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1 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 is a) eerg gaied b the box. b) eerg lost b the box. c) eerg gaied b the floor. d) eerg lost b the floor. e) Both (b) ad (c) above are true. Solutio: e The agle betwee the force ad the directio of otio is i the secod quadrat, so its cosie is egative. hece, the wor doe o the box is give b W F d cos35. Beig egative, it is lost eerg which goes to the aget of the force that is, the floor. Q: [4] Two boxes, 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 boxes has a larger wor doe b ietic frictio as the reach the botto of the rap? a) The heav oe b) The light oe c) The frictio does the sae wor i both cases d) The situatio described is ot phsicall possible: the ass is differet so a should be differet e) It depeds o the iitial speed Solutio: a The 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 box the ore wor is doe b the frictio. f g gsiθ θ Q3: [4] The adjacet graph represets the et horizotal force actig o a object as it oves i a straight lie alog x-axis. What is the et wor doe b this force i the show iterval? a) b) J c) J d) J e) Isufficiet iforatio. Solutio: c The wor is the give b the area betwee the F vs. x curve ad the x-axis. So, because we have three itervals where the force varies with x i a differet aer, we obtai 3 W W W W 5 5 J J et Q4: [4] The adjacet graph represets a vertical ap with the paths of two balls rollig fro rest dow two frictioless slopes with the idicated profile. Kowig that ball A is twice as heav as B, which ball will ove faster at the botto of its hill, ad what will be its speed? a) Ball A will ove faster, v A = 9.8 /s. b) Ball A will ove faster, v A = 39 /s. c) Ball B will ove faster, v B = 9.8 /s. d) Ball B will ove faster, but the iforatio is isufficiet to fid the speed. e) Both will ove with the sae speed, v A = v B =9.8 /s Force (N) 5 x () () 5 5 A B x () 3 4 5

2 PHYS 54 Practice Test 3 Solutios Sprig 8 Solutio: e Sice the wor doe b the weight depeds ol o the fial ad iitial altitude, we see that both balls will achieve the sae aout of ietic eerg. Usig coservatio of eerg, we see that the ass does t atter: the will ove with the sae speed. Thus, iasuch there is o frictio, for both balls: E E v fial iitial gh v gh 9.8. s Q5: [4] The elastic costat of sprig is twice the costat of sprig. The sprig is copressed b x, ad the sprig is stretched b x. The copressio x is also twice the stretch x. The the ratio PE / PE betwee the elastic potetial eergies of the two sprigs is a) b) - c) 4 d) 8 e) Noe of the above. Solutio: d Usig the expressios for the elastic potetial eerg, we get PE PE x x x x 8. Q6: [4] Two asses, = 4 g ad = g, are attached to idetical sprigs with elastic costat = 3 N/, as i the figure. What is the differece betwee the elastic potetial eergies stored i the two sstes? a) 4 J b) 98 J c).98 J d) 8 J e) Noe of the above Hit: The elastic potetial eergies are give b the sprig elogatios. How would ou calculate this elogatio at equilibriu? x Solutio: a The differece betwee the elastic potetial eergies is U U x x. e e The elogatios ca be obtaied fro the equilibriu coditios: g x x g g x x g. g 4 J Hece Ue Ue x g g Q7: [4] Two objects, oe heavier tha the other, have the sae ietic eerg. Which has the sallest liear oetu? a) The light object b) The heav object c) Both have the sae oetu. d) It depeds o the potetial eerg e) It depeds of the speed. Solutio: a The ietic eerg is give b K v v p. Therefore, deotig the asses of the two objects, sice the ietic eergies are equal, we have p p p p p p

PHYS 54 Practice Test 3 Solutios Sprig 8 Q8: [4] Two boxes, oe heavier tha the other, are iitiall at rest o a horizotal frictioless surface. The sae costat force F acts o each oe for exactl secods.

3 PHYS 54 Practice Test 3 Solutios Sprig 8 Q8: [4] Two boxes, oe heavier tha the other, are iitiall at rest o a horizotal frictioless surface. The sae costat force F acts o each oe for exactl secods. Which box has the saller oetu after the force acts? a) The heavier oe. b) The lighter oe. c) Depeds o the distace traveled i secods. d) Both have the sae oetu. e) Depeds o the local gravitatioal acceleratio. Solutio: d B Newto s d law, we have F dp dt dp Fdt. Sice the force F is applied the sae tie iterval Δt oto both boxes, which are both iitiall at rest (zero oetu), the ipulse is the sae so the chage is oetu will be the sae. (Notice however that the heavier object will ove slower tha the light oe.) Q9: [4] A 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 exerted b the wall o the ball? a) N b) 3 N p i c) N p f d) 9.8 N e) Noe of the above. Solutio: b B Ipulse-Moetu Theore ad Newto s d law, the average force exerted b the wall o the ball is give b the rate of chage of oetu i tie J Ft p F p t. The proble is oe diesioal, sice the ball oves alog the sae lie before ad after the collisio. Therefore, 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 Q: [4] Scrat, the cospicuous Ice Age 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. The sled is at rest whe Scrat throws out the acor horizotall with a speed v = 4 /s. As 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 throw awa his acor. Solutio: c Sice the frictio betwee the sled ad the slide is ver sall we ca cosider the sste as isolated. I this case, the total oetu is coserved, that is, it ust sta the sae before ad after Scrat throws his acor. Hece p before p after M v v v v M. s c s 3

4 PHYS 54 Practice Test 3 Solutios Sprig 8 P: A sall box of ass is lauched at positio x = with iitial horizotal velocit v = 3.5 /s. The box oves i a straight lie alog a frictioless surface. Besides the usual forces, a exterall applied force acts o the box give b: F F, F A x L, A, x where the variable x is the positio alog the x-axis, while A = 4.9 N ad L = 4.5 are costats. a) [5] The box is represeted o the figure at positio x = L. Figure out the orietatio of force F i that positio, ad setch the vector force diagra. Force F starts verticall, ad the it builds a horizotal copoet. At the give positio, its copoets becoe equal: xl x,, 4.9 N, 4.9 N F F F A A. xl The other forces are the weight ad the oral. F F x x b) [6] Write out Newto s d law alog the provided x ad -axes. Express the cotributio of force F i ters of A, L ad x. x-axis: Fx ax Ax L ax -axis F g a A g c) [9] Calculate explicitl the wor doe b each of the actig forces as the box oves through the iterval x = (, L). There are 3 forces actig o the box. The wor doe b each force is: W dx dxcos 9 W g dx gdx cos 9 g L L A A A WF F dx Fxdx xdx x L AL L L L J d) [5] Assuig that the box hovers o the groud (zero oral), use oe of the equatios i part (b) above to calculate the ass of the box. g A A g.5 g e) [5] Use the Wor-Eerg theore to calculate the speed of the box at positio x = L. B the Wor-Eerg theore, the chage i ietic eerg is equal to the et wor. Hece W et W Wg WF K K K v v W v v W 7.5 s. F F 4

5 PHYS 54 Practice Test 3 Solutios Sprig 8 P: A box with ass =.6 g is used to copress a sprig of costat = 5 N/ b a iitial copressio x =.. After the sprig is released, the box travels successivel through cofiguratios show o the figure. It oves a distace d =.6 alog a horizotal surface. I poit, the box starts to clib up a rap of agle θ = 37 ad legth d =.5. Evetuall, the box stops o the rap i poit, at height h with respect to the flat surface. All surfaces are rough, with coefficiet of ietic frictio μ =.7. a) [5] Calculate the echaical eerg E of the box i poit. This is the startup eerg of the box. The echaical eerg i poit is exclusivel elastic potetial eerg: E U x 3.6 J e x d v θ d h b) [6] Use the adjacet figure to setch the vector force diagra for the object i a arbitrar poit betwee. The calculate the wor W fr doe b the frictio betwee. This is the ocoservative wor doe o the box i this iterval. B defiitio, usig the otatios i the diagra, the wor doe b the frictio f is v Wfr fdcos 8 fd Wfr gd.6 J f g f g c) [6] Use the coservatio of eerg to calculate the echaical eerg of the box i poit. The use the result to copute the speed v of the box i poit. The coservatio of echaical eerg dictates that the chage i echaical eerg is equal to the wor doe b ocoservative forces (frictio i our case). Hece, we have betwee poits ad E E Wfr E Wfr E J 3. J Hece, we ca calculate the speed: E E K v v 3. s d) [7] Use the adjacet figure to setch the vector force diagra for the object i a arbitrar poit betwee. The calculate the wor W fr doe b the frictio betwee. This is the ocoservative wor doe o the box i this iterval. B defiitio, usig the otatios i the diagra, the wor doe b the frictio f is W f d cos 8 f d fr f g cos W gd cos fr.4 J e) [6] Use eerg coservatio to copute the axiu altitude h reached b the box up the rap (poit ). The box stops copletel o top of the rap, sice all the ietic eerg at the botto of the iclie is used to achieve gravitatioal potetial eerg ad couterbalace the frictio. Choosig the horizotal surface as the groud, we ca fid the eerg i poit : E3 E Wfr E3 Wfr E.4 3. J.6 J The eerg i poit is purel gravitatioal potetial eerg. Therefore E3 E3 gh h.4 g f θ g 5

6 PHYS 54 Practice Test 3 Solutios Sprig 8 P3: A woode bloc of ass M =.98 g is iitiall at rest whe it is hit horizotall b a bullet of ass =. g with speed v = 3 /s. The bullet is ibedded i the wood ad subsequetl the bullet-bloc sste copresses a sprig with elastic costat = N/ alog a frictioless surface. a) [5] Deote p bullet, p wood the respective oeta before the collisio, ad p bullet, p wood after the collisio. Write out the oetu coservatio durig the collisio, ad use it to calculate the speed of the sste bullet-box iediatel after the collisio. Net oetu is coserved durig the ielastic collisio: p p p p bullet wood bullet wood If we deote v the speed of the sste iediatel after the collisio, the oetu coservatio leads to. v v Mv v v 3. /s.3 /s M..98 v M x b) [5] What is the echaical eerg of the sste before ad after the collisio? The echaical eergies before ad iediatel after the collisio are exclusivel ietic: E v 9. J, E M v.9 J. c) [5] Calculate the percet of eerg lost durig the collisio. What happeed with the issig eerg? The eerg lost durig the perfectl ielastic collisio is give b the chage i the ietic eerg available i the sste: E E E E 8.9 J. This represets a percet of lost eerg E E E E % 99%. E E E This eerg is lost i the for of heat ad wood deforatio. d) [5] Use coservatio of echaical eerg to calculate the axiu copressio x of the sprig. The echaical eerg is coserved durig the copressio, so all the eerg available after the collisio is evetuall coverted ito elastic potetial eerg. Therefore, we get E E x x.3 3 c e) [] Kowig that the axiu copressio is achieved i a tie Δt =. s, what is the average power of the sprig? Also, use the Ipulse-Moetu Theore to calculate the average force exerted o the sste bloc-bullet b the sprig i this iterval. The power is the easured b the tie rate of eerg deliver. I our case, the sprig does wor i order to stop the box, equal to the potetial eerg evetuall stored i the sste. For a sprig, we as about a average power sice the force is ot costat so the eerg is exchaged at a variable rate. Hece, we get W E P.45 watts t t B the Ipulse-Moetu Theore, the ipulse related to the average force is give b the chage i oetu. Sice the speed of the sste varies fro v to i tie Δt, we get p M v v J Fav t p Fav t t t 3. N Notice that, sice the elastic force varies liearl with x, this could be also calculated b taig the average elastic force: Fav Fe x Fe x x 3. N 6

7 PHYS 54 Practice Test 3 Solutios Sprig 8 P4: A 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) x such that it collides with a dis # ovig with velocit v v, v v, v x where the velocit copoets are writte i the coordiate sste show o the figure. After the collisio, dis # oves with velocit v v, v v, v. x 3 6 Let s calculate the velocit v v x, v of dis # after the collisio. a) [3] Setch the et oetu o the figure before ad after collisio. The figure out how the oetu vector p is supposed to loo lie ad draw it o the figure. The 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. The write the oeta explicitl 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 x ad -directios provided o the figure. The 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 x, v. v x v x v x v x v v 3v v v v v v v v v x 6 Hece, the vector velocit of Dis # after the collisio is give b agitude v v v x v x 6 v. s 7 v 6 v.4 s.4 s, ad directio ta v v x 8 with respect to +x-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. Before the collisio: 9 K K K v v v v v v v v v.6 J x 4 8 After the collisio: K K K v v v v v v v v v.34 J x x e) [] So, is the collisio elastic or ielastic? Explai briefl. The ietic eerg after the collisio is sigificatl saller tha before, so the collisio is ielastic. before after x 7

PHY 181 Test 2 Practice Solutions Spring 2013

PHY 181 Test 2 Practice Solutions Spring 2013 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: