PHY 181 Test 2 Practice Solutions Spring 2013

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1 PHY 8 est Practice Solutios Sprig 3 Q: [4] costat et force F is applied to objects of icreasig ass. Which of the graphs describes the depedece of the objects acceleratio o the ass? a) b) c) C d) D Solutio: c Isaac s first law F a a F, that is the acceleratio varies with ass as. his depedec is best odeled b curve C. a a C a a D Q4: [4] boo of ass rests o a iclied surface of static coefficiet of frictio μ s. If the icliatio of the surface is ade larger (while the boo is still statioar) which of the followig stateets is false regardig the static frictio f s actig o the boo? a) For a icliatio, it is give b f s = μ s gcos b) For a icliatio, it is give b f s = gsi c) It icreases d) Depedig o the icliatio, f s ca be either saller or larger tha the ietic frictio whe the rap becoes steep eough for the boo to start ovig. gcos Solutio: a For ever, the static frictio atches the force pullig o the boo dow the iclie, that is, the parallel copoet of weight: f g si, s where is the icliatio agle. Cosequetl, as icreases, f s icreases fro zero through sall values (saller tha the correspodig ietic frictio) to larger ad larger values util it reaches its aiu value, f s = μ s gcos, which is larger tha the ietic frictio f = μ gcos, sice μ s > μ. f s g gsi Q3: [4] dog i a elevator levitates, barel touchig the floor. Which of the followig stateets is true? a) he dog has o weight. b) he floor pushes the dog upward with a force equal to its weight so it levitates. c) he dog s acceleratio is equal i agitude ad opposite to the elevator s acceleratio. d) he agitude of dog s acceleratio is equal to the gravitatioal acceleratio. e) his is ot possible. he observer is dru. Solutio: d a) he dog does have a weight as log as the elevator is i the proiit of the Earth. b) evitatio eas zero oral force so there is o push fro the surface. c) agitude caot be equated to a vector. e) d, fiall, the objective realit does't care about a drue stupor. Q4: [4] Plaet Earth has radius = How log should be a da i order to ae the objects o the Equator levitate? a) s b) 36 s c) 5 s d) s g Solutio: c s show o the figure, the objects are ept o the circular trajector o the Earth s surface b a cetripetal force associated with a cetripetal acceleratio a r = v /: g v, where is the radius of the Earth (sice the object is o the Equator). Sice Earth rotates with a costat speed, the speed ca be writte i ters of the period through oe coplete rotatio (that is, the tie of oe da): v = πr/.

2 PHY 8 est Practice Solutios Sprig 3 evitatio eas apparet weightlessess give b a zero iteractio with the groud, that is, =. Hece, the correspodig period of rotatio is obtaied fro 6 g v g /s 5 s Q5: [4] particle of ass is coected to oe ed of a strig of legth ad rotated verticall, as i the figure Whe the strig aes a agle with respect to the vertical directio, the tesio i the strig is ad the speed v of the ass is: a) v gsi v b) v g si g c) v g cos d) v g cos Solutio: b he role of cetripetal force is plaed b the tesio ad the radial copoet of the weight. herefore v g si v g si Q6: [4] cao fires a projectile alog a parabolic trajector above the horizotal groud as depicted i the figure. I what poit is the liear oetu aiu? a) he oetu is coserved, so it is the sae i,, C, D. b), sice here is where the ietic eerg is aiu c), sice here where the potetial eerg is aiu C d) C, sice here is where the echaical eerg is aiu e) D, sice here is where the speed is aiu D Solutio: e Sice p v ad the ass is costat, the oetu is aiu i the positio C characterized b the aiu speed. Q7: [4] wo objects, oe heavier tha the other, have the sae ietic eerg. Which has the sallest liear oetu? a) he light object b) he heav object c) oth have the sae oetu. d) It depeds o the potetial eerg Solutio: a he ietic eerg is give b K v v p. herefore, deotig the asses of the two objects, sice the ietic eergies are equal, we have p p p p p p

3 PHY 8 est Practice Solutios Sprig 3 Q8: [4] ball with ass =.5 g oves with a speed v i = 4. /s perpedicularl oto a wall, collides ielasticall with the wall ad bouces bac alog the sae lie with speed v f =. /s. If the ball was i cotact with the wall for - s, how large was the force eerted b the wall o the ball? a) N b) 3 N c) N d) 9.8 N Solutio: b Newto s d law, the average force eerted b the wall o the ball is give b the chage of oetu i tie F p t. he proble is oe diesioal, sice the ball oves alog the sae lie before ad after the collisio. herefore, choosig the positive directio to the right, the chage i oetu is p p p v v v v, f i f i f i ad the agitude of the force F v v t 3 g /s s 3 N. f i p i p f Q9: [4] Scrat, the cospicuous Ice ge squirrel, sits o a sled of ass M = 5. g, holdig tightl o his beloved acor. Scrat s ass is =. g ad acor s ass is = 3 g. he sled is at rest whe Scrat throws out the acor horizotall with a speed v = 4 /s. s a result, the sled oves with a speed a) v = /s b) v = 4 /s c) v =. /s d) v = /s e) Nosese! Scrat would ever get rid of his acor. Go, Scrat! Solutio: c Sice the frictio betwee the sled ad the slide is ver sall we ca cosider the sste as isolated. I this case, its total oetu is coserved, that is it ust sta the sae before ad after Scrat lauches his acor. Hece p before p after M v v v v M. s c s Q: [4] wo boes, oe heavier tha the other, are allowed to slide dow the sae rough iclie. It is observed that both of the have the sae costat acceleratio a. Which of the two boes has a larger wor doe b ietic frictio as the reach the botto of the rap? a) he heav oe b) he light oe c) he frictio does the sae wor i both cases d) he situatio described is ot phsicall possible: the ass is differet so a should be differet Solutio: a he et wor doe i the two cases is due to the actio of the copoet of the weight parallel with the iclie ad the opposig ietic frictio. Hece, deotig d the legth of the iclie, W ad W W W ad W et f g f g W ad gd si a g si d f So, we see that the heavier is the bo the ore wor is doe b the frictio. f g gsi 3

4 PHY 8 est Practice Solutios Sprig 3 P: wo boes of asses =.5 g ad = 3.5 g are coected b a cable passig over a ideal pulle. he boes ca slide o two setric frictioless iclies of agle = 4, as show o the figure. a) [5] Setch the free bod vector force diagras for each of the two boes. he split the weights alog the -sstes of coordiates provided o the figure. Write the respective copoets i ters of the respective ass, g ad agle. Each of the two boes is acted b its weight, a oral ad a tesio. he copoets are give o the figure. b) [5] Write out Newto s d law for each of the two boes alog the respective ad -directios. Mass : -ais: g si a -ais g cos Mass : -ais: g si a -ais g cos gsi gcos gcos gsi c) [5] Use the equatios above to calculate a sbolical epressio for the acceleratio of the two asses. Ol the calculate the acceleratio uericall. Sice the pulle is ideal, the tesio is the sae o its two sides, so let s deote both of the with :. hece we have g si a g si a ddig the equatios ter b ter (or substitutig fro oe equatio ito the other), we obtai g si g si a a a gsi.5 s. d) [5] Use the results of part (b) ad (c) to calculate a sbolical epressio for the tesio i the cable. Sa that the aiu tesio that the cable ca sustai without breaig is = N. What is the iiu agle that the two iclies ca be tilted before the cable saps? We ca substitute the acceleratio ito oe of the equatios for the tesio. For istace g si a g si g si g si g si g si g si gsi. So, we see that the agle correspodig to the aiu tesio is give b gsi si 9. g 4

5 PHY 8 est Practice Solutios Sprig 3 P: bloc of ass M ad a sall ball of ass =.5 g, are coected to each other b a light strig passed through a hole i a horizotal table. he bloc is at all ties at rest o the rough surface of the table with coefficiet of static frictio μ s =.38. he ball hags uder the table ad rotates uiforl i a horizotal circle of radius =.5 with the strig aig a agle = 35 with the vertical, as show o the figure. (We sa that ass oves as a coical pedulu.) M a) [4] O the figure below, setch the free bod force diagras for asses ad M. Cosider the portio of the strig betwee the ass M ad the hole parallel with the tabletop, as represeted. abel the forces eaigfull. M rough table f s Mg si cos g r b) [5] Write Newto s d law sbolicall for asses ad M alog the directios idicated for each object. he epressios for the ass should cotai the agle fro the copoets of the tesio. Mass M: F f s Mass : F cos g F Mg F si v c) [4] Use oe of the equatios for ass i part (b) to calculate the tesio i the strig. he substitute the tesio i the other equatio to calculate the speed v of the ball. he tesio ca be calculated fro cos g g cos 3 N. he speed coes fro the other equatio: si.3 s. si v v d) [4] Use the results of parts (b) ad (c) to calculate the frictio eepig the ass M fro ovig. What is the iiu ass M that will prevet the bloc fro slidig o the table? he frictio is static ad is equal to the tesio: f 3 N. s O the other had, the frictio caot be larger tha μ s. Hece oe ca derive the iiu ass M: f s s smg M M i g g 8. g. s s e) [3] he strig is cut at the oet show i the figure fro part (a). How will the ball ove? Circle oe: out of the page to the left dowward (freel fallig) r 5

6 PHY 8 est Practice Solutios Sprig 3 P3: child plas with a gae of idetical sall diss slidig o a flat frictioless surface. He pushes dis # with a velocit v v, v v, (where v =. /s) such that it collides with a dis # ovig with velocit v v, v v, v where the velocit copoets are writte i the coordiate sste show o the figure. fter the collisio, dis # oves with velocit v v, v v, v. 3 6 et s calculate the velocit v v, v of dis # after the collisio. a) [3] Setch the et oetu o the figure before ad after collisio. he figure out how the oetu vector p is supposed to loo lie ad draw it o the figure. he et oetu is the vector su of idividual oeta, ad is the sae before ad after the collisio sice it is coserved. Usig graphical vector subtractio we ca easil fid p : p p p p p p p. Notice that a alterative to the tail-to-tip ethod to add vectors graphicall, oe ca use the so called parallelogra ethod where the resultat of two added vectors is the ai diagoal i the parallelogra fored b the two vectors. b) [3] Use the otatio for the particle oeta before ad after the collisio give o the figure ad write out the coservatio of oetu i this collisio. he write the oeta eplicitl i ters of ass ad velocities ad fid the relatio betwee the velocities before ad after the collisio. p p p p v v v v v v v v c) [6] Write the velocit relatioship ou foud i part (b) i ters of the copoets alog ad -directios provided o the figure. he replace the copoets of v, v, ad v as give above i ters of costat v, to fid two equatios. Solve the to fid v v, v. v v v v v v 3v v v 6 v. s v v v v 7 v 6 v v v 6 v.4 s Hece, the vector velocit of Dis # after the collisio is give b agitude v v v.4 s, ad directio ta v v 8 with respect to +-directio. d) [6] Kowig that the ass of each dis is = g, calculate the et ietic eerg of the diss before ad after the collisio. efore the collisio: 9 K K K v v v v v v v v v.6 J 4 8 fter the collisio: K K K v v v v v v v v v.34 J e) [] So, is the collisio elastic or ielastic? Wh? he ietic eerg after the collisio is sigificatl saller tha before, so the collisio is ielastic. before after 6

7 PHY 8 est Practice Solutios Sprig 3 P4: bo of ass = 5. g is pulled up a iclie of agle = 43 b a force of agitude F = aig a agle α = 3 with respect to the rap, as i the figure. he bo oves a distace d = 3. fro poit, where its speed is v = 3. /s to poit, o top of the rap. he surface is rough, with coefficiet of ietic frictio μ =.47. a) [3] Setch o the figure all the other forces actig o the bo besides F. Split the forces ito ad vector copoets i the give coordiate sste. b) [4] Write out Newto s d law alog the provided ad -aes. he use the equatio alog -ais to fid the oral force i ters of give quatities (eep the result i sbolical for). -ais: F cos f g si a -ais F si g cos a g cos F si Fsiα gsi f g α F Fcosα gcos c) [4] Use the result of part (b) to write out the ietic frictio sbolicall, i ters of give quatities. defiitio, the frictio is give b f herefore, the frictio is here a positio depedet force: f g cos F si. d) [5] Calculate the wor doe b each of the actig forces as the bo oves fro to. he copute the et wor doe o the bo i this iterval. here are 4 costat forces actig o the bo, so b defiitio we obtai W dcos 9 W gd cos 9 J g W Fd cos 6 J F W f d cos8 g cos F si d J f Wet W Wg WF Wf 4 J e) [4] Use the Wor-Eerg theore to calculate the speed of the bo i poit. the Wor-Eerg theore, the chage i ietic eerg is equal to the et wor. Hece W W W W W W K K K et g F f f v v W v v W 8. s. et et 7

8 PHY 8 est Practice Solutios Sprig 3 P5: sall bo of ass = 7 g is lauched at positio = with iitial horizotal velocit v = 3.5 /s. he bo oves i a straight lie alog a rough surface with coefficiet of ietic frictio μ =.5. esides the usual forces, a eteral force acts o the bo give b: F F, F, si, where the variable is the positio alog the -ais, while = N ad =. are costats of otio. (For this proble, set our calculators to wor with radias.) a) [] he bo is represeted o the figure at positio = /. Figure out the orietatio of force F i that positio, ad setch the vector force diagra. Notice that the force F oscillates up ad dow as give b its -copoet (it is zero at = ad, ad aiu at /). t the give positio, the force F is, si, F, si. Sice the -copoet of the force F is everwhere ad the -copoet at the give positio is positive, the force is verticall upward. he other forces are the weight, the oral ad the ietic frictio. b) [4] Write out Newto s d law alog the provided ad -aes. -ais: F f a f a -ais F g a si g c) [4] Use the equatio alog -ais i part (b) to write out the ietic frictio sbolicall, i ters of give quatities. defiitio, the frictio is give b f herefore, the frictio is here a positio depedet force: si f f g. d) [5] Calculate the wor doe b each of the actig forces as the bo oves through the iterval = (, ). here are 4 forces actig o the bo. We see b ispectio that ol the frictio does wor sice the other forces are perpedicular o the displaceet. Nevertheless, for ehaustiveess, let s calculate all of the eplicitl: W d dcos 9 W g d gd cos 9 g W F d F dcos 9 F cos 8 si W f d f d f d g d f gd si d g cos g cos cos g g 69 J e) [5] Use the Wor-Eerg theore to calculate the speed of the bo at positio =. the Wor-Eerg theore, the chage i ietic eerg is equal to the et wor. Hece W W W W W W K K K et g F f f v v W v v W. s. et f f 8

9 PHY 8 est Practice Solutios Sprig 3 P6: sall bo of ass =.7 g is released fro rest i positio dow a frictioless rap with a quarter-circle profile of radius =.. esides the usual forces, a eteral force acts o the bo. he force is give i ters of the agle ade b the radius ad the -ais (as show o the figure) b: F, si, where = N is a positive costat. a) [3] he bo is represeted o the figure for a arbitrar. Setch the vector force diagra for the bo i that positio. Sice the -copoet of the force F is everwhere ad the -copoet is positive, the force is verticall upward. b) [4] Setch the vector positio r of the bo. Provide sbolical epressios (i ters of ad ) for r, ad for the respective eleetar displaceet dr. he vector positio is a vector coectig the origi of the sste of coordiates with the locatio of the object. I our case it ca be writte i ters of the paraeter ad its costat agitude : r, cos, si ; herefore, the eleetar displaceet for the eleetar chage i is dr si d, cos d. c) [6] Calculate the wor doe b each of the forces actig o the bo as it slides fro poit to poit. he calculate the et wor. he bo oves betwee = (,π/), ad we have to evaluate the wor doe b the oral, the weight ad the force F. Sice the oral is everwhere perpedicular o displaceet, its wor is zero: W dr he wor doe b the weight is the wor doe b its copoet alog the vertical: g cos si J. W g dr g d g g Siilarl, sice the force F is vertical, the wor doe b it is sipl the wor of its -copoet: F, si si, cos W F dr d d cos sid cos herefore, the et wor is Wet Wg W WF 6. J 4 J. 6. J. s we should epect, the et wor is positive cosistet with the fact that the bo accelerates dow the curve. d) [4] Use the Wor-Eerg heore to calculate the speed v of the bo i poit. K Wet K K Wet v Wet v W et 4. s e) [3] Suppose that i the show positio = 3. what aout does the force F cotribute to the cetripetal force actig o the bo i that positio? Circle oe:.5 N 4.3 N 5. N he radial copoet of force F i the respective positio is give b F F si si 3.5 N. r 9

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J 10 J W W W W PHYS 54 Practice Test 3 Solutios Sprig 8 Q: [4] A costat force is applied to a box, cotributig to a certai displaceet o the floor. If the agle betwee the force ad displaceet is 35, the wor doe b this force