Introductory Physics I - Duke University
Introductory Physics IElementary MechanicsbyRobert G. BrownDuke University Physics DepartmentDurham, NC 27708-0305rgb@phy.duke.edu
Copyright NoticeCopyright Robert G. Brown 1993, 2007, 2013
NoticeThis physics textbook is designed to support my personal teaching activities at DukeUniversity, 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.phpIt is also available in an inexpensive (really!) print version via Lulu press lwhere readers/users can voluntarily help support or reward the author by purchasingeither 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 tocheap, I enable the text can be used by students all over the world where each student canpay (or not) according to their means.Nevertheless, I am hoping that students who truly find this work useful will purchasea copy through Lulu or a bookseller (when the latter option becomes available), if onlyto help subsidize me while I continue to write inexpensive textbooks in physics or othersubjects.This textbook is organized for ease of presentation and ease of learning. In particular, they are hierarchically organized in a way that directly supports efficient learning.They are also remarkably complete in their presentation and contain moderately detailedderivations of many of the important equations and relations from first principles while notskimping on simpler heuristic or conceptual explanations as well.As a “live” document (one I actively use and frequently change, adding or deletingmaterial or altering the presentation in some way), this textbook may have errors greatand small, “stub” sections where I intend to add content at some later time but haven’t yetfinished it, and they cover and omit topics according to my own view of what is or isn’timportant to cover in a one-semester course. Expect them to change with little warning orannouncement as I add content or correct errors.Purchasers of the paper version should be aware of its probable imperfection and beprepared to either live with it or mark up their copy with corrections or additions as needbe. The latest (and hopefully most complete and correct) version is always available forfree online anyway, and people who have paid for a paper copy are especially welcometo access and retrieve it.I cherish good-hearted communication from students or other instructors pointing outerrors or suggesting new content (and have in the past done my best to implement manysuch corrections or suggestions).
Books by Robert G. BrownPhysics Textbooks Introductory Physics I and IIA lecture note style textbook series intended to support the teaching of introductoryphysics, with calculus, at a level suitable for Duke undergraduates. Classical ElectrodynamicsA lecture note style textbook intended to support the second semester (primarilythe dynamical portion, little statics covered) of a two semester course of graduateClassical Electrodynamics.Computing Books How to Engineer a Beowulf ClusterAn online classic for years, this is the print version of the famous free online book oncluster 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 LilithISBN: 978-1-4303-2245-0Web: http://www.phy.duke.edu/ rgb/Lilith/Lilith.phpLilith is the first person to be given a soul by God, and is given the job of giving allthe things in the world souls by loving them, beginning with Adam. Adam is given thejob of making up rules and the definitions of sin so that humans may one day live inan ethical society. Unfortunately Adam is weak, jealous, and greedy, and insists onbeing on top during sex to “be closer to God”.Lilith, however, refuses to be second to Adam or anyone else. The Book of Lilith isa funny, sad, satirical, uplifting tale of her spiritual journey through the ancient worldsoulgiving and judging to find at the end of that journey – herself.Poetry Who Shall Sing, When Man is GoneOriginal poetry, including the epic-length poem about an imagined end of the worldbrought about by a nuclear war that gives the collection its name. Includes many longand short works on love and life, pain and death.Ocean roaring, whipped by stormin damned defiance, hating hellwith every wave and every swell,every shark and every shelland shoreline.
Hot Tea!More original poetry with a distinctly Zen cast to it. Works range from funny andsatirical to inspiring and uplifting, with a few erotic poems thrown in.Chop water, carrywood. 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 877977The Book of Lilith is available on Amazon, Barnes and Noble and other online bookseller websites.
ContentsPrefacexiiiTextbook Layout and Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . .xivI: Getting Ready to Learn Physics3Preliminaries3See, Do, Teach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Other Conditions for Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Your Brain and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15How to Do Your Homework Effectively . . . . . . . . . . . . . . . . . . . . . . . .22The Method of Three Passes . . . . . . . . . . . . . . . . . . . . . . . . . .26Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33Homework for Week 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34II: Elementary Mechanics37Week 1: Newton’s Laws39Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .391.1: Introduction: A Bit of History and Philosophy . . . . . . . . . . . . . . . . . .471.2: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .491.3: Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .501.4: Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .571.5: Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .581.5.1: The Forces of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . .591.5.2: Force Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61i
iiCONTENTS1.6: Force Balance – Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . .63Example 1.6.1: Spring and Mass in Static Force Equilibrium . . . . . . . . .641.7: Simple Motion in One Dimension . . . . . . . . . . . . . . . . . . . . . . . .65Example 1.7.1: A Mass Falling from Height H . . . . . . . . . . . . . . . . .66Example 1.7.2: A Constant Force in One Dimension . . . . . . . . . . . . .721.7.1: Solving Problems with More Than One Object . . . . . . . . . . . . .74Example 1.7.3: Atwood’s Machine . . . . . . . . . . . . . . . . . . . . . . .75Example 1.7.4: Braking for Bikes, or Just Breaking Bikes? . . . . . . . . . .771.8: Motion in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . .791.8.1: Free Flight Trajectories – Projectile Motion . . . . . . . . . . . . . . .81Example 1.8.1: Trajectory of a Cannonball . . . . . . . . . . . . . . . . . . .811.8.2: The Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . .84Example 1.8.2: The Inclined Plane . . . . . . . . . . . . . . . . . . . . . . .841.9: Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .871.9.1: Tangential Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . .881.9.2: A Note on Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .891.9.3: Centripetal Acceleration . . . . . . . . . . . . . . . . . . . . . . . . .90Example 1.9.1: Ball on a String . . . . . . . . . . . . . . . . . . . . . . . . .92Example 1.9.2: Tether Ball/Conic Pendulum . . . . . . . . . . . . . . . . . .931.9.4: Tangential Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .941.10: Conclusion: Rubric for Newton’s Second Law Problems . . . . . . . . . . .94Homework for Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96Week 2: Newton’s Laws: Continued117Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.1: Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Example 2.1.1: Inclined Plane of Length L with Friction . . . . . . . . . . . 121Example 2.1.2: Block Hanging off of a Table . . . . . . . . . . . . . . . . . . 123Example 2.1.3: Find The Minimum No-Skid Braking Distance for a Car . . . 125Example 2.1.4: Car Rounding a Banked Curve with Friction . . . . . . . . . 1282.2: Drag Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292.2.1: Stokes, or Laminar Drag . . . . . . . . . . . . . . . . . . . . . . . . . 1332.2.2: Rayleigh, or Turbulent Drag . . . . . . . . . . . . . . . . . . . . . . . 1332.2.3: Terminal velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
CONTENTSiiiExample 2.2.1: Falling From a Plane and Surviving . . . . . . . . . . . . . . 136Example 2.2.2: Solution to Equations of Motion for Stokes’ Drag . . . . . . 1382.2.4: Advanced: Solution to Equations of Motion for Turbulent Drag . . . . 139Example 2.2.3: Dropping the Ram . . . . . . . . . . . . . . . . . . . . . . . 1402.3: Inertial Reference Frames – the Galilean Transformation . . . . . . . . . . . 1422.3.1: Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422.3.2: Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432.4: Non-Inertial Reference Frames – Pseudoforces . . . . . . . . . . . . . . . . 1462.4.1: Identifying Inertial Frames . . . . . . . . . . . . . . . . . . . . . . . . 148Example 2.4.1: Weight in an Elevator. . . . . . . . . . . . . . . . . . . . . 150Example 2.4.2: Pendulum in a Boxcar . . . . . . . . . . . . . . . . . . . . . 1522.4.2: Advanced: General Relativity and Accelerating Frames . . . . . . . . 1542.5: Just For Fun: Hurricanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Homework for Week 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Week 3: Work and Energy169Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.1: Work and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.1.1: Units of Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . 1733.1.2: Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1743.2: The Work-Kinetic Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . 1753.2.1: Derivation I: Rectangle Approximation Summation . . . . . . . . . . . 1753.2.2: Derivation II: Calculus-y (Chain Rule) Derivation . . . . . . . . . . . . 177Example 3.2.1: Pulling a Block . . . . . . . . . . . . . . . . . . . . . . . . . 179Example 3.2.2: Range of a Spring Gun . . . . . . . . . . . . . . . . . . . . 1803.3: Conservative Forces: Potential Energy . . . . . . . . . . . . . . . . . . . . . 1813.3.1: Force from Potential Energy . . . . . . . . . . . . . . . . . . . . . . . 1833.3.2: Potential Energy Function for Near-Earth Gravity . . . . . . . . . . . 1853.3.3: Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1863.4: Conservation of Mechanical Energy . . . . . . . . . . . . . . . . . . . . . . . 1873.4.1: Force, Potential Energy, and Total Mechanical Energy . . . . . . . . . 188Example 3.4.1: Falling Ball Reprise . . . . . . . . . . . . . . . . . . . . . . . 189Example 3.4.2: Block Sliding Down Frictionless Incline Reprise . . . . . . . 189Example 3.4.3: A Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . 189
ivCONTENTSExample 3.4.4: Looping the Loop . . . . . . . . . . . . . . . . . . . . . . . . 1903.5: Generalized Work-Mechanical Energy Theorem . . . . . . . . . . . . . . . . 192Example 3.5.1: Block Sliding Down a Rough Incline . . . . . . . . . . . . . 193Example 3.5.2: A Spring and Rough Incline . . . . . . . . . . . . . . . . . . 1933.5.1: Heat and Conservation of Energy . . . . . . . . . . . . . . . . . . . . 1943.6: Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Example 3.6.1: Rocket Power . . . . . . . . . . . . . . . . . . . . . . . . . . 1973.7: Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1983.7.1: Energy Diagrams: Turning Points and Forbidden Regions. . . . . . 201Homework for Week 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Week 4: Systems of Particles, Momentum and Collisions215Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.1: Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.1.1: Newton’s Laws for a System of Particles – Center of Mass . . . . . . 222Example 4.1.1: Center of Mass of a Few Discrete Particles . . . . . . . . . 2244.1.2: Coarse Graining: Continuous Mass Distributions. . . . . . . . . . . 225Example 4.1.2: Center of Mass of a Continuous Rod . . . . . . . . . . . . . 227Example 4.1.3: Center of mass of a circular wedge . . . . . . . . . . . . . . 228Example 4.1.4: Breakup of Projectile in Midflight . . . . . . . . . . . . . . . 2294.2: Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.2.1: The Law of Conservation of Momentum4.3: Impulse. . . . . . . . . . . . . . . . 231. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Example 4.3.1: Average Force Driving a Golf Ball . . . . . . . . . . . . . . . 235Example 4.3.2: Force, Impulse and Momentum for Windshield and Bug . . 2354.3.1: The Impulse Approximation . . . . . . . . . . . . . . . . . . . . . . . 2364.3.2: Impulse, Fluids, and Pressure . . . . . . . . . . . . . . . . . . . . . . 2374.4: Center of Mass Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . 2404.5: Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.5.1: Momentum Conservation in the Impulse Approximation . . . . . . . . 2424.5.2: Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2434.5.3: Fully Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 2434.5.4: Partially Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . 2444.5.5: Dimension of Scattering and Sufficient Information . . . . . . . . . . 244
CONTENTSv4.6: 1-D Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2464.6.1: The Relative Velocity Approach . . . . . . . . . . . . . . . . . . . . . 2474.6.2: 1D Elastic Collision in the Center of Mass Frame . . . . . . . . . . . 2494.6.3: The “BB/bb” or “Pool Ball” Limits . . . . . . . . . . . . . . . . . . . . . 2514.7: Elastic Collisions in 2-3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . 2534.8: Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Example 4.8.1: One-dimensional Fully Inelastic Collision (only) . . . . . . . 256Example 4.8.2: Ballistic Pendulum . . . . . . . . . . . . . . . . . . . . . . . 257Example 4.8.3: Partially Inelastic Collision . . . . . . . . . . . . . . . . . . . 2594.9: Kinetic Energy in the CM Frame . . . . . . . . . . . . . . . . . . . . . . . . . 260Homework for Week 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Week 5: Torque and Rotation in One Dimension275Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2755.1: Rotational Coordinates in One Dimension . . . . . . . . . . . . . . . . . . . 2775.2: Newton’s Second Law for 1D Rotations . . . . . . . . . . . . . . . . . . . . . 2795.2.1: The r-dependence of Torque . . . . . . . . . . . . . . . . . . . . . . . 2815.2.2: Summing the Moment of Inertia . . . . . . . . . . . . . . . . . . . . . 2835.3: The Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Example 5.3.1: The Moment of Inertia of a Rod Pivoted at One End . . . . 2845.3.1: Moment of Inertia of a General Rigid Body . . . . . . . . . . . . . . . 285Example 5.3.2: Moment of Inertia of a Ring . . . . . . . . . . . . . . . . . . 286Example 5.3.3: Moment of Inertia of a Disk . . . . . . . . . . . . . . . . . . 2865.3.2: Table of Useful Moments of Inertia . . . . . . . . . . . . . . . . . . . 2875.4: Torque as a Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287Example 5.4.1: Rolling the Spool . . . . . . . . . . . . . . . . . . . . . . . . 2895.5: Torque and the Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . 290Example 5.5.1: The Angular Acceleration of a Hanging Rod . . . . . . . . . 2915.6: Solving Newton’s Second Law Problems Involving Rolling . . . . . . . . . . 292Example 5.6.1: A Disk Rolling Down an Incline . . . . . . . . . . . . . . . . 292Example 5.6.2: Atwood’s Machine with a Massive Pulley . . . . . . . . . . . 2945.7: Rotational Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 2955.7.1: Work Done on a Rigid Object . . . . . . . . . . . . . . . . . . . . . . 2965.7.2: The Rolling Constraint and Work . . . . . . . . . . . . . . . . . . . . 297
viCONTENTSExample 5.7.1: Work and Energy in Atwood’s Machine . . . . . . . . . . . . 299Example 5.7.2: Unrolling Spool . . . . . . . . . . . . . . . . . . . . . . . . . 300Example 5.7.3: A Rolling Ball Loops-the-Loop . . . . . . . . . . . . . . . . . 3015.8: The Parallel Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303Example 5.8.1: Moon Around Earth, Earth Around Sun . . . . . . . . . . . 305Example 5.8.2: Moment of Inertia of a Hoop Pivoted on One Side . . . . . . 3055.9: Perpendicular Axis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 306Example 5.9.1: Moment of Inertia of Hoop for Planar Axis . . . . . . . . . . 308Homework for Week 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309Week 6: Vector Torque and Angular Momentum325Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3256.1: Vector Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3276.2: Total Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3286.2.1: The Law of Conservation of Angular Momentum . . . . . . . . . . . . 3296.3: The Angular Momentum of a Symmetric Rotating Rigid Object . . . . . . . . 330Example 6.3.1: Angular Momentum of a Point Mass Moving in a Circle . . . 332Example 6.3.2: Angular Momentum of a Rod Swinging in a Circle . . . . . . 333Example 6.3.3: Angular Momentum of a Rotating Disk . . . . . . . . . . . . 333Example 6.3.4: Angular Momentum of Rod Sweeping out Cone . . . . . . . 3346.4: Angular Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . 334Example 6.4.1: The Spinning Professor . . . . . . . . . . . . . . . . . . . . 3346.4.1: Radial Forces and Angular Momentum Conservation . . . . . . . . . 336Example 6.4.2: Mass Orbits On a String . . . . . . . . . . . . . . . . . . . . 3376.5: Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Example 6.5.1: Fully Inelastic Collision of Ball of Putty with a Free Rod . . . 341Example 6.5.2: Fully Inelastic Collision of Ball of Putty with Pivoted Rod . . 3456.5.1: More General Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 3476.6: Angular Momentum of an Asymmetric Rotating Rigid Obj
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 (Introduc- tory Physics for life science majors, engineers, or potential physics majors, respectively).
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The Duke MBA—Daytime Academic Calendar 2015-16 9 Preface 10 General Information 11 Duke University 11 Resources of the University 13 Technology at Fuqua 14 Programs of Study 15 The Duke MBA—Daytime 15 Concurrent Degree Programs 17 The Duke MBA—Weekend Executive 18 The Duke MBA—Global Executive 18 The Duke MBA—Cross Continent 19
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XX. Introductory Physics, High School. 411 High School Introductory Physics Test The spring 2018 high school Introductory Physics test was based on learning standards in the Introductory Physics content strand of the October 2006 version of the Massachusetts Science and Technology/Engineering Curriculum Framework. These learning
„Doris Duke of the illuminati Duke family was an heiress (at 12 years old) to the large tobacco fortune of the Duke family. She was the only child of American tobacco Co. founder James Buchanan Duke. Doris Duke, herself a member of the illuminati. Doris Duke had 5 houses (which have served as sites for illuminati rituals) – one in Beverly
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Given a power point presentation on . Ramcharitmanas: The Journey of a Text . in India International Centre, New Delhi on 7/3/2003. 7. Participated as a resource person in a refresher course for lecturers on “Hindi Public Sphere” organized by Mahatma Gandhi International University at Nainital on 13. th &14. th. June’2003. 8. Participated in the International Seminar on Fyodor J.Tyutchev .
