9 Lnear Moentu and Collsons CHAPTER OUTLINE 9. Lnear Moentu and Its Conseraton 9. Ipulse and Moentu 9.3 Collsons n One Denson 9.4 Two-Densonal Collsons 9.5 The Center o Mass 9.6 Moton o a Syste o Partcles 9.7 Deorable Systes 9.8 Rocket Propulson ANSWERS TO QUESTIONS *Q9. (a) No. Ipulse, F t, depends on the orce and the te or whch t s appled. *Q9. No. Work depends on the orce and on the dstance oer whch t acts. The oentu agntude s proportonal to the speed and the knetc energy s proportonal to the speed squared. () () The speed o the constant-ass object becoes 4 tes larger and the knetc energy 6 tes larger. Answer (a). The speed and the oentu becoe two tes larger. Answer (d). *Q9.3 () answer. For exaple, one partcle has 5 tes larger ass, t wll hae 5 tes saller speed and 5 tes saller knetc energy. () answer (d). Moentu s a ector. *Q9.4 () Equal net work nputs ply equal knetc energes. Answer. () Iagne one partcle has our tes ore ass. For equal knetc energy t ust hae hal the speed. Then ths ore asse partcle has 4() tes ore oentu. Answer (a). Q9.5 (a) It does not carry orce, or t dd, t could accelerate tsel. It cannot deler ore knetc energy than t possesses. Ths would olate the law o energy conseraton. It can deler ore oentu n a collson than t possesses n ts lght, by bouncng ro the object t strkes. *Q9.6 Mutual grataton brngs the ball and the Earth together. As the ball oes downward, the Earth oes upward, although wth an acceleraton on the order o 5 tes saller than that o the ball. The two objects eet, rebound, and separate. Moentu o the ball-earth syste s consered. Answer (d). Q9.7 (a) Lnear oentu s consered snce there are no external orces actng on the syste. The ragents go o n derent drectons and ther ector oenta add to zero. Knetc energy s not consered because the checal potental energy ntally n the explose s conerted nto knetc energy o the peces o the bob. 9 3794_9_ch9_p9-44 copy.ndd9 9 /8/7 5:47:49 PM
Chapter 9 Q9.8 Moentu conseraton s not olated we choose as our syste the planet along wth you. When you recee an pulse orward, the Earth recees the sae sze pulse backwards. The resultng acceleraton o the Earth due to ths pulse s uch saller than your acceleraton orward, but the planet s backward oentu s equal n agntude to your orward oentu. *Q9.9 () Durng the short te the collson lasts, the total syste oentu s constant. Whateer oentu one loses the other gans. Answer. () When the car oertakes the anure spreader, the aster-ong one loses ore energy than the slower one gans. Answer (a). Q9. The rle has a uch lower speed than the bullet and uch less knetc energy. Also, the butt dstrbutes the recol orce oer an area uch larger than that o the bullet. *Q9. () answer (a). The ball ges ore rghtward oentu to the block when the ball reerses ts oentu. () answer. In case (a) there s no teperature ncrease because the collson s elastc. Q9. Hs pact speed s deterned by the acceleraton o graty and the dstance o all, n g( y ). The orce exerted by the pad depends also on the unknown stness o the pad. Q9.3 The sheet stretches and pulls the two students toward each other. These eects are larger or a aster-ong egg. The te oer whch the egg stops s extended, ore or a aster ssle, so that the orce stoppng t s neer too large. *Q9.4 Thnk about how uch the ector oentu o the Frsbee changes n a horzontal plane. Ths wll be the sae n agntude as your oentu change. Snce you start ro rest, ths quantty drectly controls your nal speed. Thus s largest and d s sallest. In between the, b s larger than c and c s larger than g and g s larger than a. Also a s equal to e, because the ce can exert a noral orce to preent you ro recolng straght down when you throw the Frsbee up. The assebled answer s > b > c > g > a e > d. Q9.5 As one nger sldes towards the center, the noral orce exerted by the sldng nger on the ruler ncreases. At soe pont, ths noral orce wll ncrease enough so that statc rcton between the sldng nger and the ruler wll stop ther relate oton. At ths oent the other nger starts sldng along the ruler towards the center. Ths process repeats untl the ngers eet at the center o the ruler. Next step: Try a rod wth a nonunor ass dstrbuton. Next step: Wear a pece o sandpaper as a rng on one nger to change ts coecent o rcton. *Q9.6 (a) (d) (e) () No: echancal energy turns nto nternal energy n the couplng process. No: the Earth eeds oentu nto the boxcar durng the downhll rollng process. Yes: total energy s constant as t turns ro gratatonal nto knetc. Yes: I the boxcar starts ong north the Earth, ery slowly, starts ong south. No: nternal energy appears. Yes: Only orces nternal to the two-car syste act. 3794_9_ch9_p9-44 copy.ndd /8/7 5:47:5 PM
Lnear Moentu and Collsons Q9.7 The center o ass o the balls s n ree all, ong up and then down wth the acceleraton due to graty, durng the 4% o the te when the juggler s hands are epty. Durng the 6% o the te when the juggler s engaged n catchng and tossng, the center o ass ust accelerate up wth a soewhat saller aerage acceleraton. The center o ass oes around n a lttle closed loop wth a parabolc top and lkely a crcular botto, akng three reolutons or eery one reoluton that one ball akes. Lettng T represent the te or one cycle and F g the weght o one ball, we hae FJ6. T 3FT g and FJ 5 Fg. The aerage orce exerted by the juggler s e tes the weght o one ball. Q9.8 In epty space, the center o ass o a rocket-plus-uel syste does not accelerate durng a burn, because no outsde orce acts on ths syste. The rocket body does accelerate as t blows exhaust contanng oentu out the back. Accordng to the text s basc expresson or rocket propulson, the change n speed o the rocket body wll be larger than the speed o the exhaust relate to the rocket, the nal ass s less than 37% o the orgnal ass. Q9.9 To generalze broadly, around 74 the Englsh aored poston (a), the Gerans poston, and the French poston. But n France Ele de Chatelet translated Newton s Prncpa and argued or a ore ncluse ew. A Frenchan, Jean D Alebert, s ost responsble or showng that each theory s consstent wth the others. All the theores are equally correct. Each s useul or gng a atheatcally sple and conceptually clear soluton or soe probles. There s another coprehense echancal theory, the angular pulse angular oentu theore, whch we wll glpse n Chapter. It dentes the product o the torque o a orce and the te t acts as the cause o a change n oton, and change n angular oentu as the eect. We hae here an exaple o how scentc theores are derent ro what people call a theory n eeryday le. People who thnk that derent theores are utually excluse should brng ther thnkng up to date to around 75. SOLUTIONS TO PROBLEMS Secton 9. P9. 3. kg, (a) Lnear Moentu and Its Conseraton 3. ˆ 4. ˆj s p 9. ˆ. ˆj kg s Thus, and p x 9. kg s p y. kg s p px + py ( 9. ) + (. ) 5. kg s p y θ tan tan (. 33 p ) x 37 *P9. (a) Whoeer we consder the aggressor, brother and sster exert equal-agntude oppostelydrected orces on each other, to ge each other equal agntudes o oentu. We take the eastward coponent o the equaton total orgnal oentu total nal oentu or the two-sblng syste 65 kg (.9 s) + 4 kg 4.7 s, eanng she oes at 4.7 s east contnued on next page 3794_9_ch9_p9-44 copy.ndd /8/7 5:47:5 PM
Chapter 9 orgnal checal energy n grl s body total nal knetc energy U checal ()(65 kg)(.9 s) + ()(4 kg)(4.7 s) 77 J Syste oentu s consered wth the alue zero. The net orces on the two sblngs are o equal agntude n opposte drectons. Ther pulses add to zero. Ther nal oenta are o equal agntude n opposte drectons, to add as ectors to zero. P9.3 I hae ass 85. kg and can jup to rase y center o graty 5. c. I leae the ground wth speed gen by a( x x): ( 9. 8 s)(. 5 ). s Total oentu o the syste o the Earth and e s consered as I push the planet down and ysel up: e + 4 5. 98 kg ( ) 85. kg. s ~ e 3 s *P9.4 (a) For the syste o two blocks p, or Thereore, p p M 3M. s + Solng ges 6. s (oton toward the let). 3 kx M M 8 4 M + 3M. J FIG. P9.4 (d) The orgnal energy s n the sprng. A orce had to be exerted oer a dstance to copress the sprng, transerrng energy nto t by work. The cord exerts orce, but oer no dstance. Syste oentu s consered wth the alue zero. The orces on the two blocks are o equal agntude n opposte drectons. Ther pulses add to zero. The nal oenta o the two blocks are o equal agntude n opposte drectons. P9.5 (a) The oentu s p, so p and the knetc energy s K p p K ples K, so p K K 3794_9_ch9_p9-44 copy.ndd /8/7 5:47:5 PM
Lnear Moentu and Collsons 3 Secton 9. Ipulse and Moentu *P9.6 Fro the pulse-oentu theore, F t p, the aerage orce requred to hold onto the chld s F t ( kg )( 6 h ). 5 s s 644. 3 N. 37 h In tryng to hang onto the chld, he would hae to exert a orce o 6.44 kn (oer 4 lb) toward the back o the car, to slow down the chld s orward oton. He s not strong enough to exert so large a orce. I he were belted n and hs ars were rly ted around the chld, the chld would exert ths sze orce on h toward the ront o the car. A person cannot saely exert or eel a orce o ths agntude and a saety dece should be used. P9.7 (a) I Fdt area under cure I ( 3. 5 s)( 8 N) 3. 5 N s 3. 5 Ns F 3.5 s 9. kn Fro the graph, we see that F ax 8. kn FIG. P9.7 P9.8 The pact speed s gen by gy. The rebound speed s gen by gy. The pulse the loor parts to the ball s the change n the ball s oentu, up down + up ( gh + gh) up 5. ( 98. ) ( 96. + 5. ) 39. kg s up kg s upward P9.9 p F t p cos 6. cos6. y y y px ( sn 6. sn 6. ) sn6. ( 3. kg)(. s)(. 866) 5. kg s F ag px kg s 5. 6 N t. s FIG. P9.9 P9. Assue the ntal drecton o the ball n the x drecton. (a) Ipulse, I p p p. 6 4. ˆ. 6 ( 5.) ˆ 5. 4ˆ Ns Work K K. 6 4. 5. 7. J 3794_9_ch9_p9-44 copy.ndd3 3 /8/7 5:47:5 PM
4 Chapter 9 *P9. (a) The pulse s to the rght and equal to the area under the F-t graph: I [( + 4 N)]( s ) + (4 N)(3 s s) + ( N)( s). N s î + Ft (.5 kg)() + î N s (.5 kg) (d) Fro the sae equaton, (.5 kg)( î s) + î N s (.5 kg) F t. ˆ Ns F (5s) F. ˆ N ag ag ag 4 4.8 î s.8 î s *P9. (a) A graph o the expresson or orce shows a parabola openng down, wth the alue zero at the begnnng and end o the.8 s nteral. 8. s.8 s 9 N/s 5 N/s 8. s 3 t / t /3 3 ( 9 N/s) (. 8 s) / ( N/s )(. s) / I Fdt 9 t N/s 5 t N/s dt 944 N s 963 N s 98 N s 5 8 3 The athlete parts downward pulse to the plator, so the plator parts 98 N s o upward pulse to her. We could nd her pact speed as a ree-all calculaton, but we choose to wrte t as a conseraton-o energy calculaton: gy top () pact pact (gy top ) [(9.8 s ).6 ] 3.43 s down Graty, as well as the plator, parts pulse to her durng the nteracton wth the plator. + I plator + gt (65 kg)( 3.43 s) + 98 N s (65 kg)(9.8 s )(.8 s) 65 kg 3 N s + 98 N s 5 N s 65 kg 49 N s65 kg 3.83 s up Note that the athlete s puttng a lot o eort nto jupng and does not exert any orce on hersel. The useulness o the orce plator s to easure her eort by showng the orce she exerts on the loor. (d) Agan energy s consered n upward lght. () takeo gy top y top takeo g (3.83 s) (9.8 s ).748 P9.3 (a) Energy s consered or the sprng-ass syste: K + Us K + Us: + kx + x k Fro the equaton, a saller alue o akes x k larger. k I p x x k p (d) Fro the equaton, a larger alue o akes I x k larger. (e) For the glder, W K K kx The ass akes no derence to the work. 3794_9_ch9_p9-44 copy.ndd4 4 /8/7 5:47:5 PM
Lnear Moentu and Collsons 5 *P9.4 Ater 3 s o pourng, the bucket contans (3s)(.5 L s).75 lter o water, wth ass.75 L( kg L).75 kg, and eelng gratatonal orce.75 kg(9.8 s ) 7.35 N. The scale through the bucket ust exert 7.35 N upward on ths statonary water to support ts weght. The scale ust exert another 7.35 N to support the.75-kg bucket tsel. Water s enterng the bucket wth speed gen by gy top () pact pact (gy top ) [(9.8 s ).6 ] 7.4 s downward The scale exerts an extra upward orce to stop the downward oton o ths addtonal water, as descrbed by pact + F extra t The rate o change o oentu s the orce tsel: (ddt) pact + F extra F extra (ddt) pact (.5 kgs)( 7.4 s) +.78 N Altogether the scale ust exert 7.35 N + 7.35 N +.78 N 6.5 N Secton 9.3 Collsons n One Denson P9.5 Moentu s consered or the bullet-block syste 3 (. kg) ( 5. kg)(. 6 s) 3 s P9.6 (a) + 3 4 where 5. 4 kg 4. + 3(. ) 5. s 4 K K ( ) + ( ) 4 3 4 4 (.5 )(. 5 8. 6. ) 3. 75 J K K + E nt E nt +37.5 kj P9.7 (a) The nternal orces exerted by the actor do not change the total oentu o the syste o the our cars and the oe actor ( 4) ( 3)(. s) + ( 4. s) 6. s + 4. s 4 5. s FIG. P9.7 Wactor K K ( 3)(. s) + ( 4. s) 4 5. s 4 ( 5. kg) W actor (. + 6. 5. )( s) 37. 5 kj The eent consdered here s the te reersal o the perectly nelastc collson n the preous proble. The sae oentu conseraton equaton descrbes both processes. 3794_9_ch9_p9-44 copy.ndd5 5 /8/7 5:47:54 PM
6 Chapter 9 P9.8 Energy s consered or the bob-earth syste between botto and top o swng. At the top the st rod s n copresson and the bob nearly at rest. K + U K + U: M b + + Mg b g4 so b g Moentu o the bob-bullet syste s consered n the collson: + M( g ) 4M g P9.9 Frst we nd, the speed o at B beore collson. gh ( 9. 8)( 5. ) 9. 9 s FIG. P9.8 / Now we use the text s analyss o one-densonal FIG. P9.9 elastc collsons to nd, the speed o at B just ater collson. ( 99. ) s 33. s + 3 Now the 5-kg block bounces back up to ts hghest pont ater collson accordng to. gh ax ( 33. ) h ax ( ) 33 s. 556 98. s *P9. (a) We assue that energy s consered n the all o the basketball and the tenns ball. Each reaches ts lowest pont wth a speed gen by ( + ) K + Ug K U release g botto + gy b + gy ( 98. s)(. ) 485. s b contnued on next page 3794_9_ch9_p9-44 copy.ndd6 6 /8/7 5:47:55 PM
Lnear Moentu and Collsons 7 The two balls exert no orces on each other as they oe down. They collde wth each other ater the basketball has ts elocty reersed by the loor. We choose upward as poste. Moentu conseraton: ( 57 g) ( 4. 85 s) + ( 59 g)( 4. 85 s) ( 57 g) + ( 59 g) 4.85 /s 4.85 /s FIG. P9. To descrbe the elastc character o the collson, we use the relate elocty equaton we sole by substtuton 485. s 485. s 97. s 58 g s 57 g 59 g 97. s ( 57 g) + ( 59 g) 5 7 g s + 8 3 s. 8 s 647 Now the tenns ball-earth syste keeps constant energy as the ball rses: ( 57 g)(. 8 s) ( 57 g)( 9. 8 s ) y 65 s y 84. 98. s P9. (a), Let g and p be the x-coponents o elocty o the grl and the plank relate to the ce surace. Then we ay say that g p s the elocty o the grl relate to the plank, so that 5. () g But also we ust hae gg + pp, snce total oentu o the grl-plank syste s zero relate to the ce surace. Thereore 45. + 5, or 333. g p p g p Puttng ths nto the equaton () aboe ges 333. p p 5. or p. 346 ˆ s (answer b) FIG. P9. Then g 333. ( 346. ) 5. ˆ s (answer a) 3794_9_ch9_p9-44 copy.ndd7 7 /8/7 5:47:56 PM
8 Chapter 9 P9. We assue equal rng speeds and equal orces F requred or the two bullets to push wood bers apart. These equal orces act backward on the two bullets. For the rst, For the second, Agan, K + E K ech Substtutng or, p 3 ( 7. kg) F ( 8. ) 3 p 7. kg. 4 kg 3 ( 7. ). 4 K + E ech K : 7 (. kg) Fd (. 4 kg) 3 7 3 4 7. (. kg) Fd (. kg). 4 Fd ( ) 7 7. 3.. 4 3 3 Substtutng or, 3 7. Fd F ( 8. ). 4 d 794. c P9.3 (a) Fro the text s analyss o a one-densonal elastc collson wth an orgnally statonary target, the x-coponent o the neutron s elocty changes ro to ( ) 3 3. The x-coponent o the target nucleus elocty s 3. The neutron started wth knetc energy () The target nucleus ends up wth knetc energy () ( )( 3) Then the racton transerred s ( / 3) 48. 84 Because the collson s elastc, the other 7.6% o the 69 orgnal energy stays wth the neutron. The carbon s unctonng as a oderator n the reactor, slowng down neutrons to ake the ore lkely to produce reactons n the uel. K n (. 76). 6 J. 5 3 3 J K C (. 84). 6 J 4. 54 3 4 J 3794_9_ch9_p9-44 copy.ndd8 8 /8/7 5:47:57 PM
Lnear Moentu and Collsons 9 P9.4 (a) Usng conseraton o oentu, ( p) ( p) beore ater, ges ( 4. kg)( 5. s) + ( kg)( 3. s) + ( 3. kg) ( 4. s) [( 4. + + 3. ) kg] Thereore, +4. s, or 4. s toward the rght No. For exaple, the -kg and 3.-kg ass were to stck together rst, they would oe wth a speed gen by solng + ( ) 3kg kg 3. s 3. kg 4. s, or + 38. s Then when ths 3 kg cobned ass colldes wth the 4. kg ass, we hae ( 7 kg) ( 3 kg)(. 38 s) + ( 4. kg)( 5. s), and +4. s just as n part (a). Couplng order akes no derence to the nal elocty. P9.5 Durng pact, oentu o the clay-block syste s consered: + Durng sldng, the change n knetc energy o the clay-block-surace syste s equal to the ncrease n nternal energy: ( + ) d µ ( + ) gd (. kg). 65(. kg)( 9. 8 s )( 7. 5 ) FIG. P9.5 95. 6 s 977. s (. 3 kg) (. kg)( 9. 77 s) 9. s Secton 9.4 Two-Densonal Collsons *P9.6 (a) Oer a ery short te nteral, outsde orces hae no te to part sgncant pulse thus the nteracton s a collson. The opponent grabs the ullback and does not let go, so the two players oe together at the end o ther nteracton thus the collson s copletely nelastc. contnued on next page 3794_9_ch9_p9-44 copy.ndd9 9 /8/7 5:47:57 PM
Chapter 9 Frst, we consere oentu or the syste o two ootball players n the x drecton (the drecton o trael o the ullback). ( 9. kg)( 5. s) + ( 85 kg)v cosθ where θ s the angle between the drecton o the nal elocty V and the x axs. We nd V cos θ 43. s () Now consder conseraton o oentu o the syste n the y drecton (the drecton o trael o the opponent). ( 95. kg)( 3. s) + ( 85 kg)( V snθ ) whch ges V sn θ 54. s () Dde equaton () by () 54. tanθ. 633 43. Fro whch θ 3. 3 Then, ether () or () ges V 88. s 9 5 K (. kg)(. s) + ( 95. kg) ( 3. s) 85 kg 88 s 7 67 J K (. ). Thus, the knetc energy lost s 786 J nto nternal energy. 3 55. J P9.7 By conseraton o oentu or the syste o the two bllard balls (wth all asses equal), n the x and y drectons separately, 5. s+ ( 433. s) cos 3. + x 5. s ( 4. 33 s) sn 3. + y 6. s 5. sat 6. y x FIG. P9.7 Note that we dd not need to explctly use the act that the collson s perectly elastc. 3794_9_ch9_p9-44 copy.ndd /8/7 5:47:58 PM
Lnear Moentu and Collsons P9.8 We use conseraton o oentu or the syste o two ehcles or both northward and eastward coponents, to nd the orgnal speed o car nuber. For the eastward drecton: M( 3. s) MV cos 55. For the northward drecton: M MV sn 55. Dde the northward equaton by the eastward equaton to nd: 3. stan 55. 8. 6 s 4. 5 h FIG. P9.8 Thus, the drer o the north bound car was untruthul. Hs orgnal speed was ore than 35 h. P9.9 px p x cos 37. + Ycos 53. ( 5. s) O. 799 +. 6 5. () p y p O O Y s y sn 37. sn 53. Y. 6. 799 () O Solng () and () sultaneously, Y O 399. s and Y 3. s FIG. P9.9 P9.3 px px : Ocosθ+ Ycos ( 9. θ) cosθ+ snθ () O Y p y p : snθ sn ( 9. θ) y O Y O snθ cosθ () Y Fro equaton (), O Y cosθ snθ (3) Substtutng nto equaton (), cos θ Y Ysnθ snθ + so Y ( cos θ+ sn θ) snθ, and Y snθ FIG. P9.3 Then, ro equaton (3), O cosθ. We dd not need to wrte down an equaton expressng conseraton o echancal energy. In the proble stuaton, the requreent o perpendcular nal eloctes s equalent to the condton o elastcty. 3794_9_ch9_p9-44 copy.ndd /8/7 5:47:59 PM
Chapter 9 P9.3 + + : 3. ( 5. ) ˆ 6. ˆ 5. j 3. ˆ. ˆj s P9.3 x-coponent o oentu or the syste o the two objects: p x + px px + px: + 3 + 3 x y-coponent o oentu o the syste: + + 3 y y ( ) by conseraton o energy o the syste: + + + + 3 3 we hae also x 3 3 y y So the energy equaton becoes 4 4 9y + + 3 3 8 3 y y y x y or (a) y 3 The object o ass has nal speed and the object o ass 3 oes at 3 y y x 4 + y + 9 9 x + y 3 θ tan y x 3 θ tan 35. 3 3 7 P9.33 7. kg 7 5. kg 7 84. kg (the parent nucleus) (a) + + contnued on next page 3 3 6 6. ˆj s 6 4. ˆ s 7 where 3 36. kg FIG. P9.33 7 6 7 5. 6. ˆ 6 ( )( j) + ( 8. 4 ) 4. ˆ 7 + ( 3. 6 ) 3 6 933. ˆ 8. 33 6 ĵ s 3 3794_9_ch9_p9-44 copy.ndd /8/7 5:48: PM
Lnear Moentu and Collsons 3 E + + 33 E 7 6 ( 5 )( 6 ) 7.. + ( 8. 4 ) 4. 7 6 + 36. 5. ( ) E 439. 3 J P9.34 The ntal oentu o the syste s. Thus, and B 833. s (. ) (. s) B K ( ) + ( ) 833. s.. s 83 s K ( G) + ( )( B). 83 s or G +. B 97. s Fro conseraton o oentu, (. ) G B 6 or G. B () Solng () and () sultaneously, we nd (. ) +. 9. 7 s ( 9. 7 s /. 64) / B B B B 589. s (speed o blue puck ater collson) () and G 77. s (speed o green puck ater collson) Secton 9.5 The Center o Mass P9.35 The x-coordnate o the center o ass s x x x and the y-coordnate o the center o ass s y y + + +. kg + 3. kg + 5. kg + 4. kg y kg kg (. )( 3. ) + ( 3. ) ( 5. ) + (. 5 kg)( ) + ( 4. kg) (. 5 ). kg + 3. kg + 5. kg + 4. kg. 3794_9_ch9_p9-44 copy.ndd3 3 /8/7 5:48: PM
4 Chapter 9 P9.36 Let the x axs start at the Earth s center and pont toward the Moon. x x + x + 4 8 5. 98 kg + 7. 36 kg 384. 4 65. kg 4.67 6 ro the Earth s center The center o ass s wthn the Earth, whch has radus 6. 37 6. It s.7 M below the pont on the Earth s surace where the Moon s straght oerhead. P9.37 Let A represent the area o the botto row o squares, A the ddle square, and A 3 the top par. A A+ A + A3 M M+ M + M3 M M A A A 3 c, A c, A 3 c, A 6 c M M A M M 3 A c 6 c M M A A c M M 6 c 6 3 c M3 M A A 6 c M M 3 xm + xm + xm 3 3 5. c( M) + c M x 5. ( 6 ) +. c( 3 M ) M M x. 7 c ( 5 M. c) + + y 6 M 5. c 3 M 5. c 3. 3 c M y 3. 3 c P9.38 (a) Represent the heght o a partcle o ass d wthn the object as y. Its contrbuton to the gratatonal energy o the object-earth syste s ( d) gy. The total gratatonal energy s U g gyd g yd. For the center o ass we hae y M yd, so all ass U gmy g The olue o the rap s ( 36. )( 57. )( 648. ) 83. 3 3. Its ass s 3 3 3 6 ρv ( 3 8 kg )(. 83 ) 6. 96 kg. As shown n the chapter, ts center o ass s aboe ts base by one-thrd o ts heght, y 5 7 5 3 3... Then 6 U Mgy 696. kg 98. s 53. 357. 8 J g FIG. P9.37 3794_9_ch9_p9-44 copy.ndd4 4 /8/7 5:48: PM
Lnear Moentu and Collsons 5 P9.39 Ths object can be ade by wrappng tape around a lght st unor rod.. 3 3. (a) M λdx 5. g+. x g dx. 3 M 5. x g+. x g 5. 9 g x x xd. 3 all ass xdx M M λ M x x g+ 5 9 g 5... 3 g 3 3 5. x g +. x g dx. 3. 53 P9.4 Take the orgn at the center o curature. We hae L r 4 π, L r. An ncreental bt o the rod at angle θ ro the x axs π has ass gen by d M rdθ L, d Mr L d θ where we hae used the denton o radan easure. Now y 35 35 Mr yd r M M L d r snθ θ L snθdθ all ass θ 45 45 35 L ( cosθ ) 4 L 4 L + π L π π 45 y θ FIG. P9.4 x L The top o the bar s aboe the orgn by r, so the center o ass s below the ddle o the π bar by L 4 L L 63 5L π π π π.. Secton 9.6 P9.4 (a) Moton o a Syste o Partcles + M M. kg. ˆ s 3. ˆ j s 3. kg. ˆ s 6. ˆj s 4. ˆ + 4. ˆ j + ( + ) s 5. kg p M ( 5. kg) 4. ˆ + 4. ˆ j s 7. ˆ +. ĵ kg s 3794_9_ch9_p9-44 copy.ndd5 5 /8/7 5:48:3 PM
6 Chapter 9 *P9.4 r r + r 35. + j t+ jt ( 3 ˆ 3 ˆ) ˆ + 55. 3ˆ ˆt 6ˆjt + 35. + 55. 8. 3 +. 7t. t ˆ. 5t. 778t ˆj + ( + ) (a) At t.5 s, r ( +. 83. 7.5. 6. 5 ) ˆ + (. 5. 5 +. 778 6. 5) ˆj (.89 î.39 ĵ)c We can conenently do part on the way to part : dr ( 7. 44. t) ˆ + ( 5. + 56. t) ˆj dt att 5. s, ( 7. 44. 5. ) ˆ + ( 5. + 5. 65. ) ĵ ( 4.94 î +.39 ĵ) cs Now the total lnear oentu s the total ass tes the elocty o the center o ass: (9 g) ( 4.94 î +.39 ĵ) c s ( 44.5 î +.5 ĵ) g c s (d) Derentatng agan, d a ( 44. ) ˆ + 56. ˆj dt The center o ass acceleraton s (.44 î +.56 ĵ) c s at t.5 s and at all tes. (e) The net orce on the syste s equal to the total ass tes the acceleraton o the center o ass: (9 g) (.44 î +.56 ĵ ) c s ( î + 4 ĵ) µn P9.43 Let x dstance ro shore to center o boat length o boat x dstance boat oes as Julet oes toward Roeo The center o ass stays xed. Beore: Ater: x x + ( + ) Mx b + MJ x MR x M + M + M B J R + ( + ) + ( + ) MB x x MJ x x MR x x M + M + M 55. 77. + x ( 8. 55. 77. ) + ( 55. + 77. ) 55. 55. (. 7) x. 7 P9.44 (a) Conseraton o oentu or the two-ball syste ges us: B. kg. 5 s. 3 kg. 4 s. J R + ( ) kg +. 3 kg Relate elocty equaton: 9. s FIG. P9.43 Then. 3.. +. 3. 9 + 78. s. s. 78ˆ s. ˆ s contnued on next page 3794_9_ch9_p9-44 copy.ndd6 6 /8/7 5:48:4 PM
Lnear Moentu and Collsons 7 Beore, kg s kg (. )(. 5 ) ˆ + (. 3 ) (. 4 s)ˆ. 5 kg. 36 sˆ Aterwards, the center o ass ust oe at the sae elocty, because the oentu o the syste s consered. Secton 9.7 Deorable Systes *P9.45 (a) Yes. The only horzontal orce on the ehcle s the rctonal orce exerted by the loor, so t ges the ehcle all o ts nal oentu, (6 kg)(3 î s) 8. î kg s (d) (e) No. The rcton orce exerted by the loor on each statonary bt o caterpllar tread acts oer no dstance, so t does zero work. Yes, we could say that the nal oentu o the cart cae ro the loor or ro the planet through the loor, because the loor parts pulse. No. The loor does no work. The nal knetc energy cae ro the orgnal gratatonal energy o the eleated load, n aount ()(6 kg)(3 s) 7. J. Yes. The acceleraton s caused by the statc rcton orce exerted by the loor that preents the caterpllar tracks ro slppng backward. *P9.46 (a) Yes. The loor exerts a orce, larger than the person s weght oer te as he s takng o. (d) (e) No. The work by the loor on the person s zero because the orce exerted by the loor acts oer zero dstance. He leaes the loor wth a speed gen by () gy [(9.8 s ).5 ].7 s, so hs oentu edately ater he leaes the loor s (6 kg)(.7 s up) 3 kg s up Yes. You could say that t cae ro the planet, that ganed oentu 3 kg s down, but t cae through the orce exerted by the loor oer a te nteral on the person, so t cae through the loor or ro the loor through drect contact. ()(6 kg)(.7 s) 88. J No. The energy cae ro checal energy n the person s leg uscles. The loor dd no work on the person. 3794_9_ch9_p9-44 copy.ndd7 7 /8/7 5:48:5 PM
8 Chapter 9 *P9.47 (a) When the cart hts the buper t edately stops, and the hangng partcle keeps ong wth ts orgnal speed. The partcle swngs up as a pendulu on a xed pot, keepng constant energy. Measure eleatons ro the pot: () + g( L) + g( L cos θ) Then [gl( cosθ)] [gl( cosθ)] [(9.8 s )(. )( cos 35 )].6 s Yes. The buper ust prode the horzontal orce to the let to slow down the swng o the partcle to the rght, to reerse ts rghtward oton, and to ake t speed up to the let. When the partcle passes ts straght-down poston ong to the let, the buper stops exertng orce. It s at ths oent that the cart-partcle syste oentarly has zero horzontal acceleraton or ts center o ass. *P9.48 Dependng on the length o the cord and the te nteral t or whch the orce s appled, the sphere ay hae oed ery lttle when the orce s reoed, or we ay hae x and x nearly equal, or the sphere ay hae swung back, or t ay hae swung back and orth seeral tes. Our soluton apples equally to all o these cases. (a) The appled orce s constant, so the center o ass o the glder-sphere syste oes wth constant acceleraton. It starts, we dene, ro x and oes to (x + x ). Let and represent the horzontal coponents o elocty o glder and sphere at the oent the orce stops. Then the elocty o the center o ass s ( + ) and because the acceleraton s constant we hae (x + x ) [( + )] t t (x + x )( + ) The pulse-oentu theore or the glder-sphere syste s F t + F(x + x )( + ) ( + ) F(x + x ) ( + ) F(x + x )4 ( + ) 4 Then [F(x + x )] The appled orce does work that becoes, ater the orce s reoed, knetc energy o the constant-elocty center-o-ass oton plus knetc energy o the braton o the glder and sphere relate to ther center o ass. The appled orce acts only on the glder, so the work-energy theore or the pushng process s Fx ()( ) + E b Substtuton ges Fx ()( )F(x + x ) + E b Fx + Fx + E b Then E b Fx Fx When the cord akes ts largest angle wth the ertcal, the bratonal oton s turnng around. No knetc energy s assocated wth the braton at ths oent, but only gratatonal energy: gl( cos θ) F(x x ) Solng ges θ cos [ F(x x )gl] P9.49 A pcture one second later ders by showng e extra klogras o sand ong on the belt. (a) p x ( 5. kg )(. 75 s ) 375. N t. s The only horzontal orce on the sand s belt rcton, so ro px + t p px x ths s 375. N t The belt s n equlbru: Fx ax: + Fext and F ext 375. N contnued on next page 3794_9_ch9_p9-44 copy.ndd8 8 /8/7 5:48:6 PM
Lnear Moentu and Collsons 9 (d) W F rcos θ 375. N(.75 ) cos 8. J (e) 5. kg 75. s 4. J One-hal o the work nput becoes knetc energy o the ong sand and the other hal becoes addtonal nternal energy. The nternal energy appears when the sand does not elastcally bounce under the hopper, but has rcton elnate ts horzontal oton relate to the belt. By contrast, all o the pulse nput becoes oentu o the ong sand. Secton 9.8 Rocket Propulson P9.5 (a) The uel burns at a rate Thrust e dm dt dm dt. 7 g 668. 3 kg s.9 s : 5. 6 N 6. 68 3 kg s M e e 787 s eln M : ( 787 s) 38 s 53. 5 g+ 5. 5 g ln 53.5 g + 5. 5 g 7. g dm 3 4 7 P9.5 (a) Thrust e Thrust (. 6 s) (. 5 kgs) 3. 9 N dt Fy Thrust Mg Ma: 3. 9 7 ( 3. 6 )( 9. 8) ( 3. 6 )a a 3. s P9.5 (a) Fro the equaton or rocket propulson n the text, M M e ln e ln M M M Now, M M kt, so kt k e ln e ln M M t M Wth the denton Tp, ths becoes k t ( t) e ln T Wth e 5 s, and T p 44 s, t ( 5 s) ln 44 s t s s 4 4 488 6 88 8 78 69 3 373 p 4 35 3 5 5 5 (/s) 4 6 8 FIG. P9.5 4 t (s) contnued on next page 3794_9_ch9_p9-44 copy.ndd9 9 /8/7 5:48:7 PM
3 Chapter 9 d d at dt e t ( Tp ) ln e dt e Tp Tp t Tp t T p, or at e T t p (d) 5 s Wth e 5 s, and T p 44 s, a 44 s t t( s) a( s).4. 4 4.4 6 7.9 8 3.4 34. 6.5 3 5 4 8 6 4 a (/s ) 4 6 8 FIG. P9.5(d) 4 t (s) t t t t (e) xt + dt e dt t ln T p dt e T p ln T p T p t t t xt etp T p T p ln T p xt T t e ( p ) ln t + et T p t () Wth e 5 s. 5 k s, and T p 44 s, t x. 5 ( 44 t) ln t 44 +5. t( s) x( k) 4 6 8 3 9. 93.. 4. 7. 7 5 6 4 8 6 4 x (k) 4 6 8 53 FIG. P9.5() 4 t (s) 3794_9_ch9_p9-44 copy.ndd3 3 /8/7 5:48:8 PM
Lnear Moentu and Collsons 3 P9.53 In M ln we sole or M. e M e (a) M e M M e 3. kg 4. 45 5 3 5 3 The ass o uel and oxdzer s M M M ( 445 3. ) kg 44 etrc tons M e ( 3. etrc tons) 3. etrc tons 9. etrc tons Ths s uch less than the suggested alue o 44.5. Matheatcally, the logarth n the rocket propulson equaton s not a lnear uncton. Physcally, a hgher exhaust speed has an extra-large cuulate eect on the rocket body s nal speed, by countng agan and agan n the speed the body attans second ater second durng ts burn. Because o the exponental, a relately sall ncrease n engne ecency causes a large change n the aount o uel and oxdzer requred. kg Addtonal Probles P9.54 (a) When the sprng s ully copressed, each cart oes wth sae elocty. Apply conseraton o oentu or the syste o two glders p p : + + ( + ) + Only conserate orces act; thereore E. + ( + ) + kx Substtute or ro (a) and sole or x. x x + + ( ) ( ) ( + ) k( + ) ( + ) ( ) k + k + contnued on next page 3794_9_ch9_p9-44 copy.ndd3 3 /8/7 5:48:9 PM
3 Chapter 9 + + ( ) () Conseraton o oentu: Conseraton o energy: + + ( ) ( ) whch sples to: Factorng ges ( ) ( + ) ( ) + and wth the use o the oentu equaton (equaton ()), ths reduces to ( + ) ( + ) or + () Substtutng equaton () nto equaton () and splyng yelds: + + + Upon substtuton o ths expresson or nto equaton, one nds + + + Obsere that these results are the sae as two equatons gen n the chapter text or the stuaton o a perectly elastc collson n one denson. Whateer the detals o how the sprng behaes, ths collson ends up beng just such a perectly elastc collson n one denson. P9.55 We hope the oentu o the wrench prodes enough recol so that the astronaut can reach the shp beore he loses le support! We ght expect the elapsed te to be on the order o seeral nutes based on the descrpton o the stuaton. No external orce acts on the syste (astronaut plus wrench), so the total oentu s constant. Snce the nal oentu (wrench plus astronaut) ust be zero, we hae nal oentu ntal oentu. Thus astronaut astronaut + wrench wrench astronaut astronaut wrenchwrench. 5 kg. s 8. kg At ths speed, the te to trael to the shp s 3. t 4 s 4. nutes.5 s. 5 s The astronaut s ortunate that the wrench gae h sucent oentu to return to the shp n a reasonable aount o te! In ths proble, we dd not thnk o the astronaut as drtng away ro the shp when he threw the wrench. Howeer slowly, he ust be drtng away snce he dd not encounter an external orce that would reduce hs elocty away ro the shp. There s no ar rcton beyond earth s atosphere. In a real-le stuaton, the astronaut would hae to throw the wrench hard enough to oercoe hs oentu caused by hs orgnal push away ro the shp. 3794_9_ch9_p9-44 copy.ndd3 3 /8/7 5:48: PM
Lnear Moentu and Collsons 33 *P9.56 Proceedng step by step, we nd the real actor s speed just beore collson, usng energy conseraton n the swng-down process: a gy () a [(9.8 s )(.8 )] 5.94 s Now or the elastc collson wth a statonary target we use the specalzed equaton ro the chapter text ( )( + ) (8 kg)(5.94 s)(8 kg + ) (95 kg s)(8 kg + ) The te or the clone s all nto the ocean s gen by y y t + ()a y t 36 + ()( 9.8 s )t t.7 s so hs horzontal range s R t (.7 s )(95 kg s)(8 kg + ).58 3 kg (8 kg + ) (d) (e) By substtuton, 576 kg (8 kg + 79 kg) 6. A lttle heaer and he does not go so ar: 576 kg (8 kg + 8 kg) 6. We sole 3 58 kg (8 kg + ) 8 kg + 85.87 kg 5.87 kg The axu alue or R s 5768 3., obtaned n the lt as () we ake go to zero. (g) The nu alue o R s approachng zero, obtaned n the lt as (h) we ake go to nnty. () (j) (k) Yes, echancal energy s consered untl the clone splashes down. Ths prncple s not sucent to sole the proble. We need also conseraton o oentu n the collson. Yes, but t s not useul to nclude the planet n the analyss o oentu. We use nstead oentu conseraton or the actor-clone syste whle they are n contact. In sybols we hae [ g (.8 )] (8 kg) [ g (.8 )] (8 kg + ) t [(36 )g] and R [(36 )g] (8 kg) [ g (.8 )] (8 kg + ) Here g ddes out. At a locaton wth weaker graty, the actor would be ong ore slowly beore the collson, but the clone would ollow the sae trajectory, ong ore slowly oer a longer te nteral. P9.57 Usng conseraton o oentu ro just beore to just ater the pact o the bullet wth the block: ( M + ) or M + () The speed o the block and ebedded bullet just ater pact ay be ound usng kneatc equatons: Thus, d t and h gt h t d g gd and d g t h h M h FIG. P9.57 d M Substtutng nto () ro aboe ges + gd h. 3794_9_ch9_p9-44 copy.ndd33 33 /8/7 5:48: PM
34 Chapter 9 P9.58 (a) The ntal oentu o the syste s zero, whch reans constant throughout the oton. Thereore, when leaes the wedge, we ust hae or wedge + block + ( + ) 3. kg 5. kg 4. s wedge so wedge. 667 s wedge block 4. /s Usng conseraton o energy or the blockwedge-earth syste as the block sldes down the sooth (rctonless) wedge, we hae K + U K K U block syste + wedge + block sys te + K wedge +x FIG. P9.58 or [ + gh ]+ ( 4) + 667. + (. ) whch ges h. 95 P9.59 (a) Conseraton o oentu: + ( + ) s 5. kg( ˆ 3ˆ 8ˆ j k) s + 5. kg ( 5. ˆ 5. ˆj 4kˆ ) kg s + ( 5. ˆ 5. ˆj+ 4kˆ) 5. kg ˆ 3ˆj+ kˆ s 5. kg ˆ ˆj 3kˆ + + The orgnal knetc energy s 5. kg 5 3. kg( + + ) s + 5. kg + + 3 kg s s 4. J + derent ro the The nal knetc energy s 5 3 8. kg + + s 8. 5 J orgnal energy so the collson s nelastc. We ollow the sae steps as n part (a): ( 5. ˆ + 5. ˆj 4kˆ ) kg s 5. kg( 5. ˆ + 75. ˆj kˆ ) s+. 5 kg We see + ( + ). ˆ. ˆj 5 + 5 4kˆ kg s. 5ˆ. 375ˆj kˆ 5kg.. 5 ˆ +. 75 ˆ j. k ˆ, so the collson s perectly nelastc. s kg s contnued on next page 3794_9_ch9_p9-44 copy.ndd34 34 /8/7 5:48: PM
Lnear Moentu and Collsons 35 Conseraton o oentu: ( 5. ˆ + 5. ˆj 4kˆ ) kg s 5. kg ˆ+ 3ˆj+ akˆ s+ ( 5. ˆ 5. ˆj 4kˆ) kgs + ( 6. 7. 333a) kˆ s 5. kg 5. kg + 5. ˆ 5. ˆ j 5. akˆ kgs Conseraton o energy: 4 5 3. J. kg( + + a ) s + 5. kg( 67. +. 333a) 5. J+ 5. a + 533. J+ 33. a+. 83 3a s. 333a +. 33a 6. 67 33 33 4 333. ±. (. a ) ( 6. 67). 667 a 74. or 674.. Ether alue s possble. wth a 74., wth a 674., (. 67. 333(. 74) ) kˆ s 3. 58kˆ s (. 67. 333( 6. 74) ) kˆ s. 49kˆ s P9.6 Consder the oton o the reghter durng the three nterals: () beore, () durng, and (3) ater collson wth the plator. (a) Whle allng a heght o 4., her speed changes ro to as ound ro E K + U ( K U ), or K E U + K + U When the ntal poston o the plator s taken as the zero leel o gratatonal potental, we hae hcos ( 8 ) + + gh Solng or ges ( + ) h+ gh 3 4. 75. 98. 4. 75. FIG. P9.6 68. s contnued on next page 3794_9_ch9_p9-44 copy.ndd35 35 /8/7 5:48:3 PM
36 Chapter 9 Durng the nelastc collson, oentu s consered; and s the speed o the reghter and plator just ater collson, we hae ( + M) or 75. ( 6. 8) 538. s + M 75. +. Followng the collson and agan solng or the work done by non-conserate orces, usng the dstances as labeled n the gure, we hae (wth the zero leel o gratatonal potental at the ntal poston o the plator): E K + Ug + Us K Ug Us, or s + ( + M) g( s) + ks ( + M) Ths results n a quadratc equaton n s: s ( 93) s+ 3s 375 or s. P9.6 (a) Each prate swngs down accordng to gr MgR M gr The collson: + M + ( + M) M M + Swngng up: ( M + ) ( M + ) gr( cos 35 ) gr( cos 35 ) gr( cos35 ) ( M + ) ( M ) gr. 45M +. 45 M. 45. 575M. 43 M No change s requred the orce s derent. The nature o the orces wthn the syste o colldng objects does not aect the total oentu o the syste. Wth strong agnetc attracton, the heaer object wll be ong soewhat aster and the lghter object aster stll. Ther extra knetc energy wll all be edately conerted nto extra nternal energy when the objects latch together. Moentu conseraton guarantees that none o the extra knetc energy reans ater the objects jon to ake the swng hgher. 3794_9_ch9_p9-44 copy.ndd36 36 /8/7 5:48:4 PM
Lnear Moentu and Collsons 37 P9.6 (a) Utlzng conseraton o oentu, A + + A gh 69. s A B y Utlzng the two equatons gt y and x A t we cobne the to nd x g A x yg / y Fro the data, 66 A. s FIG. P9.6 x Most o the % derence between the alues or speed s accounted or by the uncertanty n the data, estated as... + + + + 868. 68. 8 63 57 85. 3.%. *P9.63 (a) In the sae sybols as n the text s Exaple, the orgnal knetc energy s K A () A. The exaple shows that the knetc energy edately ater latchng together s K B () A ( + ) so the racton o knetc energy reanng as knetc energy s K B K A ( + ) K B K A 9.6 kg (9.6 kg + 4 kg).49 (d) Moentu s consered n the collson so oentu ater dded by oentu beore s.. Energy s an entrely derent thng ro oentu. A coparson: When a photographer s sngle-use lashbulb lashes, a agnesu laent oxdzes. Checal energy dsappears. (Internal energy appears and lght carres soe energy away.) The easured ass o the lashbulb s the sae beore and ater. It can be the sae n spte o the % energy conerson, because energy and ass are totally derent thngs n classcal physcs. In the ballstc pendulu, conerson o energy ro echancal nto nternal does not upset conseraton o ass or conseraton o oentu. 3794_9_ch9_p9-44 copy.ndd37 37 /8/7 5:48:5 PM
38 Chapter 9 *P9.64 (a) The ass o the slegh plus you s 7 kg. Your elocty s 7.5 /s n the x drecton. You unbolt a 5.-kg seat and throw t back at the raenng woles, gng t a speed o 8. /s relate to you. Fnd the elocty o the seat relate to the ground ater your acton, and the elocty o the slegh. We substtute 8 s + + 7 kg 7.5 s 5 kg 8 s 55 kg 5 kg s kg s 7 kg 45 s 794. s 7 8 s 7. 94 s. 55 6 s + The nal elocty o the seat s. 55 6 s. ˆ That o the slegh s 7. 94 s. ˆ You do work on both the slegh and the seat, to change ther knetc energy accordng to K + W K + K + W 7 7 5 kg. s 5 kg 55 6. s 55 kg 7. 94 s 7 594 J + W. 3 J+ 8 47 J W 453 J + 3794_9_ch9_p9-44 copy.ndd38 38 /8/7 5:48:6 PM
Lnear Moentu and Collsons 39 *P9.65 The orce exerted by the sprng on each block s n agntude F kx 3. 85 N. 8. 38 N. (a) s Wth no rcton, the elastc energy n the sprng becoes knetc energy o the blocks, whch hae oenta o equal agntude n opposte drectons. The blocks oe wth constant speed ater they leae the sprng. ( K + U) ( K + U) kx + 385 (. N)( 8) ( kg) + ( kg) 5 5... + + (. 5 kg) ( ˆ ) + ( 5. kg) ˆ. 3 J. 5.. 5 5 ( kg)( ) + ( kg) ( kg) 3 8. J. s. 8 sˆ.75 kg (. 8 s). 56 s. 56 s ˆ, n For the lghter block, Fy a 5. kg 9.8 s y, n 45. N, k µ kn. ( 45. N) 45. N. We assue that the axu orce o statc rcton s a slar sze. Snce.38 N s larger than.45 N, ths block oes. For the heaer block, the noral orce and the rctonal orce are twce as large: k. 49 N. Snce.38 N s less than ths, the heaer block stands stll. In ths case, the rctonal orces exerted by the loor change the oentu o the two-block syste. The lghter block wll gan speed as long as the sprng orce s larger than the rcton orce: that s untl the sprng copresson becoes x gen by Fs kx,. 45 N ( 3. 85 N ) x,. 63 6 x. Now or the energy o the lghter block as t oes to ths axu-speed pont we hae K + U kd K + U +. 3 J. 45 N. 8. 63 6 5 ( ) (. kg) + ( 3. 85 N )(. 63 6 ). 3 J. 4 J ( 5 7 8. kg) +. J (. 55 J). 64 5. kg s Thus or the heaer block the axu elocty s and or the lghter. 64 s ˆ. For the lghter block, k. 46(. 45 N). 3 N. The orce o statc rcton ust be at least as large. The.38-N sprng orce s too sall to produce oton o ether block. Each has axu speed. 3794_9_ch9_p9-44 copy.ndd39 39 /8/7 5:48:6 PM
4 Chapter 9 P9.66 The orbtal speed o the Earth s πr π. 496 4 E 3.56 7 98. s T s In sx onths the Earth reerses ts drecton, to undergo oentu change S FIG. P9.66 E ( ) E 4 4 E E E 5. 98 kg. 98 s 3.56 9 kg s Relate to the center o ass, the sun always has oentu o the sae agntude n the opposte drecton. Its 6-onth oentu change s the sae sze, S S 356. 9 kg s. Then S 9 356. kg s 3. 99 kg. 79 s P9.67 (a) Fnd the speed when the bullet eerges ro the block by usng oentu conseraton: MV + The block oes a dstance o 5. c. Assue or an approxaton that the block quckly reaches ts axu elocty, V, and the bullet kept gong wth a constant elocty,. The block then copresses the sprng and stops. 4 /s 5. c MV kx ( 9 ) 5 V N. 5. s. kg ( 3 MV 5. kg)( 4 s). kg 5. 3 5. kg s ( s) E K + U ( ) 5 3. kg s 5. kg 4 + ( 9 N) ( 5. ) E 374 J, or there s a echancal energy loss o 374 J. FIG. P9.67 ( s) 3 3794_9_ch9_p9-44 copy.ndd4 4 /8/7 5:48:7 PM
Lnear Moentu and Collsons 4 P9.68 (a) p + Ft p : 3. kg 7. s ˆ j. Nˆ 5. s 3. kg + (. ˆ + 7. ˆ j 7. ˆj). ˆ + 7. ˆj s a : a t 5. s F a :. Nˆ a 4. ˆ s 3. kg s 4. ˆ s (d) r t + a t : r 7 j 5. sˆ. s 4. s ˆ ( 5. s) + r 5. ˆ + 35. ˆj (e) W F r: W (. Nˆ ) 5. ˆ + 35. ˆj 6 J () (g) 3. (. ˆ 7. ˆ ). ˆ kg j ( + 7. ĵ ) s (. 5 kg)( 449 s ) 674 J + 3. kg 7. s 6 J 674 J + + W (h) The acceleratons coputed n derent ways agree. The knetc energes coputed n derent ways agree. The three theores are consstent. 3794_9_ch9_p9-44 copy.ndd4 4 /8/7 5:48:8 PM
4 Chapter 9 P9.69 The orce exerted by the table s equal to the change n oentu o each o the lnks n the chan. By the calculus chan rule o derates, dp F d d dt ( ) dt dt + d dt We choose to account or the change n oentu o each lnk by hang t pass ro our area o nterest just beore t hts the table, so that d dt and d dt Snce the ass per unt length s unor, we can express each lnk o length dx as hang a ass d: d M L dx The agntude o the orce on the allng chan s the orce that wll be necessary to stop each o the eleents d. d M dx M F dt L dt L Ater allng a dstance x, the square o the elocty o each lnk gx (ro kneatcs), hence Mgx F L The lnks already on the table hae a total length x, and ther weght s supported by a orce F : Hence, the total orce on the chan s F Mgx L Ftotal F+ F Mgx L That s, the total orce s three tes the weght o the chan on the table at that nstant. 3 FIG. P9.69 ANSWERS TO EVEN PROBLEMS P9. (a) She oes at 4.7 s east. 77 J Syste oentu s consered wth the alue zero. The orces on the two sblngs are o equal agntude n opposte drectons. Ther pulses add to zero. Ther nal oenta are o equal agntude n opposte drectons. P9.4 (a) 6. ( î) s 8.4 J The orgnal energy s n the sprng. A orce had to be exerted oer a dstance to copress the sprng, transerrng energy nto t by work. The cord exerts orce, but oer no dstance. (d) Syste oentu s consered wth the alue zero. The orces on the two blocks are o equal agntude n opposte drectons. Ther pulses add to zero. The nal oenta o the two blocks are o equal agntude n opposte drectons. P9.6 In tryng to hang onto the chld, he would hae to exert a orce o 6.44 kn toward the back o the car, to slow down the chld s orward oton. He s not strong enough to exert so large a orce. I he were soldly belted n and ted to the chld, the chld would exert ths sze orce on h toward the ront o the car. 3794_9_ch9_p9-44 copy.ndd4 4 /8/7 5:48:9 PM
Lnear Moentu and Collsons 43 P9.8 39. kg s upward P9. (a) 54. N s toward the net 7. J P9. (a) 98 N s up 3.43 s 3.83 s (d).748 P9.4 6.5 N P9.6 (a) 5. s 375. 4 J P9.8 4M g P9. (a) 4.85 s 8.4 P9. 7.94 c P.4 (a) 4. s toward the rght No, couplng order akes no derence P9.6 (a) Oer a ery short te nteral, outsde orces hae no te to part sgncant pulse thus the nteracton s a collson. The opponent grabs the ullback and does not let go, so the two players oe together at the end o ther nteracton thus the collson s copletely nelastc..88 s at 3.3 783 J becoes nternal energy. P9.8 No; hs speed was 4. 5 h P9.3 Y snθ ; O cosθ P9.3 (a) ; 3 35.3 P9.34 Blue 5.89 s and Green 7.7 s P9.36 4.67 6 ro the Earth s center P9.38 (a) see the soluton 357. 8 J P9.4.63 5L P9.4 (a) (.89 î.39 ĵ)c ( 44.5 î +.5 ĵ ) g c s ( 4.94 î +.39 ĵ) c s (d) (.44 î +.56 ĵ) cs (e) ( î + 4 ĵ) µn P9.44 (a). 78î s;.î s. 36î s P9.46 (a) Yes No. The work by the loor on the person s zero. 3 kg s up (d) Yes. You could say that t cae ro the planet, that ganed oentu 3 kg s down, but t cae through the orce exerted by the loor oer a te nteral on the person, so t cae through the loor or ro the loor through drect contact. (e) 88. J () No. The energy cae ro checal energy n the person s leg uscles. The loor dd no work on the person. P9.48 (a) [F(x + x )] cos [ F(x x )gl] P9.5 (a) 787 s 38 s P9.5 see the soluton 3794_9_ch9_p9-44 copy.ndd43 43 /8/7 5:48:9 PM
44 Chapter 9 P9.54 (a) + + ( ) k( + ) + + + ; + + + P9.56 (a) R 58 kg (8 kg + ) 6. 6. (d) 5.87 kg (e) 3. () (g) (h) () Yes, untl the clone splashes down. No; we need also conseraton o oentu n the collson. (j) Yes, but t s not useul to nclude the planet n the analyss o oentu. We use nstead oentu conseraton or the actor-clone syste whle they are n contact. (k) At a locaton wth weaker graty, the actor would be ong ore slowly beore the collson, but the clone would ollow the sae trajectory, ong ore slowly oer a longer te nteral. P9.58 (a). 667 s.95 P9.6 (a) 68. s. P9.6 (a) 6.9 s 6.6 s Most o the % derence between the alues or speed s accounted or by the uncertanty n the data, estated as... + + + + 868. 68. 8 63 57 85. 3.% P9.64 (a) The ass o the slegh plus you s 7 kg and your elocty s 7.5 s n the x drecton. You unbolt a 5.-kg seat and throw t back at the woles, gng t a speed o 8. s relate to you. Fnd the elocty o the seat relate to the ground aterward, and the elocty o the slegh aterward..55 6 s n the x drecton; 7.94 s n the +x drecton 453 J P9.66. 79 s s 4 P9.68 (a). ˆ + 7. ˆj. î s 4. î s (d) 5. ˆ + 35. ˆj (e) 6 J () 674 J (g) 674 J (h) The acceleratons coputed n derent ways agree. The knetc energes coputed n derent ways agree. The three theores are consstent. 3794_9_ch9_p9-44 copy.ndd44 44 /8/7 5:48: PM