous, and greedy, and insists on being on top during sex to “be closer to God”.
Lilith, however, refuses to be second to Adam or anyone else. The Book of Lilith is a funny, sad, satirical, uplifting tale of her spiritual journey through the ancient world soulgiving and judging to find at the end of that journey – herself.
Poetry
•Who Shall Sing, When Man is Gone
Original poetry, including the epic-length poem about an imagined end of the world brought about by a nuclear war that gives the collection its name. Includes many long and short works on love and life, pain and death.
Ocean roaring, whipped by storm in damned defiance, hating hell with every wave and every swell, every shark and every shell
and shoreline.
•Hot Tea!
More original poetry with a distinctly Zen cast to it. Works range from funny and satirical to inspiring and uplifting, with a few erotic poems thrown in.
Chop water, carry wood. Ice all around,
fire is dying. Winter Zen?
All of these books can be found on the online Lulu store here:
http://stores.lulu.com/store.php?fAcctID=877977
The Book of Lilith is available on Amazon, Barnes and Noble and other online bookseller websites.
Contents
Preface |
xi |
Textbook Layout and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
xii |
I: Getting Ready to Learn Physics |
3 |
Preliminaries |
3 |
See, Do, Teach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
3 |
Other Conditions for Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
8 |
Your Brain and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
13 |
How to Do Your Homework E ectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
19 |
The Method of Three Passes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
22 |
Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
23 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
26 |
Homework for Week 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
27 |
II: Elementary Mechanics |
31 |
Week 1: Newton’s Laws |
33 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
33 |
1.1: Introduction: A Bit of History and Philosophy . . . . . . . . . . . . . . . . . . . . . . |
38 |
1.2: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
39 |
1.3: Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
41 |
1.4: Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
46 |
1.5: Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
47 |
1.5.1: The Forces of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
47 |
1.5.2: Force Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
49 |
1.6: Force Balance – Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
51 |
Example 1.6.1: Spring and Mass in Static Force Equilibrium . . . . . . . . . . . . . |
51 |
1.7: Simple Motion in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
52 |
i |
|
ii |
CONTENTS |
Example 1.7.1: A Mass Falling from Height H . . . . . . . . . . . . . . . . |
. . . . . 53 |
Example 1.7.2: A Constant Force in One Dimension . . . . . . . . . . . . . |
. . . . . 58 |
1.7.1: Solving Problems with More Than One Object . . . . . . . . . . . . . |
. . . . . 61 |
Example 1.7.3: Atwood’s Machine . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 61 |
Example 1.7.4: Braking for Bikes, or Just Breaking Bikes? . . . . . . . . . . . |
. . . . 63 |
1.8: Motion in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 64 |
1.8.1: Free Flight Trajectories – Projectile Motion . . . . . . . . . . . . . . . |
. . . . 66 |
Example 1.8.1: Trajectory of a Cannonball . . . . . . . . . . . . . . . . . . . |
. . . . 66 |
1.8.2: The Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 69 |
Example 1.8.2: The Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 69 |
1.9: Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 71 |
1.9.1: Tangential Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 72 |
1.9.2: Centripetal Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 73 |
Example 1.9.1: Ball on a String . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 74 |
Example 1.9.2: Tether Ball/Conic Pendulum . . . . . . . . . . . . . . . . . . |
. . . . 75 |
1.9.3: Tangential Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 76 |
1.10: Conclusion: Rubric for Newton’s Second Law Problems . . . . . . . . . . . . |
. . . . 77 |
Homework for Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 78 |
Week 2: Newton’s Laws: Continued |
95 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 95 |
2.1: Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 97 |
Example 2.1.1: Inclined Plane of Length L with Friction . . . . . . . . . . . . |
. . . . 98 |
Example 2.1.2: Block Hanging o of a Table . . . . . . . . . . . . . . . . . . . |
. . . . 100 |
Example 2.1.3: Find The Minimum No-Skid Braking Distance for a Car . . . |
. . . . 102 |
Example 2.1.4: Car Rounding a Banked Curve with Friction . . . . . . . . . . |
. . . . 104 |
2.2: Drag Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 106 |
2.2.1: Stokes, or Laminar Drag . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 108 |
2.2.2: Rayleigh, or Turbulent Drag . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 109 |
2.2.3: Terminal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 110 |
Example 2.2.1: Falling From a Plane and Surviving . . . . . . . . . . . . . . . |
. . . . 112 |
Example 2.2.2: Solution to Equations of Motion for Stokes’ Drag . . . . . . . |
. . . . 113 |
2.2.4: Advanced: Solution to Equations of Motion for Turbulent Drag . . . . |
. . . . 114 |
Example 2.2.3: Dropping the Ram . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 114 |
2.3: Inertial Reference Frames – the Galilean Transformation . . . . . . . . . . . . |
. . . . 117 |
2.3.1: Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 118 |
2.3.2: Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 118 |
CONTENTS |
iii |
2.4: Non-Inertial Reference Frames – Pseudoforces . . . . . . . . . . . . . . . . . . . . . . |
121 |
2.4.1: Identifying Inertial Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
122 |
Example 2.4.1: Weight in an Elevator . . . . . . . . . . . . . . . . . . . . . . . . . . |
124 |
Example 2.4.2: Pendulum in a Boxcar . . . . . . . . . . . . . . . . . . . . . . . . . . |
125 |
2.4.2: Advanced: General Relativity and Accelerating Frames . . . . . . . . . . . . . |
127 |
2.5: Just For Fun: Hurricanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
129 |
Homework for Week 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
132 |
Week 3: Work and Energy |
141 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
141 |
3.1: Work and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
143 |
3.1.1: Units of Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
145 |
3.1.2: Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
145 |
3.2: The Work-Kinetic Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
146 |
3.2.1: Derivation I: Rectangle Approximation Summation . . . . . . . . . . . . . . . |
146 |
3.2.2: Derivation II: Calculus-y (Chain Rule) Derivation . . . . . . . . . . . . . . . . |
148 |
Example 3.2.1: Pulling a Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
149 |
Example 3.2.2: Range of a Spring Gun . . . . . . . . . . . . . . . . . . . . . . . . . . |
150 |
3.3: Conservative Forces: Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . |
151 |
3.3.1: Force from Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
152 |
3.3.2: Potential Energy Function for Near-Earth Gravity . . . . . . . . . . . . . . . . |
154 |
3.3.3: Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
155 |
3.4: Conservation of Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
156 |
3.4.1: Force, Potential Energy, and Total Mechanical Energy . . . . . . . . . . . . . |
157 |
Example 3.4.1: Falling Ball Reprise . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
157 |
Example 3.4.2: Block Sliding Down Frictionless Incline Reprise . . . . . . . . . . . . |
158 |
Example 3.4.3: A Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . |
158 |
Example 3.4.4: Looping the Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
159 |
3.5: Generalized Work-Mechanical Energy Theorem . . . . . . . . . . . . . . . . . . . . . |
161 |
Example 3.5.1: Block Sliding Down a Rough Incline . . . . . . . . . . . . . . . . . . |
161 |
Example 3.5.2: A Spring and Rough Incline . . . . . . . . . . . . . . . . . . . . . . . |
162 |
3.5.1: Heat and Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . |
162 |
3.6: Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
164 |
Example 3.6.1: Rocket Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
164 |
3.7: Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
165 |
3.7.1: Energy Diagrams: Turning Points and Forbidden Regions . . . . . . . . . . . . |
168 |
Homework for Week 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
170 |
iv |
CONTENTS |
Week 4: Systems of Particles, Momentum and Collisions |
181 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . . 181 |
4.1: Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 185 |
4.1.1: Newton’s Laws for a System of Particles – Center of Mass . . . . . . . |
. . . . 186 |
Example 4.1.1: Center of Mass of a Few Discrete Particles . . . . . . . . . . . |
. . . . 188 |
4.1.2: Coarse Graining: Continuous Mass Distributions . . . . . . . . . . . . . |
. . . . 189 |
Example 4.1.2: Center of Mass of a Continuous Rod . . . . . . . . . . . . . . |
. . . . 191 |
Example 4.1.3: Center of mass of a circular wedge . . . . . . . . . . . . . . . |
. . . . 192 |
Example 4.1.4: Breakup of Projectile in Midflight . . . . . . . . . . . . . . . . |
. . . . 193 |
4.2: Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 194 |
4.2.1: The Law of Conservation of Momentum . . . . . . . . . . . . . . . . . . |
. . . . 194 |
4.3: Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 196 |
Example 4.3.1: Average Force Driving a Golf Ball . . . . . . . . . . . . . . . |
. . . . 198 |
Example 4.3.2: Force, Impulse and Momentum for Windshield and Bug . . . |
. . . . 198 |
4.3.1: The Impulse Approximation . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 199 |
4.3.2: Impulse, Fluids, and Pressure . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 200 |
4.4: Center of Mass Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 202 |
4.5: Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 204 |
4.5.1: Momentum Conservation in the Impulse Approximation . . . . . . . . |
. . . . 204 |
4.5.2: Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 204 |
4.5.3: Fully Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 205 |
4.5.4: Partially Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 205 |
4.5.5: Dimension of Scattering and Su cient Information . . . . . . . . . . . |
. . . . 205 |
4.6: 1-D Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 206 |
4.6.1: The Relative Velocity Approach . . . . . . . . . . . . . . . . . . . . . . |
. . . . 208 |
4.6.2: 1D Elastic Collision in the Center of Mass Frame . . . . . . . . . . . . |
. . . . 209 |
4.6.3: The “BB/bb” or “Pool Ball” Limits . . . . . . . . . . . . . . . . . . . . |
. . . . 211 |
4.7: Elastic Collisions in 2-3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 213 |
4.8: Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 215 |
Example 4.8.1: One-dimensional Fully Inelastic Collision (only) . . . . . . . . |
. . . . 215 |
Example 4.8.2: Ballistic Pendulum . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 217 |
Example 4.8.3: Partially Inelastic Collision . . . . . . . . . . . . . . . . . . . |
. . . . 218 |
4.9: Kinetic Energy in the CM Frame . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 219 |
Homework for Week 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 220 |
Week 5: Torque and Rotation in One Dimension |
235 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 235 |
CONTENTS |
v |
5.1: Rotational Coordinates in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . |
236 |
5.2: Newton’s Second Law for 1D Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . |
238 |
5.2.1: The r-dependence of Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
240 |
5.2.2: Summing the Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . |
242 |
5.3: The Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
243 |
Example 5.3.1: The Moment of Inertia of a Rod Pivoted at One End . . . . . . . . . |
243 |
5.3.1: Moment of Inertia of a General Rigid Body . . . . . . . . . . . . . . . . . . . . |
243 |
Example 5.3.2: Moment of Inertia of a Ring . . . . . . . . . . . . . . . . . . . . . . . |
244 |
Example 5.3.3: Moment of Inertia of a Disk . . . . . . . . . . . . . . . . . . . . . . . |
245 |
5.3.2: Table of Useful Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . |
246 |
5.4: Torque as a Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
246 |
Example 5.4.1: Rolling the Spool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
247 |
5.5: Torque and the Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
248 |
Example 5.5.1: The Angular Acceleration of a Hanging Rod . . . . . . . . . . . . . . |
249 |
5.6: Solving Newton’s Second Law Problems Involving Rolling . . . . . . . . . . . . . . . |
249 |
Example 5.6.1: A Disk Rolling Down an Incline . . . . . . . . . . . . . . . . . . . . . |
250 |
Example 5.6.2: Atwood’s Machine with a Massive Pulley . . . . . . . . . . . . . . . . |
252 |
5.7: Rotational Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
253 |
5.7.1: Work Done on a Rigid Object . . . . . . . . . . . . . . . . . . . . . . . . . . . |
253 |
5.7.2: The Rolling Constraint and Work . . . . . . . . . . . . . . . . . . . . . . . . . |
255 |
Example 5.7.1: Work and Energy in Atwood’s Machine . . . . . . . . . . . . . . . . |
256 |
Example 5.7.2: Unrolling Spool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
257 |
Example 5.7.3: A Rolling Ball Loops-the-Loop . . . . . . . . . . . . . . . . . . . . . |
258 |
5.8: The Parallel Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
259 |
Example 5.8.1: Moon Around Earth, Earth Around Sun . . . . . . . . . . . . . . . . |
261 |
Example 5.8.2: Moment of Inertia of a Hoop Pivoted on One Side . . . . . . . . . . |
261 |
5.9: Perpendicular Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
262 |
Example 5.9.1: Moment of Inertia of Hoop for Planar Axis . . . . . . . . . . . . . . |
264 |
Homework for Week 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
265 |
Week 6: Vector Torque and Angular Momentum |
277 |
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
277 |
6.1: Vector Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
278 |
6.2: Total Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
279 |
6.2.1: The Law of Conservation of Angular Momentum . . . . . . . . . . . . . . . . . |
280 |
6.3: The Angular Momentum of a Symmetric Rotating Rigid Object . . . . . . . . . . . . |
281 |
Example 6.3.1: Angular Momentum of a Point Mass Moving in a Circle . . . . . . . |
283 |
vi |
|
CONTENTS |
Example 6.3.2: Angular Momentum of a Rod Swinging in a Circle . . . . . |
. . . . . 283 |
|
Example 6.3.3: Angular Momentum of a Rotating Disk . . . . . . . . . . . . |
. . . . 284 |
|
Example 6.3.4: Angular Momentum of Rod Sweeping out Cone . . . . . . . . |
. . . . 285 |
|
6.4: Angular Momentum Conservation |
. . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 285 |
Example 6.4.1: The Spinning Professor . . . . . . . . . . . . . . . . . . . . . . |
. . . . 285 |
|
6.4.1: Radial Forces and Angular Momentum Conservation . . . . . . . . . . |
. . . . 286 |
|
Example 6.4.2: Mass Orbits On a String . . . . . . . . . . . . . . . . . . . . . |
. . . . 287 |
|
6.5: Collisions . . . . . . . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 289 |
Example 6.5.1: Fully Inelastic Collision of Ball of Putty with a Free Rod . . . |
. . . . 291 |
|
Example 6.5.2: Fully Inelastic Collision of Ball of Putty with Pivoted Rod . . |
. . . . 294 |
|
6.5.1: More General Collisions . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 296 |
6.6: Angular Momentum of an Asymmetric Rotating Rigid Object . . . . . . . . . |
. . . . 296 |
|
Example 6.6.1: Rotating Your Tires . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 299 |
|
6.7: Precession of a Top . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 300 |
Example 6.7.1: Finding ωp From |
L/ t (Average) . . . . . . . . . . . . . . . |
. . . . 302 |
Example 6.7.2: Finding ωp from |
L and t Separately . . . . . . . . . . . . |
. . . . 302 |
Example 6.7.3: Finding ωp from Calculus . . . . . . . . . . . . . . . . . . . . |
. . . . 303 |
|
Homework for Week 6 . . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 305 |
Week 7: Statics |
|
313 |
Statics Summary . . . . . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 313 |
7.1: Conditions for Static Equilibrium . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 313 |
7.2: Static Equilibrium Problems . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 315 |
Example 7.2.1: Balancing a See-Saw . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 316 |
|
Example 7.2.2: Two Saw Horses . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 317 |
Example 7.2.3: Hanging a Tavern Sign . . . . . . . . . . . . . . . . . . . . . . |
. . . . 318 |
|
7.2.1: Equilibrium with a Vector Torque . . . . . . . . . . . . . . . . . . . . . |
. . . . 319 |
|
Example 7.2.4: Building a Deck . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 320 |
7.3: Tipping . . . . . . . . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 321 |
Example 7.3.1: Tipping Versus Slipping . . . . . . . . . . . . . . . . . . . . . |
. . . . 321 |
|
Example 7.3.2: Tipping While Pushing . . . . . . . . . . . . . . . . . . . . . . |
. . . . 323 |
|
7.4: Force Couples . . . . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 324 |
Example 7.4.1: Rolling the Cylinder Over a Step . . . . . . . . . . . . . . . . |
. . . . 325 |
|
Homework for Week 7 . . . . . . . . . . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 327 |
III: Applications of Mechanics |
|
339 |
Week 8: Fluids |
339 |
CONTENTS |
vii |
Fluids Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
339 |
8.1: General Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
340 |
8.1.1: Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
341 |
8.1.2: Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
343 |
8.1.3: Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
344 |
8.1.4: Viscosity and fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
345 |
8.1.5: Properties Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
345 |
Static Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
346 |
8.1.6: Pressure and Confinement of Static Fluids . . . . . . . . . . . . . . . . . . . . |
346 |
8.1.7: Pressure and Confinement of Static Fluids in Gravity . . . . . . . . . . . . . . |
348 |
8.1.8: Variation of Pressure in Incompressible Fluids . . . . . . . . . . . . . . . . . . |
350 |
Example 8.1.1: Barometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
350 |
Example 8.1.2: Variation of Oceanic Pressure with Depth . . . . . . . . . . . . . . . |
353 |
8.1.9: Variation of Pressure in Compressible Fluids . . . . . . . . . . . . . . . . . . . |
353 |
Example 8.1.3: Variation of Atmospheric Pressure with Height . . . . . . . . . . . . |
354 |
8.2: Pascal’s Principle and Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
355 |
Example 8.2.1: A Hydraulic Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
356 |
8.3: Fluid Displacement and Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
357 |
8.3.1: Archimedes’ Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
359 |
Example 8.3.1: Testing the Crown I . . . . . . . . . . . . . . . . . . . . . . . . . . . |
360 |
Example 8.3.2: Testing the Crown II . . . . . . . . . . . . . . . . . . . . . . . . . . . |
361 |
8.4: Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
363 |
8.4.1: Conservation of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
364 |
8.4.2: Work-Mechanical Energy in Fluids: Bernoulli’s Equation . . . . . . . . . . . . |
367 |
Example 8.4.1: Emptying the Iced Tea . . . . . . . . . . . . . . . . . . . . . . . . . . |
368 |
8.4.3: Fluid Viscosity and Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . |
368 |
8.4.4: A Brief Note on Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
371 |
8.5: The Human Circulatory System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
372 |
Example 8.5.1: Atherosclerotic Plaque Partially Occludes a Blood Vessel . . . . . . . |
376 |
Example 8.5.2: Aneurisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
377 |
Example 8.5.3: The Gira e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
378 |
Homework for Week 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
379 |
Week 9: Oscillations |
389 |
Oscillation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
389 |
9.1: The Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
390 |
9.1.1: The Archetypical Simple Harmonic Oscillator: A Mass on a Spring . . . . . . |
391 |
viii |
CONTENTS |
9.1.2: The Simple Harmonic Oscillator Solution . . . . . . . . . . . . . . . . |
. . . . . 396 |
9.1.3: Plotting the Solution: Relations Involving ω . . . . . . . . . . . . . . . |
. . . . 397 |
9.1.4: The Energy of a Mass on a Spring . . . . . . . . . . . . . . . . . . . . . |
. . . . 398 |
9.2: The Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 398 |
9.2.1: The Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 400 |
9.3: Damped Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 402 |
9.3.1: Properties of the Damped Oscillator . . . . . . . . . . . . . . . . . . . . |
. . . . 404 |
Example 9.3.1: Car Shock Absorbers . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 406 |
9.4: Damped, Driven Oscillation: Resonance . . . . . . . . . . . . . . . . . . . . . |
. . . . 407 |
9.4.1: Harmonic Driving Forces . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 409 |
9.4.2: Solution to Damped, Driven, Simple Harmonic Oscillator . . . . . . . . |
. . . . 411 |
9.5: Elastic Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 414 |
9.5.1: Simple Models for Molecular Bonds . . . . . . . . . . . . . . . . . . . . |
. . . . 415 |
9.5.2: The Force Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 417 |
9.5.3: A Microscopic Picture of a Solid . . . . . . . . . . . . . . . . . . . . . . |
. . . . 418 |
9.5.4: Shear Forces and the Shear Modulus . . . . . . . . . . . . . . . . . . . |
. . . . 420 |
9.5.5: Deformation and Fracture . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 421 |
9.6: Human Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 423 |
Example 9.6.1: Scaling of Bones with Animal Size . . . . . . . . . . . . . . . |
. . . . 425 |
Homework for Week 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 427 |
Week 10: The Wave Equation |
435 |
Wave Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 435 |
10.1: Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 436 |
10.2: Waves on a String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 437 |
10.3: Solutions to the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 439 |
10.3.1: An Important Property of Waves: Superposition . . . . . . . . . . . . |
. . . . 439 |
10.3.2: Arbitrary Waveforms Propagating to the Left or Right . . . . . . . . . |
. . . . 439 |
10.3.3: Harmonic Waveforms Propagating to the Left or Right . . . . . . . . |
. . . . 439 |
10.3.4: Stationary Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 440 |
10.4: Reflection of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 442 |
10.5: Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 443 |
Homework for Week 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 446 |
Week 11: Sound |
457 |
Sound Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 457 |
11.1: Sound Waves in a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 459 |
11.2: Sound Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
. . . . 460 |
CONTENTS |
ix |
11.3: Sound Wave Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
460 |
11.3.1: Sound Displacement and Intensity In Terms of Pressure . . . . . . . . . . . . |
461 |
11.3.2: Sound Pressure and Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . |
463 |
11.4: Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
465 |
11.4.1: Moving Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
465 |
11.4.2: Moving Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
466 |
11.4.3: Moving Source and Moving Receiver . . . . . . . . . . . . . . . . . . . . . . . |
467 |
11.5: Standing Waves in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
467 |
11.5.1: Pipe Closed at Both Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
467 |
11.5.2: Pipe Closed at One End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
468 |
11.5.3: Pipe Open at Both Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
469 |
11.6: Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
470 |
11.7: Interference and Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
470 |
11.8: The Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
472 |
Homework for Week 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
475 |
Week 12: Gravity |
483 |
Gravity Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
483 |
12.1: Cosmological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
487 |
12.2: Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
491 |
12.2.1: Ellipses and Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
492 |
12.3: Newton’s Law of Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
494 |
12.4: The Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
500 |
12.4.1: Spheres, Shells, General Mass Distributions . . . . . . . . . . . . . . . . . . . |
501 |
12.5: Gravitational Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
502 |
12.6: Energy Diagrams and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
503 |
12.7: Escape Velocity, Escape Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
504 |
Example 12.7.1: How to Cause an Extinction Event . . . . . . . . . . . . . . . . . . |
505 |
12.8: Bridging the Gap: Coulomb’s Law and Electrostatics . . . . . . . . . . . . . . . . . |
506 |
Homework for Week 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
507 |
x |
CONTENTS |
Preface
This introductory mechanics text is intended to be used in the first semester of a two-semester series of courses teaching introductory physics at the college level, followed by a second semester course in introductory electricity and magnetism, and optics. The text is intended to support teaching the material at a rapid, but advanced level – it was developed to support teaching introductory calculus-based physics to potential physics majors, engineers, and other natural science majors at Duke University over a period of more than thirty years.
Students who hope to succeed in learning physics from this text will need, as a minimum prerequisite, a solid grasp of basic mathematics. It is strongly recommended that all students have mastered mathematics at least through single-variable di erential calculus (typified by the AB advanced placement test or a first-semester college calculus course). Students should also be taking (or have completed) single variable integral calculus (typified by the BC advanced placement test or a second-semester college calculus course). In the text it is presumed that students are competent in geometry, trigonometry, algebra, and single variable calculus; more advanced multivariate calculus is used in a number of places but it is taught in context as it is needed and is always “separable” into two or three independent one-dimensional integrals.
Many students are, unfortunately weak in their mastery of mathematics at the time they take physics. This enormously complicates the process of learning for them, especially if they are years removed from when they took their algebra, trig, and calculus classes (as is frequently the case for pre-medical students taking the course in their junior year of college). For that reason, a separate supplementary text intended specifically to help students of introductory physics quickly and e ciently review the required math is being prepared as a companion volume to all semesters of introductory physics. Indeed, it should really be quite useful for any course being taught with any textbook series and not just this one.
This book is located here:
http://www.phy.duke.edu/ rgb/Class/math for intro physics.php
and I strongly suggest that all students who are reading these words preparing to begin studying physics pause for a moment, visit this site, and either download the pdf or bookmark the site.
Note that Week 0: How to Learn Physics is not part of the course per se, but I usually do a quick review of this material (as well as the course structure, grading scheme, and so on) in my first lecture of any given semester, the one where students are still finding the room, dropping and adding courses, and one cannot present real content in good conscience unless you plan to do it again in the second lecture as well. Students greatly benefit from guidance on how to study, as most enter physics thinking that they can master it with nothing but the memorization and rote learning skills that have served them so well for their many other fact-based classes. Of course this is completely false – physics is reason based and conceptual and it requires a very di erent pattern of study than simply staring at and trying to memorize lists of formulae or examples.
Students, however, should not count on their instructor doing this – they need to be self-actualized in their study from the beginning. It is therefore strongly suggested that all students read this
xi
xii |
CONTENTS |
preliminary chapter right away as their first “assignment” whether or not it is covered in the first lecture or assigned. In fact, (if you’re just such a student reading these words) you can always decide to read it right now (as soon as you finish this Preface). It won’t take you an hour, and might make as much as a full letter di erence (to the good) in your final grade. What do you have to lose?
Even if you think that you are an excellent student and learn things totally e ortlessly, I strongly suggest reading it. It describes a new perspective on the teaching and learning process supported by very recent research in neuroscience and psychology, and makes very specific suggestions as to the best way to proceed to learn physics.
Finally, the Introduction is a rapid summary of the entire course! If you read it and look at the pictures before beginning the course proper you can get a good conceptual overview of everything you’re going to learn. If you begin by learning in a quick pass the broad strokes for the whole course, when you go through each chapter in all of its detail, all those facts and ideas have a place to live in your mind.
That’s the primary idea behind this textbook – in order to be easy to remember, ideas need a house, a place to live. Most courses try to build you that house by giving you one nail and piece of wood at a time, and force you to build it in complete detail from the ground up.
Real houses aren’t built that way at all! First a foundation is established, then the frame of the whole house is erected, and then, slowly but surely, the frame is wired and plumbed and drywalled and finished with all of those picky little details. It works better that way. So it is with learning.
Textbook Layout and Design
This textbook has a design that is just about perfectly backwards compared to most textbooks that currently cover the subject. Here are its primary design features:
•All mathematics required by the student is reviewed in a standalone, cross-referenced (free) work at the beginning of the book rather than in an appendix that many students never find.
•There are only twelve chapters. The book is organized so that it can be sanely taught in a single college semester with at most a chapter a week.
•It begins each chapter with an “abstract” and chapter summary. Detail, especially lecture-note style mathematical detail, follows the summary rather than the other way around.
•This text does not spend page after page trying to explain in English how physics works (prose which to my experience nobody reads anyway). Instead, a terse “lecture note” style presentation outlines the main points and presents considerable mathematical detail to support solving problems.
•Verbal and conceptual understanding is, of course, very important. It is expected to come from verbal instruction and discussion in the classroom and recitation and lab. This textbook relies on having a committed and competent instructor and a sensible learning process.
•Each chapter ends with a short (by modern standards) selection of challenging homework problems. A good student might well get through all of the problems in the book, rather than at most 10% of them as is the general rule for other texts.
•The problems are weakly sorted out by level, as this text is intended to support non-physics science and pre-health profession students, engineers, and physics majors all three. The material covered is of course the same for all three, but the level of detail and di culty of the math used and required is a bit di erent.
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•The textbook is entirely algebraic in its presentation and problem solving requirements – with very few exceptions no calculators should be required to solve problems. The author assumes that any student taking physics is capable of punching numbers into a calculator, but it is algebra that ultimately determines the formula that they should be computing. Numbers are used in problems only to illustrate what “reasonable” numbers might be for a given realworld physical situation or where the problems cannot reasonably be solved algebraically (e.g. resistance networks).
This layout provides considerable benefits to both instructor and student. This textbook supports a top-down style of learning, where one learns each distinct chapter topic by quickly getting the main points onboard via the summary, then derives them or explores them in detail, then applies them to example problems. Finally one uses what one has started to learn working in groups and with direct mentoring and support from the instructors, to solve highly challenging problems that cannot be solved without acquiring the deeper level of understanding that is, or should be, the goal one is striving for.
It’s without doubt a lot of work. Nobody said learning physics would be easy, and this book certainly doesn’t claim to make it so. However, this approach will (for most students) work.
The reward, in the end, is the ability to see the entire world around you through new eyes, understanding much of the “magic” of the causal chain of physical forces that makes all things unfold in time. Natural Law is a strange, beautiful sort of magic; one that is utterly impersonal and mechanical and yet filled with structure and mathematics and light. It makes sense, both in and of itself and of the physical world you observe.
Enjoy.
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CONTENTS |
I: Getting Ready to Learn Physics
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