Problems and Solutions in Introductory Mechanics

353 Pages·2014·2.164 MB·English

by  David Morin

Problems
and Solutions
in
INTRODUCTORY
MECHANICS
DAVID MORIN
Problems and Solutions
in
INTRODUCTORY MECHANICS
David Morin
Harvard University
'DavidMorin2014
Allrightsreserved
ISBN-10: 1482086921
ISBN-13: 978-1482086928
PrintedbyCreateSpace
Coverimage: iStockphotographbyVernonWiley
Additionalresourceslocatedat:
www.people.fas.harvard.edu/˜djmorin/book.html
Contents
Preface vii
1 Problem-solvingstrategies 1
1.1 Basicstrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Solvingproblemssymbolically . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Checkingunits/dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Checkinglimiting/specialcases . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Taylorseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Listofstrategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Gettingstarted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Solvingtheproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Finishingup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.5 Lookingahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Howtousethisbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Kinematicsin1-D 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Kinematicsin2-D(and3-D) 42
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 F=ma 68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
iii
iv CONTENTS
5 Energy 104
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Momentum 142
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7 Torque 177
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8 Angularmomentum 220
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9 Statics 251
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10 Oscillations 265
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
10.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
11 Gravity 287
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
11.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
11.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
11.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
12 Fictitiousforces 314
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
12.2 Multiple-choicequestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
12.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
12.4 Multiple-choiceanswers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
12.5 Problemsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
CONTENTS v
13 Appendices 331
13.1 AppendixA:Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
13.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
13.1.2 Cartesiancoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
13.1.3 Polarcoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
13.1.4 Addingandsubtractingvectors . . . . . . . . . . . . . . . . . . . . . . 334
13.1.5 Usingcomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
13.1.6 Dotproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
13.1.7 Crossproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
13.2 AppendixB:Taylorseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
13.2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
13.2.2 Howmanytermstokeep? . . . . . . . . . . . . . . . . . . . . . . . . . 340
13.2.3 Dimensionlessquantities . . . . . . . . . . . . . . . . . . . . . . . . . . 341
13.3 AppendixC:Limitingcases,scientificmethod . . . . . . . . . . . . . . . . . . . 342
13.4 AppendixD:Problemsrequiringcalculus . . . . . . . . . . . . . . . . . . . . . 344

Preface
This problem book grew out of a “freshman physics” mechanics course taught at Harvard
Universityduringthepastdecadeandahalf. Mostoftheproblemsarefromexamsorproblem
sets,althoughIhaveaddedotherstoroundoutthedistributionoftopics. Someoftheproblems
arestandardones,butmanyareoffthebeatenpath. Intheend,thereisafinitenumberofprin-
ciplesinintroductorymechanics,sotheproblemsinevitablystartlookingfamiliarafterawhile.
Two topics from the course that aren’t included in this book are relativity and damped/driven
oscillatorymotion. Perhapsthesewillappearinafutureedition,alongwithothertopicssuchas
fluidsandprecessionalangularmomentum.
Thisbookwillbehelpfultobothhigh-schoolstudentsandcollegestudentstakingcoursesin
introductoryphysics(justmechanics,notelectricityandmagnetism). Calculusisusedthrough-
out the book, although it turns out that only a sixth of the problems actually require it. This
subsetofproblemsislistedinAppendixD.Ifyouhaven’tstudiedcalculusyet,juststeerclearof
thoseproblems,andyoucanviewthisbookasanalgebra-basedone. Theproblemsaregener-
allyontheleveloftheone-starortwo-starproblemsinmyIntroductiontoClassicalMechanics
textbook,1whichcoversanumberofmoreadvancedtopicssuchasLagrangians,normalmodes,
gyroscopicmotion,etc. Iwilloccasionallyreferyoutothatbookifyouareinterestedindelving
furtherintovarioustopics.
It is important to note that this book should not be thought of as a textbook. Although
thereisanintroductiontoeachchapterwherethebasicsarepresented,thisintroductionisbrief.
It is no substitute for the text in a chapter in a standard introductory textbook. This book is
thereforedesignedtobeusedintandemwithanormaltextbook. Youcanthinkofthisbookas
supplementingatextbookbyprovidingastockpileofadditionalproblems. Oryoucanthinkof
atextbookassupplementingthisbookbyprovidingadditionalbackground.
Inmostchaptersthefirstfewproblemsarefoundationalones. Theseproblemscoverbasic
resultsandtheoremsthatyoucanusewhensolvingotherproblems.Whenabasicresultisstated
intheintroductiontoeachchapter,youwillgenerallybereferredtoafoundationalproblemfor
theproof. Thebookisselfcontained,inthatwederiveeverythingweneed. It’sjustthatmany
ofthederivationsareshiftedtotheproblems.
Asetofmultiple-choicequestionsprecedestheproblemsineachchapter. Thesequestions
areusuallyconceptualonesthatyoucandoinyourhead. Intherarecasewheretheyrequirea
calculation, it is a very minor one. The book contains about 150 multiple-choice questions, in
additiontonearly250free-responseproblems.
Dependingonhowyouusethisbook,itcanbeaninvaluableresource—oracompletewaste
oftime. Sohereissomecriticaladviceonusingthesolutionstotheproblems: Ifyouarehaving
trouble solving a problem, it is imperative that you don’t look at the solution too soon. Brood
overitforawhile. Ifyoudofinallylookatthesolution,don’tjustreaditthrough. Instead,cover
itupwithapieceofpaperandreadonelineatatimeuntilyougetahinttogetstarted. Then
setthebookasideandworkouttheproblemforreal. Repeatthisprocessasnecessary. Actively
solvingtheproblemistheonlywayitwillsinkin. Thispieceofadviceonhowtousethisbook
issoimportantthatI’mgoingtorepeatitanddisplayitprominentlyinabox:
1Introduction to Classical Mechanics, With Problems and Solutions, David Morin, Cambridge University Press,
2008.Thiswillbereferredtoas“Morin(2008).”
vii
viii CONTENTS
Ifyouneedtolookatthesolutiontoaproblemtogetahint(afterhavingthoughtaboutit
forawhile),coveritupwithapieceofpaperandreadonelineatatimeuntilyoucanget
started. Thensetthebookasideandworkthingsoutforreal. Youwilllearnagreatdealthis
way. Ifyouinsteadreadasolutionstraightthroughwithouthavingfirstsolvedtheproblem,
youwilllearnverylittle.
Theonlyscenarioinwhichyoushouldeverreadasolutionstraightthroughiswhereyou’ve
alreadysolvedtheproblem. However,eveninthiscaseyoushouldbecareful. IfI’vegivenan
alternativesolution,thenyoushouldagainjustreadonelineatatimeuntilyoucangetstarted
andsolveitthatwaytoo.
To belabor the point, it is quite astonishing how unhelpful it is to simply read a solution
instead of solving a problem. You’d think it would do some good, but in fact it is completely
ineffective in raising your understanding to the next level. Of course, a careful reading of the
introductionsisnecessarytogetthebasicsdown.Butoncethatisaccomplished,it’stimetostart
solving problems. If Level 1 is understanding the basic concepts, and Level 2 is being able to
apply those concepts, then you can read and read until the cows come home, and you’ll never
getpastLevel1.
A few informational odds and ends: We’ll use the standard mks (meter-kilogram-second)
systemofunitsinthisbook. Concerningnotation,adotabovealetter,suchas x˙,denotesatime
derivative. A boldface letter, such as v, denotes a vector. Chapter 13 consists of appendices:
AppendixAgivesareviewofvectors,AppendixBcoversTaylorseries,AppendixCisanaside
onthescientificmethod,andAppendixDliststheproblemsthatrequirecalculus. Thereare364
figuresinthebook,whichcoincidentallyisthetotalnumberofgiftsgivenduringthe12daysof
Christmas,andwhichironicallyisonegiftforeverydayoftheyearexceptChristmas!
Itwasthefallsemesterof2000whenIfirsttaughtthecourseonwhichthisbookisbased,so
itwouldbeanunderstatementtosaythatIhavebenefittedovertheyearsfromtheinputofmany
people, including roughly 1,000 students. I would particularly like to thank Carey Witkov for
carefullyreadingthroughtheentirebookandofferingmanyvaluablesuggestions. Otherfriends
andcolleagueswhoseinputIamgratefulforare(withmymemorybeingskewedtowardmore
recentyears): JacobBarandes,AllenCrockett,HowardGeorgi,DougGoodale,TheresaMorin
Hall,RobHart,PaulHorowitz,RandyKelley,AndrewMilewski,PraharMitra,JoonPahk,Dave
Patterson,JoePeidle,CourtneyPeterson,DanielRosenberg,WolfgangRueckner,AlexiaSchulz,
NilsSorensen,JoeSwingle,CorriTaylor,andRebeccaTaylor.
Despitecarefulediting,thereiszeroprobabilitythatthisbookiserrorfree. Ifanythinglooks
amiss, please check the webpage www.people.fas.harvard.edu/˜djmorin/book.html for a list of
typos,updates,additionalmaterial,etc. Andpleaseletmeknowifyoudiscoversomethingthat
isn’talreadyposted. Suggestionsarealwayswelcome. Happyproblemsolving!
DavidMorin
Cambridge,MA
Chapter 1
Problem-solving strategies
TO THE READER: This book is available as both a paperback and an eBook. I have made a
few chapters available on the web, but it is possible (based on past experience) that a pirated
versionofthecompletebookwilleventuallyappearonfile-sharingsites. Intheeventthatyou
arereadingsuchaversion,Ihavearequest:
If you don’t find this book useful (in which case you probably would have returned it, if you
hadboughtit),orifyoudofinditusefulbutaren’tabletoaffordit,thennoworries;carryon.
However, if you do find it useful and are able to afford the Kindle eBook1 (priced somewhere
between $7 and $10), then please consider purchasing it (available on Amazon). I chose to
self-publishthisbooksothatIcouldkeepthecostlow. Theresultingpriceofaround$10,which
is very inexpensive for a 350-page physics book, is less than a movie and a bag of popcorn,
withtheaddedbonusthatthebooklastsformorethantwohoursandhaszerocalories(ifused
properly!).
–DavidMorin
1.1 Basic strategies
In view of the fact that this is a problem book, it makes sense to start off by arming you with
somestrategiesforsolvingproblems. Thisisthesubjectofthepresentchapter. We’llbeginwith
afewstrategiesthatarediscussedsomewhatindepth,andthenwe’llprovidealonglistof30-ish
strategies. Youobviouslyshouldn’ttrytomemorizeallofthem. Justrememberthatthelistis
there,andreferbacktoiteverynowandthen.
1.1.1 Solvingproblemssymbolically
If you are solving a problem where the given quantities are specified numerically, it is highly
advantageoustoimmediatelychangethenumberstolettersandthensolvetheprobleminterms
of the letters. After you obtain a symbolic answer in terms of these letters, you can plug in
theactualnumericalvaluestoobtainanumericalanswer. Therearemanyadvantagestousing
letters:
(cid:136) It is quicker. It’s much easier to multiply a g by an ℓ by writing them down on a piece
of paper next to each other, than it is to multiply their numerical values on a calculator.
If solving a problem involves five or ten such operations, the time would add up if you
performedalltheoperationsonacalculator.
(cid:136) Youarelesslikelytomakeamistake.It’sveryeasytomistypean8fora9inacalculator,
butyou’reprobablynotgoingtomiswriteaqforanaonapieceofpaper. Butevenifyou
1Ifyoudon’talreadyhavetheKindlereadingappforyourcomputer,youcandownloaditfreefromAmazon.
1