Review Problems for Introductory Physics 1

1 Review ProblemsforIntroductory Physics 1 May 20, 2019 Robert G. Brown, InstructorDuke University Physics DepartmentDurham, NC NoticeThis document is Copyright Robert G. Brown 2011. The specific dateof last modification is determinableby examining the original document Preface32 Short Math Review Long Problems .. 193 Essential Laws, Theorems, and Principles214 Problem Solving275 Newton s Kinematics .. Multiple Choice .. Short Answer .. Long Problems .. Dynamics .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. Circular Motion .. Multiple Choice .. Short Answer .. Long Problems .. 1146 Work and Work and Kinetic Energy .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. Work and Mechanical Energy .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. Power .. Multiple Choice .. Long Problems .. 1737 Center of Mass and Center of Mass.

2 Multiple Choice .. Short Answer .. Long Problems .. Momentum .. Multiple Choice .. Short Answer .. Long Problems .. 2098 One Dimensional Rotation and Rotation .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. Moment of Inertia .. Ranking/Scaling .. Short Answer .. Long Problems .. 2839 vector Torque and Angular Angular Momentum .. Multiple Choice .. Short Answer .. Long Problems .. vector Torque .. Multiple Choice .. Short Answer .. Long Problems .. 31710 Statics .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. 35211 Fluids .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. 41412 Oscillations .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. 48013 Waves .. Multiple Choice .. Short Answer .. Short Answer .. Long Problems .. 52914 Sound Waves .. Multiple Choice.

3 Ranking/Scaling .. Short Answer .. Long Problems .. 55715 Gravity .. Multiple Choice .. Ranking/Scaling .. Short Answer .. Long Problems .. 5862 CONTENTSC hapter 1 PrefaceThe Problems in this Review guide are provided as iswithout any guarantee of being correct!That snot to suggest that they are all broken on the contrary, most of them are well-tested and have beenused as homework, quiz and exam Problems for decades if not centuries. It is to suggest that they havetypos in them, errors of other sorts, bad figures, and one or twoof them are really too difficult for thiscourse but haven t been sorted out or altered to make them these in just adds to the fun. Physics Problems arenotall cut and dried; Physics itself isn thing you should be building up as you work is an appreciation for what is easy, what is difficult,what is correct and what is incorrect. If you find an error and bringit to my attention, I ll do my bestto correct it, of course, but in the meantime, be warned!

4 Afewof the Problems have rather detailed solutions (due to Prof. RonenPlesser and myself), providedas examples of how a really good solution might develop, with considerable annotation. However, mostproblems do not have included solutions andnever will have. I am actually philosophically opposed toproviding students with solutions that they are then immediately tempted to memorize. This guide isprovided so that you can learn to solve Problems and work sufficientlycarefully that they can trust invariably then ask: But how are we to know if we ve solved the Problems correctly? The answer is simple. The same way you wouldin the real world!Work on them in groups and checkyour algebra, your approach, and your answers against one another s. Build a consensus. Solve themwith mentoring (course TAs, professors, former students, tutors all arehappyto help you). Find answersthrough research on the web or in the be honest, almost any of the ways that involvehard work on your partare good ways to learn to solvephysics Problems .

5 The onlybadway to (try to) learn is to have the material all laid out, cut and dried,so that you don t have tostruggleto learn, so that you don t have towork hardand therebypermanentlyimprint the knowledge on your brain as you go. Physics requires engagement and investment of timeand energy like no subject you have ever taken, if only because it is one of the most difficult subjectsyou ve ever tried to learn (at the same time it is remarkably simple, paradoxically enough).In any event, to use this guide most effectively, first skim through the whole thing to see what is there,then start in at the beginning and work through it, again and again, reviewing repeatedly all of theproblems and material you ve covered so far as you go on to what you are working on currently in classand on the homework and for the upcoming exam(s). Don t be afraidto solve Problems more than once,34 CHAPTER 1. PREFACEor even more than three or four in groups!Seriously! With pizza and 2 Short Math Review ProblemsThe Problems below are a diagnostic for what you are likely to need in order to work Physics aren t really enough of them to constitute practice , but if you have difficulty withanyof them,you should probably find a math Review (there is usually one in almost any Introductory Physics textand there are a number available online) and work through in geometry, trigonometry, algebra, calculus , solving simultaneous equations, or general vi-sualization and graphing will all negatively impact your Physics performance and, if uncorrected, 2.

6 SHORT MATH Review PROBLEMSP roblem down thebinomial expansionfor the following expressions, given the conditions indicated. FYI,the binomial expansion is:(1 +x)n= 1 +nx+n(n 1)2!x2+n(n 1)(n 2)3!x3+..wherexcan be positive or negative and wherenis any real number and only converges if|x|<1. Writeat least the firstthree non-zeroterms in the expansion:a) Forx > a:1(x+a)2b) Forx > a:1(x+a)3/2c) Forx > a:(x+a)1/2d) Forx > a:1(x+a)1/2 1(x a)1/2e) Forr > a:1(r2+a2 2arcos( ))1/27 Problem the following expressions, whereddtmeans differentiate with respect tot :a)ddtsin( t) =b)ddtcos( t) =c)ddtln(at) =d)ddt(at5+bt2+c)=e)ddte t=f)ddt(1 +at2)3=8 CHAPTER 2. SHORT MATH Review PROBLEMSP roblem zABa) Express the dot product in terms of its Cartesian components ~A=Ax x+Ay y+Az z:~A ~B=b) Express the dot product in terms of the magnitudesA,Band :~A ~B=c) Express themagnitudeof cross product in terms of the magnitudesA,Band : ~A ~B =d) Express the cross product in terms of its Cartesian components ~A=Ax x+Ay y+Az z(thishas a lot of terms):~A ~B=9 Problem the following short Problems :a)10 245=b) 3.

7 103+ ) 32 1 6=d)cos( /6)/9 =e)sin(30 ) =10 CHAPTER 2. SHORT MATH Review PROBLEMSP roblem the following indefinite and definite integrals:a) /20sin( )d =b) cos( t)dt=c) xndx=d) ba1xdx=e) ( gt+v0)dt=11 Problem the firstthree nonzero termsfor theTaylor Seriesfor the following expressions. Recall thatthe radius of convergence for the binomial expansion (another name for the first taylor series in the listbelow) is|x|<1 this gives you two ways to consider the expansions of the form (x+a) ) Expand aboutx= 0:(1 +x) 2 b) Expand aboutx= 0:ex c) Forx > a(expand aboutxor use the binomial expansion after factoring):(x+a) 2 d) Estimate within 1% without a calculator, if you can. Explain your 2. SHORT MATH Review PROBLEMSP roblem fort. Your answer should be an equation, although you may give a numberanswer for the lastone aswellas the algebraic answer if you have a calculator handy. You may find ln(2) to be auseful thing to know if )v0t x0= 0t=b) 12gt2+v0t= 0t=c) 12gt2+v0t+x0= 0t=d)A/2 =Ae tt=(forA= 5).

8 13 Problem the following system of simultaneous equations your workand givealgebraicanswers in terms ofm1,m2, andg:m1gsin( ) T=m1aT m2g=m2aa=T=14 CHAPTER 2. SHORT MATH Review PROBLEMSP roblem vector ~A= 4 x+ 6 yand vector ~B= 9 x+ 6 y. Then the vector ~C=~A+~B:a) is in the first quadrant (x+,y+) and has magnitude ) is in the fourth quadrant (x+,y-) and has magnitude ) is in the first quadrant (x+,y+) and has magnitude ) is in the second quadrant (x-,y+) and has magnitude ) is in the third quadrant (x-,y-) and has magnitude ) Suppose vector ~A= 3 x+ 6 yand bector~B= 7 x 3 y. Then the vector ~C=~A+~B:A) is in the first quadrant (x+,y+) and has magnitude ) is in the second quadrant (x-,y+) and has magnitude ) is in the second quadrant (x-,y+) and has magnitude ) is in the fourth quadrant (x+,y-) and has magnitude ) is in the third quadrant (x-,y-) and has magnitude ) Suppose vector ~A= 4 x+ 6 yand vector ~B= 9 x+ 6 y. Then the vector ~C=~A ~B:A) is in thex-direction and has magnitude ) is in they-direction and has magnitude ) is in the y-direction and has magnitude ) is in thex-direction and has magnitude ) is in the x-direction and has magnitude 2.

9 SHORT MATH Review PROBLEMSP roblem the firstthree nonzero termsfor theTaylor seriesfor the following expressions. Expand aboutthe indicated point:a) Expand aboutx= 0:(1 +x)n b) Expand aboutx= 0:sin(x) c) Expand aboutx= 0:cos(x) d) Expand aboutx= 0:ex e) Expand aboutx= 0 (note:i2= 1):eix Verify that the expansions of both sides of the following expressionmatch:ei = cos( ) +isin( )17 Problem hypoppadjFill in the following in terms of the marked sides. For example, one of the answers below might be (butprobably isn t)hypopp:sin( ) =cos( ) =tan( ) =18 CHAPTER 2. SHORT MATH Review PROBLEMSP roblem (a)(b)Two simple Problems in vector analysis are presented above. You mayleave your answers in terms ofradical fractions ( 7/13) where appropriate. You may not use calculators!a) Find thecartesiancoordinate components (X, Y) of the vector ) Find thepolarcoordinate components (V, ) of the vector given.|V|= = Long ProblemsProblem position of a particle as a function of time is given by:~x(t) =x0cos( t) x+y0sin( t) ywherex0> ) What is~v(t) for this particle?

10 B) What is~a(t) for this particle?c) Draw a generic plot of the trajectory function for the kind of shape is this? Inwhat direction/sense is the particle moving (indicate with arrow on trajectory)?d) Draw separate plots ofx(t) andy(t) on the same 2. SHORT MATH Review PROBLEMSC hapter 3 Essential Laws, Theorems, andPrinciplesThe questions below guide you through basic physical laws and concepts. They are the stuff that oneway or another you should know going into any exam or quiz following the lecture in which they arecovered. Note that there aren t really all that many of them, and alot of them are actually easily derivedfrom the most important is absolutely no point in memorizing solutions to all of the Problems in this guide. In fact, forall but truly prodigious memories, memorizing them all would be impossible (presuming that one couldwork out all of the solutions into an evenlargerbook to memorize!). However,every studentshouldmemorize, internalize, learn,knowthe principles, laws, and and theorems covered in this section (andperhaps a few that haven t yet been added).