Chapter 4 Newton s Laws

Transcription

1 Chpte 4 Newton s Lws Conceptul Pobles While on ve sooth level tnscontinentl plne fliht, ou coffee cup sits otionless on ou t. Ae thee foces ctin on the cup? If so, how do the diffe fo the foces tht would be ctin on the cup if it st on ou kitchen tble t hoe? Deteine the Concept Yes, thee e foces ctin on it. he e the nol foce of the tble nd the vittionl pull of Eth (weiht). Becuse the cup is not cceletin eltive to the ound, the foces e the se s those tht would ct on it if it ws sittin on ou tble t hoe. You e pssin nothe c on hihw nd deteine tht, eltive to ou, the c ou pss hs n cceletion to the west. Howeve, the dive of the othe c is intinin constnt speed nd diection eltive to the od. Is the efeence fe of ou c n inetil one? If not, in which diection (est o west) is ou c cceletin eltive to the othe c? Deteine the Concept No. You e in non-inetil fe tht is cceletin to the est, opposite the othe c s ppent cceletion. 3 [SSM] You e idin in liousine tht hs opque windows tht do not llow ou to see outside. he c is on flt hoizontl plne, so the c cn ccelete b speedin up, slowin down, o tunin. Equipped with just sll hev object on the end of stin, how cn ou use it to deteine if the liousine is chnin eithe speed o diection? Cn ou deteine the liousine s velocit? Deteine the Concept In the lio ou hold one end of the stin nd suspend the object fo the othe end. If the stin eins veticl, the efeence fe of the lio is n inetil efeence fe. 4 If onl sinle nonzeo foce cts on n object, does the object ccelete eltive to ll inetil efeence fes? Is it possible fo such n object to hve zeo velocit in soe inetil efeence fe nd not in nothe? If so, ive specific eple. Deteine the Concept An object cceletes when net foce cts on it. he fct tht n object is cceletin tells us nothin bout its velocit othe thn tht it is lws chnin. 95

2 96 Chpte 4 Yes, the object ust hve cceletion eltive to ll inetil fes of efeence. Accodin to Newton s fist nd second lws, n object ust ccelete, eltive to n inetil efeence fe, in the diection of the net foce. If thee is onl sinle nonzeo foce, then this foce is the net foce. Yes, the object s velocit be oentil zeo in soe inetil efeence fe nd not in nothe. Duin the peiod in which the foce is ctin, the object be oentil t est, but its velocit cnnot ein zeo becuse it ust continue to ccelete. hus, its velocit is lws chnin. 5 A bsebll is cted upon b sinle known foce. o this infotion lone, cn ou tell in which diection the bsebll is ovin eltive to soe efeence fe? Eplin. Deteine the Concept No. Pedictin the diection of the subsequent otion coectl equies dditionl infotion (knowlede of the initil velocit s well s the cceletion). While the cceletion cn be obtined fo the net foce thouh Newton s second lw, the velocit cn onl be obtined b intetin the cceletion. 6 A tuck oves diectl w fo ou t constnt velocit (s obseved b ou while stndin in the iddle of the od). It follows tht () no foces ct on the tuck, (b) constnt foce cts on the tuck in the diection of its velocit, (c) the net foce ctin on the tuck is zeo, (d) the net foce ctin on the tuck is its weiht. Deteine the Concept An object in n inetil efeence fe cceletes if thee is net foce ctin on it. Becuse the object is ovin t constnt velocit, the net foce ctin on it is zeo. (c) is coect. 7 Sevel spce pobes hve been lunched tht e now f out in spce Pionee 0, fo eple, ws lunched in the 970s nd is still ovin w fo the Sun nd its plnets. Is the ss of Pionee 0 chnin? Which of the known fundentl foces continue to ct on it? Does it hve net foce on it? Deteine the Concept No. he ss of the pobe is constnt. Howeve, the sol sste will ttct the pobe with vittionl foce. As the distnce between Pionee 0 nd the sol sste becoes le, the nitude of the vittionl foce becoes slle. hee is net foce on the pobe becuse no othe foces ct on it.

3 Newton s Lws 97 8 Astonuts in ppent weihtlessness duin thei st on the Intentionl Spce Sttion ust cefull onito thei sses becuse sinificnt loss of bod ss is known to cuse seious edicl pobles. Give n eple of how ou iht desin equipent to esue the ss of n stonut on the obitin spce sttion. Deteine the Concept You could use clibted spin ( spin with known stiffness constnt) to pull on ech stonut nd esue thei esultin cceletion. hen ou could use Newton s second lw to clculte thei ss. 9 [SSM] You e idin in n elevto. Descibe two situtions in which ou ppent weiht is ete thn ou tue weiht. Deteine the Concept You ppent weiht is the edin of scle. If the cceletion of the elevto (nd ou) is diected upwd, the nol foce eeted b the scle on ou is ete thn ou weiht. You could be ovin down but slowin o ovin up nd speedin up. In both cses ou cceletion is upwd. 0 Suppose ou e in tin ovin t constnt velocit eltive to the ound. You toss bll to ou fiend sevel sets in font of ou. Use Newton s second lw to eplin wh ou cnnot use ou obsevtions of the tossed bll to deteine the tin s velocit eltive to the ound. Deteine the Concept Becuse ou e ovin with constnt velocit, ou fe of efeence is n inetil efeence fe. In n inetil efeence fe thee e no fictitious foces. hus ovin o not ovin, the bll will follow the se tjecto in ou efeence fe. he net foce on the bll is the se, so its cceletion is the se. Eplin wh, of the fundentl intections, the vittionl intection is the in concen in ou eved lives. One othe on this list lso pls n incesinl sinificnt ole in ou pidl dvncin technolo. Which one is tht? Wh e the othes not obviousl ipotnt? Deteine the Concept he ost sinificnt foce in ou eved wold is vit. It litell keeps us on o ne the ound. he othe ost coon foce is the electonetic foce. It povides the lue to hold solids toethe nd ke the iid. It is of et ipotnce in electic cicuits. Give n eple of n object tht hs thee foces ctin on it, nd () cceletes, (b) oves t constnt (non-zeo) velocit, nd (c) eins t est. () An object fo which the vecto su of the thee foces doesn t dd to zeo. o eple, sled on fictionless sufce pulled hoizontll. he nol foce plus the weiht plus the pullin foce do not dd to zeo, so the sled cceletes.

4 98 Chpte 4 (b) Pullin fish veticll upwd t constnt velocit while it is still in the wte. he foces ctin on the fish e the pull, the vittionl foce (weiht of the fish), nd wte d foces. hese foces dd up to zeo. (c) he thee foces need to dd vectoill to zeo. An eple is pictue hun b two wies. 3 [SSM] Suppose block of ss ests on block of ss nd the cobintion ests on tble s shown in iue ell the ne of the foce nd its cteo (contct vesus ction-t--distnce) fo ech of the followin foces; () foce eeted b on, (b) foce eeted b on, (c) foce eeted b on the tble, (d) foce eeted b the tble on, (e) foce eeted b Eth on. Which, if n, of these foces constitute Newton s thid lw pi of foces? Deteine the Concept () he foce eeted b on. (b) he foce eeted b on. (c) he foce eeted b on the tble. (d) he foce eeted b the tble on. (e) he foce eeted b Eth on. Nol foce, contct Nol foce, contct Nol foce, contct Nol foce, contct Gvittionl foce, ction-t--distnce he Newton s thid lw foce pis e the two nol foces between the two blocks nd the two nol foces between the tble nd the botto block. he vittionl foce hs thid lw foce pi tht cts on Eth nd so is not in the question set. 4 You nk fish ou hve just cuht on ou line upwd fo est into ou bot. Dw fee-bod di of the fish fte it hs left the wte nd s it ins speed s it ises. In ddition, tell the tpe (tension, spin, vit, nol, fiction, etc.) nd cteo (contct vesus ction-t--distnce) fo ech foce on ou di. Which, if n, pis of the foces on ou di constitute Newton s thid lw pi? Cn ou tell the eltive nitudes of the foces fo the infotion iven? Eplin.

5 Newton s Lws 99 Deteine the Concept A fee-bod di showin the foces ctin on the fish is shown to the iht. he foces do not constitute Newton s 3 d lw pi. A tble suizin the tpe nd cteo of the foces is shown below. fish stin on fish Eth on fish w oce pe Cteo ension Contct stin on fish Gvit Action-t--distnce Eth on fish Becuse the fish cceletes upwd, the tension foce ust be ete in nitude thn the vittionl foce ctin on the fish. 5 If ou entl set fnc plte on the tble, it will not bek. Howeve if ou dop it fo heiht, it iht ve well bek. Discuss the foces tht ct on the plte (s it contcts the tble) in both these situtions. Use kinetics nd Newton s second lw to descibe wht is diffeent bout the second sitution tht cuses the plte to bek. Deteine the Concept When the plte is sittin on the tble, the nol foce n ctin upwd on it is eeted b the tble nd is the se size s the vittionl foce ctin on the plte. Hence, the plte does not ccelete. Howeve, to slow the plte down s it hits the tble equies tht n > (o n >> if the tble is hd nd the plte slows quickl). A le nol foce eeted on delicte chin cn esil bek it. 6 o ech of the followin foces, ive wht poduces it, wht object it cts on, its diection, nd the ection foce. () he foce ou eet on ou biefcse s ou hold it while stndin t the bus stop. (b) he nol foce on the soles of ou feet s ou stnd befooted on hoizontl wood floo. (c) he vittionl foce on ou s ou stnd on hoizontl floo. (d) he hoizontl foce eeted on bsebll b bt s the bll is hit stiht up the iddle towds cente field fo sinle. Deteine the Concept () he foce ou eet on ou biefcse to hold it while stndin t the bus stop: You poduce this foce. It cts on the biefcse. It cts upwd. he ection foce is the foce the biefcse eets on ou hnd.

6 300 Chpte 4 (b) he nol foce on the soles of ou feet s ou stnd befooted on hoizontl wood floo. (c) he vittionl foce on ou s ou stnd on hoizontl floo. (d) he hoizontl foce eeted on bsebll b bt s the bll is hit stiht up the iddle towds cente field fo sinle. he floo poduces this foce. It cts on ou feet. It cts upwd. he ection foce is the foce ou feet eet on the floo. Eth poduces this foce. It cts on ou. It cts downwd. he ection foce is the vittionl foce ou eet on Eth. he bt poduces this foce. It cts on the bll. It cts hoizontll. he ection foce is the foce the bll eets on the bt. 7 o ech cse, identif the foce (includin its diection) tht cuses the cceletion. () A spinte t the ve stt of the ce. (b) A hocke puck skiddin feel but slowl coin to est on the ice. (c) A lon fl bll t the top of its c. (d) A bunee jupe t the ve botto of he descent. Deteine the Concept () A spinte t the ve stt of the ce: (b) A hocke puck skiddin feel but slowl coin to est on the ice: (c) A lon fl bll t the top of its c: (d) A bunee jupe t the ve botto of he descent: he nol foce of the block on the spinte, in the fowd diection. he fictionl foce b the ice on the puck, in the opposite diection to the velocit. he vittion foce b Eth on the bll, in the downwd diection. he foce eeted b the bunee cod on the jupe, in the upwd diection. 8 ue o flse: () (b) If two etenl foces tht e both equl in nitude nd opposite in diection ct on the se object, the two foces cn neve be Newton s thid lw pi. he two foces of Newton s thid lw pi e equl onl if the objects involved e not cceletin. () ue. B definition, thid lw pis cnnot ct on the se object.

7 Newton s Lws 30 (b) lse. Action nd ection foces e equl independentl of n otion of the involved objects. 9 An 80-k n on ice sktes is pushin his 40-k son, lso on sktes, with foce of 00 N. oethe, the ove coss the ice stedil inin speed. () he foce eeted b the bo on his fthe is () 00 N, () 00 N, (3) 50 N, o (4) 40 N. (b) How do the nitudes of the two cceletions cope? (c) How do the diections of the two cceletions cope? Deteine the Concept () () hese foces e Newton 3 d lw foce pi, nd so the foce eeted b the bo on his fthe is 00 N. (b) Becuse the fthe nd son ove toethe, thei cceletions will be the se. (c) he diections of thei cceletion e the se. 0 A il holds stone in he hnd nd cn ove it up o down o keep it still. ue o flse: () he foce eeted b he hnd on the ock is lws the se nitude s the weiht of the stone. (b) he foce eeted b he hnd on the ock is the ection foce to the pull of vit on the stone. (c) he foce eeted b he hnd is lws the se size the foce he hnd feels fo the stone but in the opposite diection. (d) If the il oves he hnd down t constnt speed, then he upwd foce on the stone is less thn the weiht of the stone. (e) If the il oves he hnd downwd but slows the stone to est, the foce of the stone on the il s hnd is the se nitude s the pull of vit on the stone. () lse. If the ock is cceletin, the foce the il eets ust be ete thn the weiht of the stone. (b) lse. he ection foce to the pull of vit is the foce the ock eets on Eth. (c) ue. hese foces constitute Newton s thid lw pi. (d) lse. If she oves the stone downwd t constnt speed, the net foce ctin on the stone ust be zeo. (e) lse. If she is slowin the stone, it is epeiencin cceletion nd the net foce ctin on it cn not be zeo. he foce of he hnd on the stone, which hs the se nitude s the foce of the stone on he hnd, is ete thn the foce of vit on the stone.

8 30 Chpte 4 [SSM] A.5-k object hns t est fo stin ttched to the ceilin. () Dw fee bod di of the object, indicte the ection foce to ech foce dwn, nd tell wht object the ection foce cts on. (b) Dw fee bod di of the stin, indicte the ection foce to ech foce dwn, nd tell wht object ech ection foce cts on. Do not nelect the ss of the stin. Deteine the Concept he foce dis will need to include foces eeted b the ceilin, on the stin, on the object, nd foces eeted b Eth. () b stin on object.5 k Object b Eth on object oce b stin on object b Eth on object hid-lw Pi b object on stin b object on Eth (b) b ceilin on stin Stin b Eth on stin b object on stin oce b ceilin on stin b Eth on stin b object on stin hid-lw Pi b stin on ceilin b b stin on Eth stin on object () Which of the fee-bod dis in iue epesents block slidin down fictionless inclined sufce? (b) o the coect fiue, lbel the foces nd tell which e contct foces nd which e ction-t--distnce foces. (c) o ech foce in the coect fiue, identif the ection foce, the object it cts on nd its diection. Deteine the Concept Identif the objects in the block s envionent tht e eetin foces on the block nd then decide in wht diections those foces ust be ctin if the block is slidin down the inclined plne. () ee-bod di (c) is coect.

9 (b) Becuse the incline is fictionless, the foce n the incline eets on the block ust be nol to the sufce nd is contct foce. he second object cpble of eetin foce on the block is Eth nd its foce; the vittionl foce ctin on the block cts diectl downwd nd is n ction-t--distnce foce. he nitude of the nol foce is less thn tht of the weiht becuse it suppots onl potion of the weiht. Newton s Lws 303 n (c) he ection to the nol foce is the foce the block eets pependicull on the sufce of the incline. he ection to the vittionl foce is the upwd foce the block eets on Eth. 3 A wooden bo on the floo is pessed inst copessed, hoizontl spin tht is ttched to wll. he hoizontl floo beneth the bo is fictionless. Dw the fee bod di of the bo in the followin cses. () he bo is held t est inst the copessed spin. (b) he foce holdin the bo inst the spin no lone eists, but the bo is still in contct with the spin. (c) When the bo no lone hs contct with the spin. Deteine the Concept In the followin fee-bod dis we ll ssue tht the bo is initill pushed to the left to copess the spin. () Note tht, in the fee-bod di to the iht, tht n b floo b Eth nd b. spin b hnd (b) Note tht while n b floo b Eth, b spin is now the net foce ctin on the bo. As the spin decopesses, b spin will becoe slle. b hnd n b floo b Eth b spin n b floo b spin b Eth

10 304 Chpte 4 (c) When the bo septes fo the spin, the foce eeted b the spin on the bo oes to zeo. Note tht it is still tue tht n. b floo b Eth n b floo b Eth 4 Iine ouself seted on wheeled desk chi t ou desk. Conside n fiction foces between the chi nd the floo to be neliible. Howeve, the fiction foces between the desk nd the floo e not neliible. When sittin t est, ou decide ou need nothe cup of coffee. You push hoizontll inst the desk, nd the chi olls bckwd w fo the desk. () Dw ou fee-bod di of ouself duin the push nd clel indicte which foce ws esponsible fo ou cceletion. (b) Wht is the ection foce to the foce tht cused ou cceletion? (c) Dw the fee-bod di of the desk nd eplin wh it did not ccelete. Does this violte Newton s thid lw? Eplin. Deteine the Concept In the followin fee-bod dis we ll ssue tht the desk is to the left nd tht ou otion is to the iht. () Newton s thid lw ccounts fo this s follows. When ou push with ou hnds inst the desk, the desk pushes bck on ou hnds with foce of the se nitude but opposite diection. his foce cceletes ou bckwd. n b desk (b) he ection foce to the foce tht cused ou cceletion is the foce tht ou eeted on the desk. (c) When ou pushed on the desk, ou did not ppl sufficient foce to ovecoe the foce of fiction between the desk nd the floo. In tes of foces on the desk, ou pplied foce, nd the floo pplied fiction foce tht, when dded s vectos, cncelled. he desk, theefoe, did not ccelete nd Newton s thid lw is not violted. he foces in the di do not constitute Newton s thid lw pi of foces, even thouh the e equl in nitude nd opposite in diection.

11 Newton s Lws 305 b ou n b floo b floo on desk 5 he se (net) hoizontl foce is pplied fo fied tie intevl Δt to ech of two objects, hvin sses nd, tht sit on flt, fictionless sufce. (Let >.) () Assuin the two objects e initill t est, wht is the tio of thei cceletions duin the tie intevl in tes of, nd? (b) Wht is the tio of thei speeds v nd v t the end of the tie intevl? (c) How f pt e the two objects (nd which is hed) the end of the tie intevl? Pictue the Poble We cn ppl Newton s second lw to find the tios of the cceletions nd speeds of the two objects nd constnt-cceletion equtions to epess the seption of the objects s function of the elpsed tie. () Use Newton s second lw to epess the cceletions of the two objects: nd Dividin the fist of these equtions b the second nd siplifin ields: (b) Becuse both objects stted fo est, thei speeds fte tie Δt hs elpsed e: v Δt nd v Δt Dividin the fist of these equtions b the second nd siplifin ields: v v Δt Δt (c) he seption of the two objects t the end of the tie intevl is iven b: Δ Δ () Δ

12 306 Chpte 4 Usin constnt cceletion eqution, epess the distnces tveled b the two objects when tie Δt hs elpsed: Substitute fo Δ nd Δ in eqution () nd siplif to obtin: Δ nd Δ ( t) Δ Δ ( t) Δ ( Δt) ( Δt) ( ) Δt nd, becuse >, the object whose ss is is hed. Estition nd Appoition 6 Most cs hve fou spins ttchin the bod to the fe, one t ech wheel position. Devise n epeientl ethod of estitin the foce constnt of one of the spins usin ou known weiht nd the weihts of sevel of ou fiends. Assue the fou spins e identicl. Use the ethod to estite the foce constnt of ou c s spins. Pictue the Poble Suppose ou put in 800 lbs (o bout 3600 N) of weiht nd the c ss sevel inches (o 6.00 c). hen ech spin suppots bout 900 N nd we cn use the definition of the foce constnt k to deteine its vlue. he foce constnt is the tio of the copessin (o stetchin) foce to the copession (o stetch): k Δ evlute k: k 4 ( 3600 N) 6.00 c 50 N/c 7 [SSM] Estite the foce eeted on the olie s love b the puck when he ctches hd slp shot fo sve. Pictue the Poble Suppose the olie s love slows the puck fo 60 /s to zeo s it ecoils distnce of 0 c. uthe, ssue tht the puck s ss is 00. Becuse the foce the puck eets on the olie s love nd the foce the olie s love eets on the puck e ction-nd-ection foces, the e equl in nitude. Hence, if we use constnt-cceletion eqution to find the puck s cceletion nd Newton s second lw to find the foce the love eets on the puck, we ll hve the nitude of the foce eeted on the olie s love.

13 Newton s Lws 307 Appl Newton s second lw to the puck s it is slowed b the olie s love to epess the nitude of the foce the love eets on the puck: Use constnt-cceletion eqution to elte the initil nd finl speeds of the puck to its cceletion nd stoppin distnce: () love on puck v v0 + puck Δ puck puck ( ) puck Solvin fo puck ields: puck v v 0 ( Δ ) puck Substitute fo puck in eqution () to obtin: love on puck puck ( v v ) ( Δ ) puck 0 evlute love on puck : love on puck ( ) ( 0.00 k) 0 ( 60 /s) ( 0.0 ) 3.6 kn Reks: he foce on the puck is bout 800 ties its weiht. 8 A bsebll ple slides into second bse duin stel ttept. Assuin esonble vlues fo the lenth of the slide, the speed of the ple t the beinnin of the slide, nd the speed of the ple t the end of the slide, estite the vee foce of fiction ctin on the ple. Pictue the Poble Let s ssue tht the ple s ss is 00 k, tht he ets oin fil quickl down the bse pth, nd tht his speed is 8.0 /s when he beins his slide. uthe, let s ssue tht he ppoches the bse t the end of the slide t 3.0 /s. o these speeds, nd the lenth of the slide, we cn use Newton s second lw nd constnt-cceletion eqution to find the foce due to fiction (which cuses the slowin down). Appl Newton s second lw to the slidin unne: Usin constnt-cceletion eqution, elte the unne s initil nd finl speeds to his cceletion nd the lenth of his slide: Substitutin fo in eqution () ields: fiction () v vf v vi Δ Δ f + fiction vf v Δ i i

14 308 Chpte 4 Assuin the ple slides.0, substitute nueicl vlues nd evlute fiction : fiction ( ) ( ) 3.0 /s ( 8.0 /s 00 k ) (.0 ).4 kn whee the inus sin indictes tht the foce of fiction opposes the unne s otion. 9 A ce c skiddin out of contol nes to slow down to 90 k/h befoe cshin hed-on into bick wll. otuntel, the dive is wein sfet hness. Usin esonble vlues fo the ss of the dive nd the stoppin distnce, estite the vee foce eeted on the dive b the sfet hness, includin its diection. Nelect n effects of fictionl foces on the dive b the set. Pictue the Poble Assue cush distnce of.0 t 90 k/h (5 /s) nd dive s ss of 55 k. We cn use constnt-cceletion eqution (the definition of vee cceletion) to find the cceletion of the dive nd Newton s second lw to find the foce eeted on the dive b the set belt. Appl Newton s second lw to the dive s she is bouht to est b he sfet hness: Use constnt-cceletion eqution to elte the initil nd finl speeds of the dive to he cceletion nd stoppin distnce: () sfet hness on dive v v0 + dive dive dive ( Δ ) dive Solvin fo dive ields: dive v v 0 ( Δ ) dive Substitute fo dive in eqution () to obtin: sfet hness on dive dive v v 0 ( Δ ) dive evlute : sfet hness ( 55 k) sfet hness on dive on dive ( 5 /s) ( ) kn whee the inus sin indictes tht the foce eeted b the sfet hness is in the opposite diection fo the dive s otion.

15 Newton s Lws 309 Reks: he vee foce on the sfet hness is bout 3 ties he weiht. Newton s ist nd Second Lws: Mss, Ineti, nd oce 30 A pticle is tvelin in stiht line t constnt speed of 5.0 /s. Suddenl constnt foce of 5.0 N cts on it, binin it to stop in distnce of 6.5. () Wht is the diection of the foce? (b) Deteine the tie it tkes fo the pticle to coe to stop. (c) Wht is its ss? Pictue the Poble he cceletion of the pticle, its stoppin tie, nd its ss cn be found usin constnt-cceletion equtions nd Newton s second lw. A convenient coodinte sste is shown in the followin di. v 0 () Becuse the constnt foce slows the pticle, we cn conclude tht, s shown in the di, its diection is opposite the diection of the pticle s otion. (b) Use constnt-cceletion eqution to elte the initil nd finl velocities of the pticle to its cceletion nd stoppin tie: v v0 + Δt o, becuse v 0, 0 v0 + Δt Δ v () 0 t Use constnt-cceletion eqution to elte the initil nd finl velocities of the pticle to its cceletion nd stoppin distnce: v v0 + Δ o, becuse v 0, 0 v Δ 0 + v0 Δ Substitutin fo in eqution () nd siplifin ields: evlute Δt: Δ Δ t v 0 ( 6.5 ) Δ t 5.0 /s 5.00 s (c) Appl Newton s second lw to the pticle to obtin: Solvin fo ields: net net ()

16 30 Chpte 4 Substitute fo in eqution () to Δ obtin: v 0 net (6.5 )(5.0 N) evlute : ( 5.0 /s ) 3.00 k 3 An object hs n cceletion of 3.0 /s when sinle foce of nitude 0 cts on it. () Wht is the nitude of its cceletion when the nitude of this foce is doubled? (b) A second object hs n cceletion of 9.0 /s unde the influence of sinle foce of nitude 0. Wht is the tio of the ss of the second object to tht of the fist object? (c) If the two objects e lued toethe to fo coposite object, wht cceletion nitude will sinle foce of nitude 0 ctin on the coposite object poduce? Pictue the Poble he cceletion of n object is elted to its ss nd the net foce ctin on it b. net 0 () Use Newton s second lw of otion to elte the cceletion of the object to the net foce ctin on it: net When net 0 : evlute : 0 0 ( 3.0 /s ) 6.0 /s (b) Let the subscipts nd distinuish the two objects. he tio of the two sses is found fo Newton s second lw: /s 9.0/s 3 (c) he cceletion of the coposite object is the net foce divided b the totl ss + of the coposite object: net Substitute fo nd evlute : ( ) /s.3 /s 3 A tubot tows ship with constnt foce of nitude. he incese in the ship s speed duin 0-s intevl is 4.0 k/h. When second tubot pplies n dditionl constnt foce of nitude in the se diection, the speed inceses b 6 k/h duin 0-s intevl. How do the 4

17 Newton s Lws 3 nitudes of nd cope? (Nelect the effects of wte esistnce nd i esistnce.) Pictue the Poble he cceletion of n object is elted to its ss nd the net foce ctin on it b net. Let be the ss of the ship, be the cceletion of the ship when the net foce ctin on it is, nd be its cceletion when the net foce is +. Usin Newton s second lw, epess the net foce ctin on the ship when its cceletion is : Epess the net foce ctin on the ship when its cceletion is : + Divide the second of these equtions b the fist nd solve fo the tio / : + Substitute fo the cceletions to deteine the tio of the cceletin foces nd solve fo to obtin: 6 k/h 0s k/h 0s 33 A sinle constnt foce of nitude N cts on pticle of ss. he pticle stts fo est nd tvels in stiht line distnce of 8 in 6.0 s. ind. Pictue the Poble he ss of the pticle is elted to its cceletion nd the net foce ctin on it b Newton s second lw of otion. Becuse the foce is constnt, we cn use constnt-cceletion fouls to clculte the cceletion. Choose coodinte sste in which the + diection is the diection of otion of the pticle. he ss is elted to the net foce nd the cceletion b Newton s second lw: () Becuse the foce is constnt, the cceletion is constnt. Use constnt-cceletion eqution to elte the displceent of the pticle to it s cceletion: Substitute fo in eqution () to obtin: ( t) Δ ( t) Δ v0 t + Δ o, becuse v 0 0, Δ Δ ( t) Δ Δ ( Δt)

18 3 Chpte 4 evlute : ( N)( 6.0s) ( 8) k 34 A net foce of (6.0 N) ˆ i (3.0 N) j ˆ cts on n object of ss.5 k. ind the cceletion. Pictue the Poble he cceletion of n object is elted to its ss nd the net foce ctin on it ccodin to net. Appl Newton s second lw to the object to obtin: evlute : net ( 6.0 N) iˆ ( 3.0 N).5k ˆj ( ) iˆ 4.0 /s (.0 /s )j ˆ 35 [SSM] A bullet of ss k ovin t 500 /s ipcts tee stup nd penettes 6.00 c into the wood befoe coin to est. () Assuin tht the cceletion of the bullet is constnt, find the foce (includin diection) eeted b the wood on the bullet. (b) If the se foce cted on the bullet nd it hd the se speed but hlf the ss, how f would it penette into the wood? Pictue the Poble Choose coodinte sste in which the + diection is in the diection of the otion of the bullet nd use Newton s second lw nd constnt- cceletion eqution to epess the eltionship between stoppin nd the ss of the bullet nd its displceent s it is bouht to est in the block of wood. () Appl Newton s second lw to the bullet to obtin: stoppin () Use constnt-cceletion eqution to elte the bullet s initil nd finl speeds, cceletion, nd stoppin distnce: vf vi + Δ o, becuse v f 0, 0 v Δ i + vi Δ Substitute fo in eqution () to obtin: stoppin vi () Δ

19 Newton s Lws 33 evlute stoppin : ( ) ( 500 /s.80 0 k ) 3 stoppin ( 6.00 c) 3.8 kn whee the inus sin indictes tht stoppin opposes the otion of the bullet. (b) Solvin eqution () fo Δ ields: o nd Δ Δ : Evlute this epession fo ' to obtin: vi Δ (3) stoppin v Δ' ' i stoppin vi Δ' (4) 4 stoppin Dividin eqution (4) b eqution (3) ields: v Δ' 4 Δ o ' Δ Δ i stoppin vi stoppin evlute Δ : ( 6.00 c) 3.00 c Δ' 36 A ct on hoizontl, line tck hs fn ttched to it. he ct is positioned t one end of the tck, nd the fn is tuned on. Sttin fo est, the ct tkes 4.55 s to tvel distnce of.50. he ss of the ct plus fn is 355. Assue tht the ct tvels with constnt cceletion. () Wht is the net foce eeted on the ct-fn cobintion? (b) Mss is dded to the ct until the totl ss of the ct-fn cobintion is 7, nd the epeient is epeted. How lon does it tke fo the ct, sttin fo est, to tvel.50 now? Inoe the effects due to fiction.

20 34 Chpte 4 Pictue the Poble Choose the coodinte sste shown in the di to the iht. he foce ctin on the ct-fn cobintion is the consequence of the fn blowin i to the left. We cn use Newton s second lw nd constnt-cceletion eqution to epess the eltionship between nd the ss of the ct-fn cobintion nd the distnce it tvels in iven intevl of tie. n () Appl Newton s second lw to the ct-fn cobintion to obtin: Usin constnt-cceletion eqution, elte the distnce the ctfn cobintion tvels to its initil speed, cceletion, nd the elpsed tie: () ( t) Δ ( t) Δ v0 Δt + Δ o, becuse v 0 0, Δ Δ ( Δt) Substitute fo in eqution () to obtin: ( ) Δ () Δt ( ) (.50 ) evlute : k ( 4.55 s) N N (b) Solve eqution () fo Δt to obtin: Δ t Δ evlute Δt: ( 0.7 k)(.50 ) Δ t N 6.49 s 37 A hoizontl foce of nitude 0 cuses n cceletion of 3.0 /s when it cts on n object of ss slidin on fictionless sufce. ind the nitude of the cceletion of the se object in the cicustnces shown in iue 4-35 nd 4-35b. Pictue the Poble he cceletion of n object is elted to its ss nd the net foce ctin on it thouh Newton s second lw. Choose coodinte sste in which the diection of 0 in (b) is the positive diection nd the diection of the left-ost 0 in () is the positive diection. ind the esultnt foce in ech cse nd then find the esultnt cceletion.

21 Newton s Lws 35 () Appl Newton s second lw to the object to obtin: net () he nitude of the net foce is iven b: net + 0 Substitute fo net in eqution () nd 0 siplif to obtin: 0 Substitute the nueicl vlue of 0 nd evlute : ( 3.0 /s ) 4. /s (b)he nitude of net foce is iven b: ( 0 sin 45 ) + ( cos 45 ). 0 net + 80 Substitute fo net in eqution () nd 0 siplif to obtin: Substitute the nueicl vlue of 0 nd evlute : ( ) 8.4 /s /s 38 Al nd Bet stnd in the iddle of le fozen lke (fictionless sufce). Al pushes on Bet with foce of 0 N fo.5 s. Bet s ss is 00 k. Assue tht both e t est befoe Al pushes Bet. () Wht is the speed tht Bet eches s he is pushed w fo Al? (b) Wht speed does Al ech if his ss is 80 k? Pictue the Poble he speed of eithe Al o Bet cn be obtined fo thei cceletions; in tun, the cn be obtined fo Newtons second lw pplied to ech peson. he fee-bod dis to the iht show the foces ctin on Al nd Bet. he foces tht Al nd Bet eet on ech othe e ction-nd-ection foces. Al on Bet n,bet Bet n, Al Bet on Al Al () Appl to Bet: Alon Bet Bet Bet Bet Al on Bet Bet evlute Bet : Bet 0 N 0.00/s 00k

22 36 Chpte 4 Usin constnt-cceletion eqution, elte Bet s speed to his initil speed, speed fte.5 s, nd cceletion: evlute Bet s speed t the end of.5 s: v v v Bet, Δt ( 0.00 /s )(.5 s) 0.30 /s (b) o Newton's 3 d lw, n equl but oppositel diected foce cts on Al while he pushes Bet. Becuse the ice is fictionless, Al speeds off in the opposite diection. Appl to Al:, Al Bet on Al, AlAl, Solvin fo Al s cceletion ields: Al, Bet on Al, Al evlute Al, : Al, 0 N 80k 0.50/s Usin constnt-cceletion eqution, elte Al s speed to his initil speed, speed fte.5 s, nd cceletion: v v 0 + Al, Δt evlute Al s speed t the end of.5 s: v (.5 s) 0 + ( 0.50 /s )(.5 s) 0.38 /s 39 If ou push block whose ss is coss fictionless floo with hoizontl foce of nitude 0, the block hs n cceletion of /s. If ou push on diffeent block whose ss is with hoizontl foce of nitude 0, its cceletion is 3.0 /s. () Wht cceletion will hoizontl foce of nitude 0 ive to sinle block with ss? (b) Wht cceletion will hoizontl foce of nitude 0 ive to sinle block with ss +? Pictue the Poble he fee-bod dis show the foces ctin on the two blocks. We cn ppl Newton s second lw to the foces ctin on the blocks nd eliinte to obtin eltionship between the sses. Additionl pplictions of Newton s second lw to the su nd diffeence of the sses will led us to vlues fo the cceletions of these cobintions of ss.

23 n, n, 0 0 Newton s Lws 37 () Appl to the two blocks:, 0, nd, 0, Eliinte 0 between the two, equtions nd solve fo : Substitute nueicl vlues to /s obtin: /s, Epess the cceletion of n object whose ss is when the net foce ctin on it is 0 : , 4 3 evlute : 3 ( /s ) 4.0 /s (b) Epess the cceletion of n object whose ss is + when the net foce ctin on it is 0 : , evlute : 5 ( /s ).4 /s 40 o d 75.0-k lo lon the ound t constnt velocit, ou tcto hs to pull it with hoizontl foce of 50 N. () Dw the fee bod di of the lo. (b) Use Newton s lws to deteine the foce of fiction on the lo. (c) Wht is the nol foce of the ound on the lo? (d) Wht hoizontl foce ust ou eet if ou wnt to ive the lo n cceletion of.00 /s ssuin the foce of fiction does not chne. Redw the lo s fee bod di fo this sitution. Pictue the Poble Becuse the velocit is constnt, the net foce ctin on the lo ust be zeo. Choose coodinte sste in which the positive diection is the diection of otion of the lo nd ppl Newton s second lw to the lo.

24 38 Chpte 4 () he fee-bod di shows the foces ctin on the lo when it is bein ded in the + diection t constnt velocit. n b ound pull (b) Appl to the lo when it is ovin t constnt speed: o b ound pull b ound pull 0 Substitute fo pull nd evlute the foce of fiction b ound : b ound pull 50 N (c) Appl to the lo to obtin: Becuse the vittionl foce is iven b : evlute n : o n n n ( 75.0 k)( 9.8 /s ) 736 N n 0 (d) he fee-bod di shows the foces ctin on the lo when it is cceletin in the positive diection. b ound n pull Appl to the lo when it is cceletin to the iht: pull b ound Solvin fo pull ields: pull + b ound

25 evlute pull : pull ( 75.0 k)(.00 /s ) 400 N Newton s Lws N 4 A 4.0-k object is subjected to two constnt foces, (.0 N) ˆ i + ( 3.0 N) j ˆ nd (4.0 N) ˆ i ( N) j ˆ. he object is t est t the oiin t tie t 0. () Wht is the object s cceletion? (b) Wht is its velocit t tie t 3.0 s? (c) Whee is the object t tie t 3.0 s? Pictue the Poble he cceletion cn be found fo Newton s second lw. Becuse both foces e constnt, the net foce nd the cceletion e constnt; hence, we cn use the constnt-cceletion equtions to nswe questions concenin the otion of the object t vious ties. () Appl Newton s second lw to the 4.0-k object to obtin: net + siplif to evlute : (.0 N) iˆ + ( 3.0 N) ˆj + ( 4.0 N) iˆ + ( N) ˆj ( 6.0 N) iˆ + ( 4 N) 4.0 k ( ) iˆ.5 /s + ( 3.5 /s )j ˆ 4.0 k ˆj (b) Usin constnt-cceletion eqution, epess the velocit of the object s function of tie: v v 0 + t evlute v ( 3.0 s) : v [ ˆj ]( 3.0s) ( 4.5 /s) iˆ + ( 0.5/s) ( 4.5/s) iˆ + ( /s)j ˆ ( ) ( ) iˆ 3.0 s.5/s + ( 3.5/s ) ˆj (c) Epess the position of the object in tes of its vee velocit: v v v t 0 + v t vt Substitute fo v nd evlute this epession t t 3.0 s: ( 3.0 s) [ ( ) iˆ ( ) ˆj ]( ) ( ) iˆ 4.5/s + 0.5/s 3.0 s ( 5.8) ( 6.8) iˆ + ( 6)j ˆ ˆj

26 30 Chpte 4 Mss nd Weiht 4 On the oon, the cceletion due to vit is onl bout /6 of tht on Eth. An stonut, whose weiht on Eth is 600 N, tvels to the lun sufce. His ss, s esued on the oon, will be () 600 k, (b) 00 k, (c) 6. k, (d) 9.8 k, (e) 360 k. Pictue the Poble he ss of the stonut is independent of vittionl fields nd will be the se on the oon o, fo tht tte, out in deep spce. Epess the ss of the stonut in tes of his weiht on Eth nd the vittionl field t the sufce of Eth: w eth 600 N 9.8N/k eth nd (c) is coect. 6. k 43 ind the weiht of 54-k student in () newtons nd (b) pounds. Pictue the Poble he weiht of n object is elted to its ss nd the vittionl field thouh. () he weiht of the student is: w ( 54k)( 9.8N/k) 530 N N (b) Convet newtons to pounds: N w 530 9lb. 0 lb 4.45 N/lb 44 ind the ss of 65-lb eninee in kilos. Pictue the Poble he ss of n object is elted to its weiht nd the vittionl field. Convet the weiht of the n into newtons: ( 65lb)( 4.45 N/lb) 734 N 65 lb Clculte the ss of the n fo his weiht nd the vittionl field: w 734 N 9.8N/k 74.8k 45 [SSM] o tin stonuts to wok on the oon, whee the cceletion due to vit is onl bout /6 of tht on Eth, NASA subees the in tnk of wte. If n stonut, who is cin bckpck, i conditionin unit, oen suppl, nd othe equipent, hs totl ss of 50 k, deteine the followin quntities. () he weiht, includin he bckpck, etc., on Eth, (b) he weiht on the oon, (c) the equied upwd

27 Newton s Lws 3 buonc foce of the wte duin he tinin fo the oon s envionent on Eth. Pictue the Poble We cn use the eltionship between weiht (vittionl foce) nd ss, toethe with the iven infotion bout the cceletion due to vit on the oon, to find the stonut s weiht on Eth nd on the oon. () He weiht on Eth is the poduct of he ss nd the vittionl field t the sufce of Eth: evlute w: (b) He weiht on the oon is the poduct of he ss nd the vittionl field t the sufce of the oon: Substitute fo he weiht on Eth nd evlute he weiht on the oon: (c) he equied upwd buonc foce of the wte equls the diffeence between he weiht on Eth nd on the oon: w Eth w Eth oon ( 50 k)( 9.8 /s ).45 kn w w w oon 6 6 w.453 kn eth (.453 kn) 409 N oon 6 buonc w w Eth.45 kn 0.4 kn.04kn oon 46 It is the e 075 nd spce tvel is coon. A phsics pofesso bins his fvoite techin deonsttion with hi to the oon. he pptus consists of ve sooth hoizontl (fictionless) tble nd n object to slide on it. On Eth, when the pofesso ttches spin (spin constnt 50 N/) to the object nd pulls hoizontll so the spin stetches.0 c, the object cceletes t.5 /s. () Dw the fee-bod di of the object nd use it nd Newton s lws to deteine the object s ss. (b) Wht would the object s cceletion be unde identicl conditions on the oon? Pictue the Poble he foces ctin on the object e the nol foce eeted b the tble, the vittionl foce eeted b Eth, nd the foce eeted b the stetched spin.

28 3 Chpte 4 () he fee-bod di shown to the iht ssues tht the spin hs been stetched to the iht. Hence the foce tht the spin eets on the object is to the left. Note tht the + diection hs been chosen to be in the se diection s the foce eeted b the spin. s n Appl to the object to obtin: he foce eeted b the spin on the object is iven b: s s () kδ whee Δ is the ount b which the spin hs been stetched o copessed nd k is the foce constnt. s Substitutin fo s in eqution () ields: evlute : kδ ( 50 N/)(.0 c).5 /s 0.67 k (b) Becuse the object s ss is the se on the oon s on Eth nd the foce eeted b the spin is the se, its cceletion on the oon would be the se s on Eth. ee-bod Dis: Sttic Equilibiu 47 A 35.0-k tffic liht is suppoted b two wies s in iue () Dw the liht s fee-bod di nd use it to nswe the followin question qulittivel: Is the tension in wie ete thn o less thn the tension in wie? (b) Pove ou nswe b pplin Newton s lws nd solvin fo the two tensions. Pictue the Poble Becuse the tffic liht is not cceletin, the net foce ctin on it ust be zeo; i.e.,

29 Newton s Lws 33 () Constuct fee-bod di showin the foces ctin on the suppot point: Appl to the suppot point: Solve fo in tes of : (b) Appl to the suppot point: cos30 cos60 cos () cos60 hus is ete thn. sin 30 sin Substitute fo to obtin: 3 (.73 ) 0 + Solvin fo ives:.000 evlute : ( 35.0 k)( 9.8 /s ) 7 N.000 o eqution () we hve:.73 (.73)( 7 N) 98 N 48 A 4.6-k lp is hnin fo wies s shown in iue he in hs neliible ss. he tension in the veticl wie is () 09 N, (b) 48 N, (c) 570 N, (d) 360 N, (e) 730 N. Pictue the Poble o the fiue, it is cle tht suppots the full weiht of the lp. Dw fee-bod di showin the foces ctin on the lp nd ppl 0. Appl 0 to the lp to obtin: 0

30 34 Chpte 4 Solve fo nd substitute fo to obtin: evlute : ( 4.6 k)( 9.8/s ) 48N nd (b) is coect. 49 [SSM] In iue 4-38, k block is suspended t the idpoint of.5--lon stin. he ends of the stin e ttched to the ceilin t points septed b.00. () Wht nle does the stin ke with the ceilin? (b) Wht is the tension in the stin? (c) he k block is eoved nd two 0.50-k blocks e ttched to the stin such tht the lenths of the thee stin seents e equl (iue 4-38b). Wht is the tension in ech seent of the stin? Pictue the Poble he fee-bod dis fo Pts (), (b), nd (c) e shown below. In both cses, the block is in equilibiu unde the influence of the foces nd we cn use Newton s second lw of otion nd eoet nd tionoet to obtin eltionships between θ nd the tensions. () nd (b) (c) () Refein to the fee-bod di fo Pt (), use tionoet to deteine θ : (b) Notin tht, ppl to the k block nd solve fo the tension : θ cos sinθ 0 becuse 0 nd sinθ 37 evlute : ( 0.500k)( 9.8/s ) 4.N sin36.9

31 Newton s Lws 35 (c) he lenth of ech seent is: ind the distnce d: d Epess θ in tes of d nd solve fo its vlue: Appl to the 0.50-k block: evlute 3 : Appl to the 0.50-k block nd solve fo the tension : evlute : θ cos cos 0.47 d sinθ 0 3 sinθ 3 ( 0.50 k)( 9.8/s ) sin N 3 cosθ 0 since 0. nd 3 cosθ N ( N) cos N B set: N 50 A bll weihin 00 N is shown suspended fo sste of cods (iue 4-39). Wht e the tensions in the hoizontl nd nled cods?

32 36 Chpte 4 Pictue the Poble he suspended bod is in equilibiu unde the influence of the foces ho, 45, nd. ht is, ho Dw the fee-bod di of the foces ctin on the knot just bove the 00-N bod. Choose coodinte sste with the positive diection to the iht nd the positive diection upwd. Appl the conditions fo tnsltionl equilibiu to deteine the tension in the hoizontl cod. ho Appl to the knot: sin Solvin fo 45 ields: 00 N sin 45 sin N Appl to the knot: cos ho Solvin fo ho ives: cos 45 ( 4 N) ho N cos 45 5 [SSM] A 0-k object on fictionless tble is subjected to two hoizontl foces, nd, with nitudes 0 N nd 30 N, s shown in iue ind the thid foce 3 tht ust be pplied so tht the object is in sttic equilibiu. Pictue the Poble he cceletion of n object is diectl popotionl to the net foce ctin on it. Choose coodinte sste in which the positive diection is the se s tht of nd the positive diection is to the iht. Add the two foces to deteine the net foce nd then use Newton s second lw to find the cceletion of the object. If 3 bins the sste into equilibiu, it ust be tue tht Epess 3 in tes of nd : 3 ()

33 Newton s Lws 37 Epess nd in unit vecto nottion: nd (0 N) iˆ {( 30 N)sin 30 } iˆ + {(30 N)cos30 } ˆj ( 5 N)ˆ i (6 N) ˆj + Substitute fo nd in eqution () nd siplif to obtin: ( 0 N) iˆ [( 5 N)ˆ i + (6 N) ˆj ] ( 5.0 N) iˆ + ( 6 N)j ˆ 3 5 o the sstes to be in equilibiu in iue 4-4, iue 4-4b, nd iue 4-4c find the unknown tensions nd sses. Pictue the Poble he fee-bod dis fo the sstes in equilibiu e shown below. Appl the conditions fo tnsltionl equilibiu to find the unknown tensions. () (b) (c) () Appl 0 nd 0 to the knot bove the suspended ss to obtin: cos60 30 N 0 () nd sin 60 0 () Solvin eqution () fo ields: 30 N cos60 60 N Solvin eqution () fo ields: sin 60 ( 60 N) 5.96 N 5 N sin 60 Becuse is the weiht of the object whose ss is :

34 38 Chpte 4 evlute : 5.96 N 9.8 /s 5.3 k (b) Appl 0 nd 0 to the knot bove the suspended ss to obtin: Solvin the fist of these equtions fo ields: Solvin the second of these equtions fo ields: nd ( 80 N) cos60 sin 60 0 ( N) sin 60 cos60 0 ( 80 N) 46 N 80 cos N sin 60 ( N) sin 60 cos evlute : ( 80 N) sin 60 ( 46.9 N) 46.9 N 46 N cos60 Becuse is the weiht of the object whose ss is : evlute : 46.9 N 9.8 /s 4.7 k (c) Appl 0 nd 0 to the knot bove the suspended ss to obtin: Solvin the fist of these equtions fo ields: Solvin the second of these equtions fo ields: evlute nd 3 : nd 3 cos cos60 0 sin sin sin 60 sin N ( 6.0 k)( 9.8 /s ) sin N Becuse : ( 6.0 k)( 9.8 /s ) 59 N 58.9 N

35 Newton s Lws 39 Becuse the effect of the pulle is to chne the diection cts, is the weiht of the object whose ss is : evlute : N 9.8 /s 3.5 k 53 You c is stuck in ud hole. You e lone, but ou hve lon, ston ope. Hvin studied phsics, ou tie the ope tutl to telephone pole nd pull on it sidews, s shown in iue 4-4. () ind the foce eeted b the ope on the c when the nleθ is 3.00º nd ou e pullin with foce of 400 N but the c does not ove. (b) How ston ust the ope be if it tkes foce of 600 N to ove the c whenθ is 4.00º? Pictue the Poble Constuct the fee-bod di fo tht point in the ope t which ou eet the foce nd choose the coodinte sste shown in the fee-bod di. We cn ppl Newton s second lw to the ope to elte the tension to. () Notin tht nd tht the c s cceletion is zeo, ppl to the c: θ sinθ 0 θ sinθ evlute : 400 N sin kn (b) Poceed s in Pt () to obtin: 600 N sin kn 54 Blloon ches e often seen t festivls o celebtions; the e de b ttchin heliu-filled blloons to ope tht is fied to the ound t ech end. he lift fo the blloons ises the stuctue into the ch shpe. iue 4-43 shows the eoet of such stuctue: N blloons e ttched t equll spced intevls lon ssless ope of lenth L, which is ttched to two suppots t its ends. Ech blloon povides lift foce. he hoizontl nd veticl coodintes of the point on the ope whee the ith blloon is ttched e i nd i, nd i is the tension in the ith seent. (Note seent 0 is the seent between the point of ttchent nd the fist blloon, nd seent N is the seent between the lst blloon nd the othe point of ttchent). () iue 4-43b shows fee-bod di fo the ith blloon. o this di, show tht the hoizontl coponent of the foce i (cll it H ) is the se fo ll the

36 330 Chpte 4 stin seents. (b) B considein the veticl coponent of the foces, use Newton s lws to deive the followin eltionship between the tension in the ith nd (i )th seents: i sin θ i i sin θ i (b) Show tht tnθ 0 tn θ N+ N/ H. (c) o the di nd the two epessions bove, i L show tht tn θ i (N i)/ H nd tht i cosθ N + j, j 0 i L i sin θ N + j. j 0 Pictue the Poble () Applin 0 to the blloon shown in iue 4-43 (b) will show tht the hoizontl coponent of the foce i is the se fo ll stin seents. In Pt (b) we cn ppl 0 to obtin the iven epession fo. In (c) we cn use set uent to find n epession fo tn θ 0. inll, in (d) we cn use ou esults obtined in () nd (b) to epess i nd i. () Applin 0 to the blloon shown in iue 4-43 (b) ives: cosθ cosθ i i i i 0 Becuse H i cosθi : H i cosθ i 0 H i cosθ i independentl of the stin seent. (b) Appl 0 to the blloon shown in iue 4-43 (b) ives: Solvin fo ields: (c) B set, ech suppot ust blnce hlf of the foce ctin on the entie ch. heefoe, the veticl coponent of the foce on the suppot ust be N/. he hoizontl coponent of the tension ust be H. Epess tnθ 0 in tes of N/ nd H : B set, θ N+ θ 0. heefoe, becuse the tnent function is odd: sinθi i sinθi + i sin i θi tnθ 0 tnθ N H sinθ N tnθ 0 N+ i H i N H 0

37 (d) Usin H i cosθ i i cosθ i, divide both sides of the esult in (b) b H nd siplif to obtin: H i i tnθ sinθ cosθ i i i tnθ Newton s Lws 33 i sinθi cosθ i i i Usin this esult, epess tn θ : tnθ tnθ0 H Substitute fo tn θ 0 fo (): N tnθ H H ( N ) H Genelize this esult to obtin: tnθ i ( N i) H Epess the lenth of ope between L two blloons: l between blloons N + Epess the hoizontl coodinte of the point on the ope whee the ith blloon is ttched, i, in tes of i nd the lenth of ope between two blloons: i L i + cosθ N + i Su ove ll the coodintes to obtin: L i θ N + i cos j j 0 Poceed siill fo the veticl coodintes to obtin: L i θ N + i sin j j 0 55 () Conside nueicl solution to Poble 54. Wite spedsheet po to ke ph of the shpe of blloon ch. Use the followin petes: N 0 blloons ech povidin lift foce.0 N nd ech ttched to ope lenth L, with hoizontl coponent of tension H 0 N. How f pt e the two points of ttchent? How hih is the ch t its hihest point? (b) Note tht we hven t specified the spcin between the suppots it is deteined b the othe petes. V H while keepin the othe petes the se until ou cete n ch tht hs spcin of 8.0 between the suppots. Wht is H then? As ou incese H, the ch should et fltte nd oe sped out. Does ou spedsheet odel show this?

38 33 Chpte 4 Pictue the Poble () A spedsheet po is shown below. he fouls used to clculte the quntities in the coluns e s follows: Cell Content/oul Alebic o C7 ($B$ *B7)/(*$B$4) ( N i) D7 SIN(AAN(C7)) sin( tn θ i ) E7 COS(AAN(C7)) cos( tn θ ) G $B$/($B$+)*E7 L i + cosθi N + G8 G7+$B$/($B$+)*D7 L i + cosθi N + A B C D E G L N 0 3 N 4 H 3.7 N 5 6 i tnθ i sinθ i cosθ i i i H i

39 Newton s Lws 333 (b) A hoizontl coponent of tension 3.7 N ives spcin of 8. At this spcin, the ch is.63 hih, tll enouh fo soeone to wlk thouh. A ph of i s function of i fo the conditions specified follows: i, i, he spedsheet ph shown below shows tht chnin the hoizontl coponent of tension to 5 N bodens the ch nd deceses its heiht s pedicted b ou theticl odel i, i,

40 334 Chpte 4 ee-bod Dis: Inclined Plnes nd the Nol oce 56 A le bo whose ss is 0.0 k ests on fictionless floo. A ove pushes on the bo with foce of 50 N t n nle 35.0º below the hoizontl. Dw the bo s fee bod di nd use it to deteine the cceletion of the bo. Pictue the Poble he fee-bod di shows the foces ctin on the bo s the n pushes it coss the fictionless floo. We cn ppl Newton s second lw to the bo to find its cceletion. n θ Appl to the bo: cosθ cosθ evlute : ( 50 N) cos k 0. /s 57 A 0-k bo ests on fictionless p with 5.0º slope. he ove pulls on ope ttched to the bo to pull it up the incline (iue 4-44). If the ope kes n nle of 40.0º with the hoizontl, wht is the sllest foce the ove will hve to eet to ove the bo up the p? Pictue the Poble he fee-bod di shows the foces ctin on the bo s the n pushes it up the fictionless incline. We cn ppl Newton s second lw to the bo to deteine the sllest foce tht will ove it up the incline t constnt speed. n θ φ Lettin in, ppl to the bo s it oves up the incline with constnt speed: ( φ θ ) sinθ 0 in cos Solve fo in to obtin: in sinθ cos ( φ θ )

41 Becuse : evlute in : in in sinθ cos ( φ θ ) ( 0.0 k)( 9.8/s ) cos 56.0 N Newton s Lws 335 ( ) sin5 58 In iue 4-45, the objects e ttched to spin scles clibted in newtons. Give the edins of the blnce(s) in ech cse, ssuin tht both the scles nd the stins e ssless. Pictue the Poble he blnce(s) indicte the tension in the stin(s). Dw fee-bod dis fo ech of these sstes nd ppl the condition(s) fo equilibiu. () hook (b) blnce ' (c) (d) blnce ' () Appl 0 to the hook to obtin: 0 o, becuse, evlute : ( 0k)( 9.8/s ) 98 N (b) Appl 0 to the blnce to obtin: ' 0 o, becuse, '

42 336 Chpte 4 evlute nd : ' ( 0k)( 9.8/s ) 98N (c) Appl 0 to the suspended object to obtin: 0 o, becuse, 0 evlute : ( 0k)( 9.8/s ) 49 N (d) Appl 0 to the blnce to obtin: ' 0 o, becuse, ' evlute nd : ' ( 0k)( 9.8/s ) 98 N Reks: Note tht (), (b), nd (d) ive the se nswes the supisin esult until one hs lened to dw BDs nd ppl the conditions fo tnsltionl equilibiu. 59 A bo is held in position on fictionless incline b cble (iue 4-46). () If θ 60º nd 50 k, find the tension in the cble nd the nol foce eeted b the incline. (b) ind the tension s function of θ nd, nd check ou esult fo plusibilit in the specil cses when θ 0º nd θ 90º. Pictue the Poble Becuse the bo is held in plce (is in equilibiu) b the foces ctin on it, we know tht + n + 0 Choose coodinte sste in which the + diection is in the diection of nd the + diection is in the diection of n. Appl Newton s second lw to the block to obtin epessions fo nd n. n θ () Appl 0 to the bo: sinθ 0 o, becuse, sin θ 0 sinθ ()

43 evlute : ( 50k)( 9.8/s ) 0.4 kn Newton s Lws 337 sin 60 Appl 0 to the bo: n cosθ 0 o, becuse, n cosθ 0 n cosθ evlute n : n ( 50k)( 9.8/s ) 0.5kN cos60 (b) he tension s function of θ nd is iven b eqution (): sinθ o θ 90 : 90 sin 90 esult we know to be coect. o θ 0 : sin know to be coect. esult we 60 A hoizontl foce of 00 N pushes -k block up fictionless incline tht kes n nle of 5º with the hoizontl. () Wht is the nol foce tht the incline eets on the block? (b) Wht is the cceletion of the block? Pictue the Poble Dw fee-bod di fo the bo. Choose coodinte sste in which the positive -is is pllel to the inclined plne nd the positive -is is in the diection of the nol foce the incline eets on the block. Appl Newton s second lw of otion to find the nol foce n tht the incline eets on the block nd the cceletion of the block. n N () Appl to the block: n cos5 ( 00 N) sin 5 0 o, becuse, cos5 00 N ( ) sin 5 0 n Solvin fo n ields: + ( ) n cos 5 00 N sin 5

44 338 Chpte 4 evlute n : n ( k)( 9.8/s ) ( 00 N) N sin 5 cos5 (b) Appl block: to the ( N) cos5 sin 5 00 o, becuse, 00 N cos5 sin 5 ( ) Solve fo to obtin: ( 00 N) cos5 sin 5 evlute : ( 00 N) cos5 ( 9.8/s ) k 3.4 /s sin 5 6 [SSM] A 65-k student weihs hiself b stndin on scle ounted on sktebod tht is ollin down n incline, s shown in iue Assue thee is no fiction so tht the foce eeted b the incline on the sktebod is nol to the incline. Wht is the edin on the scle if θ 30º? Pictue the Poble he scle edin (the bo s ppent weiht) is the foce the scle eets on the bo. Dw fee-bod di fo the bo, choosin coodinte sste in which the positive -is is pllel to nd down the inclined plne nd the positive -is is in the diection of the nol foce the incline eets on the bo. Appl Newton s second lw of otion in the diection. 30 n Appl to the bo to find n. Reebe tht thee is no cceletion in the diection: n cos30 0 o, becuse, cos30 0 n Solvin fo n ields: n cos30 evlute n : n ( 65k)( 9.8/s ) 0.55kN cos30

45 Newton s Lws A block of ss slides coss fictionless floo nd then up fictionless p (iue 4-48). he nle of the p is θ nd the speed of the block befoe it stts up the p is v 0. he block will slide up to soe iu heiht h bove the floo befoe stoppin. Show tht h is independent of θ b deivin n epession fo h in tes of v 0 nd. Pictue the Poble he fee-bod di fo the block slidin up the incline is shown to the iht. Applin Newton s second lw to the foces ctin in the diection will led us to n epession fo. Usin this epession in constnt-cceletion eqution will llow us to epess h s function of v 0 nd. θ n θ he heiht h is elted to the distnce Δ tveled up the incline: Usin constnt-cceletion eqution, elte the finl speed of the block to its initil speed, cceletion, nd distnce tveled: Substitutin fo Δ in eqution () ields: Appl to the block nd solve fo its cceletion: h Δsinθ () v v0 + Δ o, becuse v 0, v 0 v0 + Δ Δ 0 v0 h sinθ () sinθ Becuse sinθ sinθ : Substitute fo in eqution () nd siplif to obtin: v v0 h 0 sin sin θ θ which is independent of the p s nle θ. ee-bod Dis: Elevtos 63 [SSM] () Dw the fee bod di (with ccute eltive foce nitudes) fo n object tht is hun b ope fo the ceilin of n elevto tht is scendin but slowin. (b) Repet Pt () but fo the sitution in which the

46 340 Chpte 4 elevto is descendin nd speedin up. (c) Cn ou tell the diffeence between the two dis? Eplin wh the dis do not tell nthin bout the object s velocit. Pictue the Poble () he fee bod di fo n object tht is hun b ope fo the ceilin of n scendin elevto tht is slowin down is shown to the iht. Note tht becuse >, the net foce ctin on the object is downwd; s it ust be if the object is slowin down s it is ovin upwd. (b) he fee bod di fo n object tht is hun b ope fo the ceilin of n elevto tht is descendin nd speedin up is shown to the iht. Note tht becuse >, the net foce ctin on the object is downwd; s it ust be if the object is speedin up s it descends. (c) No, thee is no diffeence. In both cses the cceletion is downwd. You cn onl tell the diection of the cceletion, not the diection of the velocit. 64 A 0.0-k block is suspended fo the ceilin of n elevto b cod ted to withstnd tension of 50 N. Shotl fte the elevto stts to scend, the cod beks. Wht ws the iniu cceletion of the elevto when the cod boke? Pictue the Poble he fee-bod di shows the foces ctin on the 0.0-k block s the elevto cceletes upwd. Appl Newton s second lw to the block to find the iniu cceletion of the elevto equied to bek the cod. Appl to the block: o, becuse, Solve fo to deteine the iniu bekin cceletion:

47 Newton s Lws 34 evlute : 50 N 0.0 k 9.8/s 5.9/s 65 A.0-k block hns fo spin scle clibted in newtons tht is ttched to the ceilin of n elevto (iue 4-49).Wht does the scle ed when () the elevto is scendin with constnt speed of 30 /s, (b) the elevto is descendin with constnt speed of 30 /s, (c) the elevto is scendin t 0 /s nd inin speed t te of 3.0 /s? (d) Suppose tht fo t 0 to t 5.0 s, the elevto scends t constnt speed of 0 /s. Its speed is then stedil educed to zeo duin the net 4.0 s, so tht it is t est t t 9.0 s. Descibe the edin of the scle duin the intevl 0 < t < 9.0 s. Pictue the Poble he fee-bod di shows the foces ctin on the -k block s the elevto scends t constnt velocit. Becuse the cceletion of the elevto is zeo, the block is in equilibiu unde the influence of nd. Appl Newton s second lw of otion to the block to deteine the scle edin. () Appl obtin: to the block to o otion with constnt velocit, 0 nd:.0 k o, becuse, () 0 evlute : (.0k)( 9.8/s ) 0 N (b) As in Pt (), fo constnt velocit, 0. Hence: (c) Solve eqution () fo nd siplif to obtin: Becuse the elevto is scendin nd its speed is incesin, we hve 3.0 /s. Substitute nueicl vlues nd evlute : 0 nd (.0k)( 9.8/s ) 0 N ( ) + + () (.0k)( 9.8/s /s ) 6 N

48 34 Chpte 4 (d) Duin the intevl 0 < t < 5.0 s, 0. Hence: 0 5.0s 0 N Usin its definition, clculte fo 5.0 s < t < 9.0 s: Substitute in eqution () nd evlute : Δv 0 0/s.5/s Δ t 4.0s 5s 9s (.0k)( 9.8/s.5/s ) 5 N ee-bod Dis: Sevel Objects nd Newton s hid Lw 66 wo boes of ss nd connected b ssless stin e bein pulled lon hoizontl fictionless sufce, s shown in iue () Dw the fee bod di of both boes septel nd show tht ( + ). (b) Is this esult plusible? Eplin. Does ou nswe ke sense both in the liit tht >> nd in the liit tht <<? Eplin. Pictue the Poble Dw fee-bod di fo ech bo nd ppl Newton s second lw. Solve the esultin siultneous equtions fo the tio of to. () he fee-bod dis fo the two boes e shown below: n,,, n,,, Appl to the bo on the left: Appl to the bo on the iht:, o, becuse,,, o, becuse,,

49 Newton s Lws 343 Becuse the boes hve the se cceletion, we cn divide the second eqution b the fist to obtin: + (b) Divide the nueto nd denointo of the epession inside the pentheses b to obtin: + o << : 0, s epected. o << :, s epected. 67 A bo of ss 3.5 k ests on fictionless hoizontl shelf nd is ttched b stins to boes of sses.5 k nd 3.5 k s shown in iue 4-5. Both pulles e fictionless nd ssless. he sste is elesed fo est. Afte it is elesed, find () the cceletion of ech of the boes nd (b) the tension in ech stin. Pictue the Poble Cll the coon cceletion of the boes. Assue tht bo oves upwd, bo to the iht, nd bo 3 downwd nd tke this diection to be the positive diection. Dw fee-bod dis fo ech of the boes, ppl Newton s second lw of otion, nd solve the esultin equtions ' ' siultneousl. Note tht nd., ' n,, 3 ',3 () Appl to the bo whose ss is :, o, becuse,, ()

50 344 Chpte 4 Appl to the bo whose ss is : ' Notin tht, ppl to the bo whose ss is 3 : Add equtions (), (), nd (3) to obtin: ' ' o, becuse, (),3 3 o, becuse,3 3, (3) Solvin fo ives: ( ) evlute : (.5k.5k)( 9.8/s ).5k + 3.5k +.5k.3 /s.3/s (b) o eqution () we hve: + ( ) evlute : + (.5 k)(.3 /s /s ) 7 N o eqution (3) we hve: ( ) evlute : (.5 k)( 9.8 /s.3 /s ) N 68 wo blocks e in contct on fictionless hoizontl sufce. he blocks e cceleted b sinle hoizontl foce pplied to one of the (iue 4-5). ind the cceletion nd the contct foce of block on block () in tes of,, nd, nd (b) fo the specific vlues 3. N,.0 k, nd 6.0 k.

51 Newton s Lws 345 Pictue the Poble Choose coodinte sste in which the + diection is to the iht nd the + diection is upwd. Let, be the contct foce eeted b the block whose ss is on the block whose ss is nd, be the foce eeted b the block whose ss is on the block whose ss is. hese foces e equl nd opposite so,,. he fee- bod dis fo the blocks e shown below. Appl Newton s second lw to ech block septel nd use the fct tht thei cceletions e equl., n, n,,,, () Appl to the block whose ss is : Appl to the block whose ss is :, o, becuse the blocks hve coon cceletion, (),, () Addin equtions () nd () ields: + ( ) + Solve fo to obtin: + Substitute fo in eqution () to obtin:, + + (b) Substitute nueicl vlues in the equtions deived in Pt () nd evlute nd, : 3. N.0k + 6.0k nd ( 3. N)( 6.0k).0k + 6.0k, 0.40 /s.4 N

52 346 Chpte 4 Reks: Note tht ou esults fo the cceletion e the se s if the foce hd cted on sinle object whose ss is equl to the su of the sses of the two blocks. In fct, becuse the two blocks hve the se cceletion, we cn conside the to be sinle sste with ss Repet Poble 68, but with the two blocks intechned. Ae ou nswes the se s in Poble 68? Eplin. Pictue the Poble Choose coodinte sste in which the + diection is to the iht nd the + diection is upwd. Let, be the contct foce eeted b the block whose ss is on the block whose ss is nd, be the foce eeted b the block whose ss is on the block whose ss is. hese foces e equl nd opposite so,,. he fee- bod dis fo the blocks e shown. We cn ppl Newton s second lw to ech block septel nd use the fct tht thei cceletions e equl. n,,, n,,, () Appl to the block whose ss is : Appl to the block whose ss is : Add these equtions to eliinte, nd, : Solve fo to obtin:, o, becuse the blocks hve coon cceletion, (),, (), +, + o, becuse,,, ( + ) +

53 Newton s Lws 347 Substitutin fo into eqution () ields:, + + (b) Substitute nueicl vlues in the equtions deived in Pt () nd evlute nd, : 3. N.0k + 6.0k nd ( 3. N)(.0k).0k + 6.0k, 0.40 /s 0.80 N Reks: Note tht ou esults fo the cceletion e the se s if the foce hd cted on sinle object whose ss is equl to the su of the sses of the two blocks. In fct, becuse the two blocks hve the se cceletion, we cn conside the to be sinle sste with ss wo 00-k boes e ded lon hoizontl fictionless sufce t constnt cceletion of.00 /s, s shown in iue Ech ope hs ss of.00 k. ind the foce nd the tension in the opes t points A, B, nd C. Pictue the Poble he fee-bod dis fo the boes nd the opes e below. Becuse the veticl foces hve no bein on the poble the hve not been included. Let the nuel denote the 00-k bo to the left, the nuel the ope connectin the boes, the nuel 3 the bo to the iht nd the nuel 4 the ope to which the foce is pplied. 3, 4 is the tension foce eeted b the bo whose ss is 3 on the ope whose ss is 4, 4, 3 is the tension foce eeted b the ope whose ss is 4 on the bo whose ss is 3,, 3 is the tension foce eeted b the ope whose ss is on the bo whose ss is 3, 3, is the tension foce eeted b the bo whose ss is 3 on the ope whose ss is,, is the tension foce eeted b the bo whose ss is on the ope whose ss is, nd, is the tension foce eeted b the ope whose ss is on the bo whose ss is. he equl nd opposite pis of foces e,,, 3,,3, nd 4, 3 3,4. We cn ppl Newton s second lw to ech bo nd ope septel nd use the fct tht thei cceletions e equl. Appl to the bo whose ss is : (),

54 348 Chpte 4

55 Newton s Lws 349 Appl to the ope whose ss is : Appl to the bo whose ss is 3 : Appl to the ope whose ss is 4 : Add these equtions to eliinte,,,, 3,,,3, 4,3, nd 3,4 nd solve fo : 3,, () (3) 4,3, , ( + + ) evlute : ( 0k)(.00 /s ) 0 N Use eqution () to find the tension t point A: Use eqution () to epess the tension t point B: evlute 3, : Use eqution (3) to epess the tension t point C: evlute 4,3 : ( 00k)(.00 /s ) 00 N, 3,, + 3, 00 N + 0N 4,3, ,3 0N + 0N (.00 k)(.00 /s ) ( 00k)(.00 /s ) 7 [SSM] A block of ss is bein lifted veticll b ope of ss M nd lenth L. he ope is bein pulled upwd b foce pplied t its top end, nd the ope nd block e cceletin upwd with n cceletion of nitude. he distibution of ss in the ope is unifo. Show tht the tension in the ope t distnce (whee < L) bove the block is ( + )[ + (/L)M].

56 350 Chpte 4 Pictue the Poble Becuse the distibution of ss in the ope is unifo, we cn epess the ss of lenth of the ope in tes of the totl ss of the ope M nd its lenth L. We cn then epess the totl ss tht the ope ust suppot t distnce bove the block nd use Newton s second lw to find the tension s function of. L ope Set up popotion epessin the ss of lenth of the ope s function of M nd L nd solve fo : Epess the totl ss tht the ope ust suppot t distnce bove the block: Appl to the block nd lenth of the ope: Substitutin fo + ields: Solve fo nd siplif to obtin: ' M L ' + ' + M L M L ( + ' ) ( ' ) + + M L + M ( + ) + L M L 7 A chin consists of 5 links, ech hvin ss of 0.0 k. he chin is bein pulled upwd b foce pplied b ou hnd to its top link, ivin the chin n upwd cceletion of.5 /s. ind () the foce nitude eeted on the top link b ou hnd; (b) the net foce on ech link; nd (c) the nitude of the foce ech link eets on the link below it. Pictue the Poble Choose coodinte sste with the + diection upwd nd denote the top link with the nuel, the second with the nuel, etc. he fee-bod dis show the foces ctin on links nd. We cn ppl Newton s second lw to ech link to obtin sste of siultneous equtions tht we cn solve fo the foce ech link eets on the link below it. Note tht the net foce on ech link is the poduct of its ss nd cceletion.

57 Newton s Lws 35 () Appl to the top link nd solve fo : evlute : 5 5 nd ( ) ( 0.0 k)( 9.8/s +.5/s ) 6. N (b) Appl to sinle link: link link 0.5 N ( 0.0 k)(.5/s ) (c) Appl to the st thouh 5 th links to obtin: Add equtions () thouh (5) to obtin:, () 3, () 3 4, (3) 4 5, nd (4) 5 (5) 4 4 Solve fo to obtin: ( + ) evlute : 4 ( 0.0 k)( 9.8/s +.5/s ) 4.9 N Substitute fo to find 3, nd then substitute fo 3 to find 4 : 3.7 N nd.5n 3 4 Solve eqution (5) fo 5 : ( + ) 5

58 35 Chpte 4 evlute 5 : 5 ( 0.0 k)( 9.8/s +.5/s ). N 73 [SSM] A 40.0-k object is suppoted b veticl ope. he ope, nd thus the object, is then cceleted fo est upwd so tht it ttins speed of 3.50 s in s. () Dw the object s fee bod di with the eltive lenths of the vectos showin the eltive nitudes of the foces. (b) Use the fee-bod di nd Newton s lws to deteine the tension in the ope. Pictue the Poble A net foce is equied to ccelete the object. In this poble the net foce is the diffeence between nd ( ). () he fee-bod di of the object is shown to the iht. A coodinte sste hs been chosen in which the upwd diection is positive. he nitude of is ppoitel.5 ties the lenth of. (b) Appl to obtin: to the object Solvin fo ields: + ( + ) Usin its definition, substitute fo to obtin: Δv + Δt evlute : ( 40.0 k) 59 N 3.50 /s /s s 74 A k helicopte is lowein 4000-k tuck b cble of fied lenth. he tuck, helicopte, nd cble e descendin t 5.0 /s nd ust be slowed to 5.00 /s in the net 50.0 of descent to pevent din the tuck. Assue constnt te of slowin. () Dw the fee-bod di of the tuck. (b) Deteine the tension in the cble. (c) Deteine the lift foce of the helicopte bldes.

59 Newton s Lws 353 Pictue the Poble A net foce in the upwd diection is equied to slow the tuck s descent. his net foce is the diffeence between nd. We cn use Newton s second lw nd constnt-cceletion eqution to find the tension in the cble tht suppots the tuck. () ee-bod dis showin the foces ctin on the tuck nd on the helicopte e shown to the iht. A coodinte sste in which the downwd diection is positive hs been chosen. Note tht, becuse it is hovein, the helicopte is in equilibiu unde the influence of the foces, lift, nd. lift (b) Appl obtin: to the tuck to Solve fo the tension in the lowe potion of the cble: t o, becuse t, t t ( ) t t t () Use constnt-cceletion eqution to elte the tuck s initil nd finl speeds to its displceent nd cceletion: v v + Δ 0 v v Δ 0 evlute : Substitute nueicl vlues in eqution () nd evlute : (c) Appl helicopte to obtin: to the ( 5.00 /s) ( 5.0 /s) ( 50.0 ).000 /s ( 4000 k)( 9.8 /s /s ) 47.4 kn 47. kn h + lift h Solvin fo lift ields: + ( ) lift h h h +

60 354 Chpte 4 Substitute fo fo Pt (): lift h ( ) + t ( ) ( + )( ) evlute lift : ( 4000 k k)( 9.8 /s +.00 /s ) 4 kn lift 75 wo objects e connected b ssless stin, s shown in iue he incline nd the ssless pulle e fictionless. ind the cceletion of the objects nd the tension in the stin fo () in tes of θ, nd, nd (b) θ 30º nd 5.0 k. Pictue the Poble Becuse the stin does not stetch o becoe slck, the two objects ust hve the se speed nd theefoe the nitude of the cceletion is the se fo ech object. Choose coodinte sste in which up the incline is the + diection fo the object of ss nd downwd is the + diection fo the object of ss. his idelized pulle cts like piece of polished pipe; i.e., its onl function is to chne the diection the tension in the ssless stin cts. Dw fee-bod di fo ech of the two objects, ppl Newton s second lw to both objects, nd solve the esultin equtions siultneousl. t h () he fee-bod dis fo the two objects e shown to the iht: n, ', θ, Appl to the object whose ss is : Appl to the object whose ss is to obtin: Addin equtions () nd () ields:, sin θ o, becuse, nd the two objects hve coon cceletion, sin θ () () sin θ + Solve fo to obtin: ( θ ) + sin (3)

61 Newton s Lws 355 Substitute fo in eithe of the equtions continin the tension nd solve fo to obtin: (b) Substitute nueicl vlues in eqution (3) nd evlute : ( + θ ) + sin (4) ( 9.8 /s ) 5.0 k ( 5.0 k).5 /s ( sin 30 ) 5.0 k k Substitute nueicl vlues in eqution (4) nd evlute : ( 9.8 /s )( 5.0 k) ( + sin30 ) 37 N 5.0 k k 76 Duin ste poduction of Pete Pn, the 50-k ctess plin Pete hs to fl in veticll (descend). o be in tie with the usic, she ust be loweed, sttin fo est, distnce of 3. in. s t constnt cceletion. Bckste, sooth sufce sloped t 50º suppots counteweiht of ss, s shown in iue Show the clcultions tht the ste ne ust pefo to find () the ss of the counteweiht tht ust be used nd (b) the tension in the wie. Pictue the Poble he nitudes of the cceletions of Pete nd the counteweiht e the se. Choose coodinte sste in which up the incline is the + diection fo the counteweiht nd downwd is the + diection fo Pete. he pulle chnes the diection the tension in the ope cts. Let Pete s ss be P. Inoin the ss of the ope, dw fee-bod dis fo the counteweiht nd Pete, ppl Newton s second lw to ech of the, nd solve the esultin equtions siultneousl. () Usin constnt-cceletion eqution, elte Pete s displceent to he cceletion nd descent tie: ( Δ ) ( Δ ) Δ v0 Δt + t o, becuse v 0 0, Δ t Δ ( Δt) he fee-bod di fo the counteweiht is shown to the iht: n θ

62 356 Chpte 4 Appl to the counteweiht: he fee-bod di fo Pete is shown to the iht: sin 50 ' P Notin tht ', ppl to Pete: Addin the two equtions nd solvin fo ields: P P ( ) P + sin50 Substitutin fo ields: Δ P ( ) Δt Δ + sin 50 ( Δt) evlute : ( 50 k) ( 3.) (.s) 9.8 /s /s 48.0 k 48 k ( 3.) (.s) sin 50 (b) Substitute fo in the foce eqution fo the counteweiht nd solve fo to obtin: ( + sin ) 50 evlute : ( 48.0 k)[.3 /s + ( 9.8/s ) sin50 ] 0.4 kn 77 An 8.0-k block nd 0-k block, connected b ope tht psses ove fictionless pe, slide on fictionless inclines (iue 4-56). () ind the cceletion of the blocks nd the tension in the ope. (b) he two blocks e eplced b two othes of sses nd such tht thee is no cceletion. ind whteve infotion ou cn bout the sses of these two new blocks.

63 Newton s Lws 357 Pictue the Poble he nitude of the cceletions of the two blocks e the se. Choose coodinte sste in which up the incline is the + diection fo the 8.0-k object nd downwd is the + diection fo the 0-k object. he pe chnes the diection the tension in the ope cts. Dw fee-bod dis fo ech object, ppl Newton s second lw of otion to both of the, nd solve the esultin equtions siultneousl. ' n,8 n, () Appl to the 8.0-k block: Notin tht, ppl to the 0-k block: Add equtions () nd () to obtin: 8 8 sin 40 () 0 sin 50 0 () sin 50 sin ( ) Solvin fo ields: ( sin 50 sin ) evlute : ( 9.8 /s )( ( 0 k) sin 50 ( 8.0 k) sin 40 ) 8.0 k + 0 k.37 /s.4 /s Solvin the fist of the two foce equtions fo ields: ( sin ) evlute : ( 8.0k)( [ 9.8/s ) sin /s ] 6N (b) Becuse the sste is in equilibiu, 0, nd equtions () nd () becoe: sin 40 0 (3) nd sin 50 0 (4)

64 358 Chpte 4 Addin equtions (3) nd (4) ields: sin 50 sin 40 0 Solve fo nd evlute the tio / to obtin: sin50 sin A hev ope of lenth 5.0 nd ss 4.0 k lies on fictionless hoizontl tble. One end is ttched to 6.0-k block. he othe end of the ope is pulled b constnt hoizontl 00-N foce. () Wht is the cceletion of the sste? (b) Give the tension in the ope s function of position lon the ope. Pictue the Poble he pictoil epesenttions shown below suize the infotion iven in this poble. While the ss of the ope is distibuted ove its lenth, the ope nd the 6.0-k block hve coon cceletion. Choose coodinte sste in which the diection of the 00-N foce is the + diection. Becuse the sufce is hoizontl nd fictionless, the onl foce tht influences ou solution is the 00-N foce. Pt ( ) Pt ( b) 4.0 k 00 N 6.0 k 6.0 k 5.0 () Appl to the sste shown fo Pt (): Solve fo to obtin: 00 N + 00 N + ( ) evlute : 00 N 4.0 k k 0 /s (b) Let epesent the ss of lenth of the ope. Assuin tht the ss of the ope is unifol distibuted lon its lenth: Appl to the block in Pt (b) to obtin: L ope ( ) + 4.0k k 5.0 Substitutin fo ields: 4.0k + 5.0

65 N + ( 8.0 N/) Newton s Lws k evlute : 6.0k + ( 0/s ) 79 [SSM] A 60-k housepinte stnds on 5-k luinu pltfo. he pltfo is ttched to ope tht psses thouh n ovehed pulle, which llows the pinte to ise heself nd the pltfo (iue 4-57). () o ccelete heself nd the pltfo t te of 0.80 /s, with wht foce ust she pull down on the ope? (b) When he speed eches.0 /s, she pulls in such w tht she nd the pltfo o up t constnt speed. Wht foce is she eetin on the ope now? (Inoe the ss of the ope.) Pictue the Poble Choose coodinte sste in which the upwd diection is the positive diection. Note tht is the foce eeted b the pinte on the ope nd tht is the esultin tension in the ope. Hence the net upwd foce on the pinte-plus-pltfo is. () Lettin +, tot fe pinte ppl to the fe-pluspinte: Solvin fo ives: ( ) evlute : tot tot + tot ( 75k)( 0.80 /s + 9.8/s ) 398 N Becuse : 398 N 0.40kN (b) Appl 0 to obtin: tot 0 tot evlute : ( 75k)( 9.8/s ) 0.37 kn 80 iue 4-58 shows 0-k block slidin on 0-k block. All sufces e fictionless nd the pulle is ssless nd fictionless. ind the cceletion of ech block nd the tension in the stin tht connects the blocks.

66 360 Chpte 4 Pictue the Poble Choose coodinte sste in which up the incline is the + diection nd dw fee-bod dis fo ech block. Notin tht 0 0, ppl Newton s second lw to ech block nd solve the esultin equtions siultneousl. Dw fee-bod di fo the 0-k block: n on Appl to the block to obtin: 0 sin 0 00, () Dw fee-bod di fo the 0-k block. Becuse ll the sufces, includin the sufces between the blocks, e fictionless, the foce the 0-k block eets on the 0-k block ust be nol to thei sufces s shown to the iht. n on on 0 Appl to the block to obtin: Becuse the blocks e connected b tut stin: Substitutin fo 0, in eqution () ields: 0 sin 0 00, () 0, 0, 0 sin 0 00, (3) Eliinte between equtions 0 0 () nd (3) to obtin: 0, sin

67 Newton s Lws 36 0 k 0 k evlute 0, : 0, ( 9.8 /s ) 0 k + 0 k. /s sin 0 Becuse 0, 0, : 0,./s Substitute fo eithe of the cceletions in the foce equtions nd solve fo : 45N 8 A 0-k block with pulle ttched slides lon fictionless lede. It is connected b ssless stin to 5.0-k block vi the neent shown in iue ind () the cceletion of ech block nd (b) the tension in the connectin stin. Pictue the Poble Choose coodinte sste in which the + diection is to the iht nd dw fee-bod dis fo ech block. Becuse of the pulle, the stin eets foce of. Appl Newton s second lw of otion to both blocks nd solve the esultin equtions siultneousl. () Dw fee-bod di fo the 0-k block: n,0,0 0 Appl to the block to obtin: 00, Dw fee-bod di fo the 5.0-k block: Appl to the block to obtin: 5 5 5,,5 5

68 36 Chpte 4 Usin constnt-cceletion eqution, elte the displceent of the 5.0-k block to its cceletion nd the tie duin which it is cceleted: Usin constnt-cceletion eqution, elte the displceent of the 0-k block to its cceletion nd cceletion tie: Δ Δ ( t) 5 5, Δ ( t) 0 0, Δ Divide the fist of these equtions b the second to obtin: Δ Δ 5 0 5, 0, ( Δt) 5, ( Δt) 0, Use the esult of Pt () to obtin: Let 0,. hen 5, nd the foce equtions becoe: Eliinte between the two equtions to obtin: evlute 0, nd 5, : 5, 0, 0 nd 5 0, nd 0, 5 ( ) ( 5.0k)( 9.8/s ) ( 5.0k) + ( 0k).5/s (.45/s ) /s.45/s Substitute fo eithe of the cceletions in eithe of the foce equtions nd solve fo : 5N 8 he pptus in iue 4-60 is clled n Atwood s chine nd is used to esue the fee-fll cceletion b esuin the cceletion of the two blocks connected b stin ove pulle. Assue ssless, fictionless pulle nd ssless stin. () Dw fee-bod di of ech block. (b) Use the fee-bod dis nd Newton s lws to show tht the nitude of the cceletion of eithe block nd the tension in the stin e + +. (c) Do these epessions ( ) ( ) nd ( ) ive plusible esults if, in the liit tht, <<? Eplin. >> nd in the liit tht

69 Newton s Lws 363 Pictue the Poble Assue tht >. Choose coodinte sste in which the + diection is downwd fo the block whose ss is nd upwd fo the block whose ss is nd dw fee-bod dis fo ech block. Appl Newton s second lw to both blocks nd solve the esultin equtions siultneousl. () he fee-bod dis fo the two blocks e shown to the iht: (b) Appl whose ss is : to the block () Appl whose ss is : to the block () Becuse the blocks e connected b tut stin, the hve the se cceletion. Let epesent thei coon cceletion. hen equtions () nd () becoe: Addin these equtions eliintes nd ields: Solve fo to obtin: nd + (3) + Substitutin fo in eithe of the foce equtions nd solvin fo ields: + (4) (c) o, equtions (3) nd (4) becoe: nd s epected.

70 364 Chpte 4 Divide the nuetos nd denointos if equtions (3) nd (4) b to obtin: + nd + o >> these equtions becoe: nd s epected. Divide the nuetos nd denointos if equtions (3) nd (4) b to obtin: + nd + o << these equtions becoe: nd s epected. 83 If one of the sses of the Atwood s chine in iue 4-60 is. k, wht should be the othe ss so tht the displceent of eithe ss duin the fist second followin elese is 0.30? Assue ssless, fictionless pulle nd ssless stin. Pictue the Poble he cceletion cn be found fo the iven displceent duin the fist second. he tio of the two sses cn then be found fo the cceletion usin the fist of the two equtions deived in Poble 8 eltin the cceletion of the Atwood s chine to its sses. o Poble 8 we hve: Solvin fo ields: Usin constnt-cceletion eqution, elte the displceent of the sses to thei cceletion nd solve fo the cceletion: + + () ( t) Δ ( t) Δ v0 t + Δ o, becuse v 0 0, Δ Δ ( Δt) evlute : ( 0.30) (.0s) 0.600/s

71 Newton s Lws 365 Substitute nueicl vlues in eqution () to obtin: 9.8 /s 9.8 /s /s /s ind the second vlue fo fo. k:.4k o.k 84 he cceletion of vit cn be deteined b esuin the tie t it tkes fo ss in n Atwood s chine descibed in Poble 8 to fll distnce L, sttin fo est. () Usin the esults of Poble 8 (Note the cceletion is constnt.), find n epession fo in tes of,, L, nd t. (b) Show tht sll eo in the tie esueent dt, will led to n eo in b n ount d iven b d/ dt/t. (c) Assue tht the onl sinificnt uncetint in the epeientl esueents is the tie of fll. If L 3.00 nd is.00 k, find the vlue of such tht cn be esued with n ccuc of ±5% with tie esueent tht is ccute to ±0. s. Pictue the Poble Use constnt-cceletion eqution to elte the displceent of the descendin (o isin) ss s function of its cceletion nd then use one of the esults fo Poble 8 to elte to. Diffeentition of ou epession fo will llow us to elte uncetint in the tie esueent to uncetint in the esued vlue fo nd to the vlues of tht would ield n epeientl vlue fo tht is ood to within ±5%. () o Poble 8 we hve: Solvin fo ields: Usin constnt-cceletion eqution, epess the displceent, L, s function of t: + + () ( t) Δ L v0 Δt + Δ o, becuse v 0 0 nd Δt t, L L t () t Substitute fo in eqution () to obtin: L + t

72 366 Chpte 4 (b) Evlute d/dt to obtin: d dt 3 + 4Lt L + t t t Divide both sides of this epession b nd ultipl both sides b dt to septe the vibles: d dt t d (c) Becuse ±0. 05 Substitute nueicl vlues in eqution () nd evlute : : dt ( ± 0.s) t d ( 3.00) ( 4s) ± s 0.375/s Solve eqution () fo to obtin: + Evlute with.00 k: 9.8/s 9.8/s 0.9k 0.375/s /s (.00 k) Solve eqution () fo to obtin: + Substitute nueicl vlues to obtin: (.96 k) k 9.8/s 9.8/s /s 0.375/s 0 Becuse the sses e intechneble: Genel Pobles 0.9k o k 85 A pebble of ss ests on the block of ss of the idel Atwood s chine in iue ind the foce eeted b the pebble on the block of ss. Pictue the Poble We cn ppl Newton s second lw to the pebble nd use the epession fo the cceletion deived in Poble 8 to find the foce b

73 Newton s Lws 367 eeted b the block of ss on the pebble. he foces on eeted b the pebble on the block of ss nd b eeted b the block on the pebble constitute thid lw pi nd e equl in nitude. () he upwd diection hs been chosen s the + diection in the feebod di fo the pebble shown to the iht. he foces ctin on the pebble e the foce eeted b the object whose ss is ( b ) nd the vittionl foce ( ) eeted b Eth. (b) Appl to obtin: to the pebble Becuse b nd on thid lw pi: constitute o Poble 8, the cceletion of the blocks is iven b: With pebble of ss estin on the block of ss, the epession fo the cceletion becoes: b b b ( + ) ( ) on + () + + ( + ) ( + ) Substitutin fo in eqution () nd siplifin ields: on + ( + ) ( + ) A siple cceleoete cn be de b suspendin sll ssive object fo stin ttched to fied point on n cceletin object. Suppose such n cceleoete is ttched to point P on the ceilin of n utoobile tvelin in stiht line on flt sufce t constnt cceletion. Due to the cceletion, the stin will ke n nle θ with the veticl. () Show tht the nitude of the cceletion is elted to the nle θ b tn θ. (b) Suppose the utoobile bkes stedil to est fo 50 k/h ove distnce of 60. Wht nle will the stin ke with the veticl? Will the suspended

74 368 Chpte 4 object be positioned below nd hed o below nd behind point P duin the bkin? Pictue the Poble he fee-bod di shown below nd to the left shows the foces ctin on n object suspended fo the ceilin of c tht is inin speed to the iht. Choose the coodinte sste shown nd use Newton s second lw nd constnt- cceletion equtions to descibe the influence of the foces ctin on the suspended object on its otion. he foces ctin on the object e the vittionl foce nd the tension in the stin. he fee-bod di shown to the iht shows the foces ctin on n object suspended fo the ceilin of c tht is bkin while ovin to the iht. θ P θ θ () Appl to the object: sin θ nd cos θ 0 ke the tio of these two equtions to eliinte nd : sinθ cosθ o tnθ tnθ (b) Solve the eqution deived in () fo θ : Usin constnt-cceletion eqution, epess the velocit of the c in tes of its cceletion nd solve fo the cceletion: θ tn () v v0 + Δ o, becuse v 0, v0 0 v0 + Δ Δ

75 Newton s Lws 369 Substitute fo in eqution () nd siplif to obtin: evlute θ : θ tn v 0 Δ tn k 50 h θ tn 9.8 s 9.3 v 0 Δ h 3600 s ( 60 ) he suspended object will be positioned below nd hed (b 9.3 ) of point P duin the bkin. 87 [SSM] he st of silbot is suppoted t bow nd sten b stinless steel wies, the foest nd bckst, nchoed 0 pt (iue 4-6). he.0--lon st weihs 800 N nd stnds veticll on the deck of the bot. he st is positioned 3.60 behind whee the foest is ttched. he tension in the foest is 500 N. ind the tension in the bckst nd the foce tht the st eets on the deck. Pictue the Poble he fee-bod di shows the foces ctin t the top of the st. Choose the coodinte sste shown nd use Newton s second nd thid lws of otion to nlze the foces ctin on the deck of the silbot. Appl to the top of the st: ind the nles tht the foest nd bckst ke with the veticl: Solvin the -diection eqution fo B ields: sinθ B sinθ B 0 θ tn nd B θ tn B sinθ sinθ B

76 370 Chpte 4 sin6.7 sin 8. evlute B : B ( 500 N) 305N Appl 0 to the st: cosθ cosθ 0 Solve fo st to obtin: st cosθ + B cosθ B evlute st : st ( 500 N) cos6.7 + ( 305N) cos N st B B he foce tht the st eets on the deck is the su of its weiht nd the downwd foces eeted on it b the foest nd bckst: st on the deck 748 N N.55kN 88 A 50-k block is suspended fo unifo chin tht is hnin fo the ceilin. he ss of the chin itself is 0 k, nd the lenth of the chin is.5. Deteine the tension in the chin () t the point whee the chin is ttched to the block, (b) idw up the chin, nd (c) t the point whee the chin is ttched to the ceilin. Pictue the Poble Let be the ss of the block nd M be the ss of the chin. he fee-bod dis shown below displ the foces ctin t the loctions identified in the poble. We cn ppl Newton s second lw with 0 to ech of the seents of the chin to deteine the tensions. () (b) (c) () Appl to the block nd solve fo : o, becuse 0, evlute : ( 50k)( 9.8/s ) 0.49kN

77 Newton s Lws 37 (b) Appl to the block nd hlf the chin nd solve fo b : evlute b : (c) Appl to the block nd chin nd solve fo c : evlute c : M + o, becuse 0, M b + b b c ( 50k + 0k)( 9.8/s ) 0.59 kn ( + M ) o, becuse 0, c ( + M ) c ( 50k + 0k)( 9.8/s ) 0.69 kn 89 he speed of the hed of edheded woodpecke eches 5.5 /s befoe ipct with the tee. If the ss of the hed is k nd the vee foce on the hed duin ipct is N, find () the cceletion of the hed (ssuin constnt cceletion), (b) the depth of penettion into the tee, nd (c) the tie it tkes fo the hed to coe to stop. Pictue the Poble () Choose coodinte sste in which the + diection is the diection in which the woodpecke s hed is ovin nd ppl Newton s second lw to the woodpecke s hed to find its cceletion. (b) Becuse we ve ssued constnt cceletion, we cn use constnt-cceletion eqution to find the depth of penettion in the tee. In Pt (c), we cn the constntcceletion eqution v v0 + Δt to find the tie it tkes fo the hed to coe to stop. () Appl to the woodpecke s hed to obtin: on the hed on the hed evlute : 6.0 N k 0.0 k/s (b) Use constnt-cceletion eqution to elte the depth of penettion to the initil speed of the hed nd its cceletion: v v v v0 + Δ Δ 0

78 37 Chpte 4 Becuse v 0: evlute Δ: v Δ 0 ( 5.5 /s) Δ ( 0.0 k/s ) 5 c (c) Appl constnt-cceletion eqution to elte the stoppin tie of the hed to its initil nd finl speeds nd to its cceletion: v v v0 v0 + Δt Δt evlute Δt: /s Δt 0.0 k/s 55 s 90 A fictionless sufce is inclined t n nle of 30.0º to the hoizontl. A 70- block on the p is ttched to block usin pulle, s shown in iue 4-6. () Dw two fee-bod dis, one fo the 70- block nd the othe fo the block. (b) ind the tension in the stin nd the cceletion of the 70- block. (c) he 70- block is elesed fo est. How lon does it tke fo it to slide distnce of.00 lon the sufce? Will it slide up the incline, o down the incline? Pictue the Poble he ppliction of Newton s second lw to the block nd the hnin weiht will led to siultneous equtions in thei coon cceletion nd the tension in the cod tht connects the. Once we know the cceletion of this sste, we cn use constnt-cceletion eqution to pedict how lon it tkes the block to tvel.00 fo est. Note tht the nitudes of nd ' e equl. () he fee-bod dis e shown to the iht. 70 epesents the ss of the 70- block nd 75 the ss of the block. n ' 70 θ 75 (b) Appl to the block nd the suspended ss: θ nd 70 sin

79 Newton s Lws 373

80 374 Chpte 4 Lettin epesent the coon cceletion of the two objects, eliinte between the two equtions nd solve : sinθ evlute : ( 0.70 k) sin30 ( ) k 9.8/s.706 /s.7/s k k whee the inus sin indictes tht the cceletion is down the incline. Substitute fo in eithe of the foce equtions to obtin: N (c) Usin constnt-cceletion eqution, elte the displceent of the block down the incline to its initil speed nd cceletion: evlute Δt: ( t) Δ v o, becuse v 0 0, 0 Δt + Δ ( t) Δ Δ t Δ (.00) Δt.706 /s Δ.08s Becuse the block is elesed fo est nd its cceletion is netive, it will slide down the incline. 9 A bo of ss is pulled lon fictionless hoizontl sufce b hoizontl foce tht is pplied to the end of ope of ss (see iue 4-63). Nelect n s of the ope. () ind the cceletion of the ope nd block, ssuin the to be one object. (b) Wht is the net foce ctin on the ope? (c) ind the tension in the ope t the point whee it is ttched to the block. Pictue the Poble Note tht, while the ss of the ope is distibuted ove its lenth, the ope nd the block hve coon cceletion. Becuse the sufce is hoizontl nd sooth, the onl foce tht influences ou solution is. he fiue isepesents the sitution in tht ech seent of the ope epeiences vittionl foce; the cobined effect of which is tht the ope ust s. () Appl / net tot to the opeblock sste to obtin: +

81 (b) Appl to the ope, substitute the cceletion of the sste obtined in (), nd siplif to obtin: net Newton s Lws (c) Appl to the block, substitute the cceletion of the sste obtined in (), nd siplif to obtin: A.0-k block ests on fictionless wede tht hs 60º incline nd n cceletion to the iht such tht the ss eins sttion eltive to the wede (iue 4-64). () Dw the fee bod di of the block nd use it to deteine the nitude of the cceletion. (b) Wht would hppen if the wede wee iven n cceletion le thn this vlue? Slle thn this vlue? Pictue the Poble We cn ppl Newton s second lw to the -0-k block to deteine the nitude of its cceletion. () he fee-bod di of the.0-k block is shown to the iht. Becuse the sufce of the wede is fictionless, the foce it eets on the block ust be nol to its sufce. he foces ctin on the block e the nol foce n eeted b the fictionless sufce nd the vittionl foce. he diection of the cceletion of the wede hs been chosen s the + diection n () Appl to the block to obtin: n sin 30 o, becuse 0 nd, n sin 30 () Appl to the block: n cos30 () Divide eqution () b eqution () to obtin: cot30 tn30 evlute : 9.8/s tn 30 7 /s

82 376 Chpte 4 (b) An cceletion of the wede ete thn /tn30 would equie tht the nol foce eeted on the bod b the wede be ete thn tht iven in Pt (); tht is, n > /sin30. Unde this condition, thee would be net foce in the diection nd the block would ccelete up the wede. With n cceletion less thn /tn30, the block would ccelete down the wede. 93 [SSM] he sses ttched to ech side of n idel Atwood s chine consist of stck of five wshes, ech of ss, s shown in iue he tension in the stin is 0. When one of the wshes is eoved fo the left side, the einin wshes ccelete nd the tension deceses b N. () ind. (b) ind the new tension nd the cceletion of ech ss when second wshe is eoved fo the left side. Pictue the Poble Becuse the sste is initill in equilibiu, it follows tht 0 5. When one wshe is eoved fo the left side the einin wshes on the left side will ccelete upwd (nd those on the iht side downwd) in esponse to the net foce tht esults. he fee-bod dis show the foces unde this unblnced condition. Applin Newton s second lw to ech collection of wshes will llow us to deteine both the cceletion of the sste nd the ss of sinle wshe. ' () Appl wshes: to the isin 4 5 ( ) 4 4 () Notin tht, ppl to the descendin sses: Eliinte between these equtions to obtin: ( ) 5 5 () 9 Use this cceletion in eqution () o eqution () to obtin: 40 9 Epessin the diffeence Δ between 0 nd ields: Δ Δ

83 Newton s Lws 377 evlute : 9 5 ( N) 9.8 /s 55.0 (b) Poceed s in () to obtin: Add these equtions to eliinte nd solve fo to obtin: 3 3 nd evlute : 4 ( 9.8/s ).45 /s Eliinte in eithe of the otion equtions nd solve fo to obtin: evlute : ( k)( 9.8/s ).03N 94 Conside the idel Atwood s chine in iue When N wshes e tnsfeed fo the left side to the iht side, the iht side dops 47. c in 0.40 s. ind N. Pictue the Poble he fee-bod di epesents the Atwood s chine with N wshes oved fo the left side to the iht side. Appliction of Newton s second lw to ech collection of wshes will esult in two equtions tht cn be solved siultneousl to elte N,, nd. he cceletion cn then be found fo the iven dt. ( 5 N ) ' ( 5 + N ) Appl wshes: to the isin ( 5 N ) ( N ) 5 Notin tht, ppl to the descendin wshes: ( 5 + N ) ( 5 + N ) Add these equtions to eliinte : ( 5 + N ) ( 5 N ) ( 5 N ) + ( 5 + N )

84 378 Chpte 4 Solvin fo N ields: N 5 Usin constnt-cceletion eqution, elte the distnce the wshes fell to thei tie of fll: ( t) Δ ( t) Δ v0 Δt + Δ o, becuse v 0 0, Δ Δ ( Δt) evlute : ( 0.47) ( 0.40s) 5.89/s Substitute in the epession fo N: N 5.89 /s 9.8/s Blocks of ss nd e on hoizontl fictionless sufce (iue 4-66). he blocks e connected b stin. In ddition, foces nd e pplied to the blocks s shown. () If the foces shown e constnt, find the tension in the connectin stin. (b) If the foces v with tie s Ct nd Ct, whee C 5.00 N/s nd t is tie, find the tie t 0 t which the tension in the stin equls 0.0 N. Pictue the Poble Dw the feebod di fo the block of ss nd ppl Newton s second lw to obtin the cceletion of the sste nd then the tension in the ope connectin the two blocks. n () Appl to the block of ss : () Appl + 3 to both blocks: ( ) Solvin fo ives: Substitute fo in the eqution () to obtin: 3 3

85 Newton s Lws 379 Solvin fo ields: ( + ) 3 (b) Substitute fo nd in the eqution deived in Pt (): 4 3 ( Ct + ( Ct) ) 3 Ct evlute t t 0 : t t 4 3( 0.0 N) ( 5.00 N/s) 0.50 s 3 4C 97 Elvis Pesle, hs supposedl been sihted nueous ties fte he pssed w on Auust 6, 977. he followin is cht of wht Elvis weiht would be if he wee sihted on the sufce of othe objects in ou sol sste. Use the cht to deteine: () Elvis ss on Eth, (b) Elvis ss on Pluto, nd (c) the cceletion due to vit on Ms. (d) Cope the fee-fll cceletion on Pluto to the fee-fll cceletion on the oon. plnet Elvis s weiht (N) Mecu 43 Venus 03 Eth 33 Ms 43 Jupite 880 Stun Pluto 58 Moon 9 Pictue the Poble Elvis ss is the tio of his weiht on iven plnet to the cceletion of vit on tht plnet. In (b), (c), nd (d) we cn use the eltionship between the vittionl foce (weiht) ctin on n object, its ss (independent of loction), nd the locl vlue of the fee-fll cceletion. () Elvis ss on Eth is iven b: w Eth Eth evlute : 3 N 5.3 k 9.8 /s 5 k (b) Becuse his ss is independent of his loction, Elvis ss on Pluto is 5 k.

86 380 Chpte 4 (c) he fee-fll cceletion on Ms is the tio of the weiht of n object on Ms to the ss of the object: Ms won Ms Substitute nueicl vlues fo Elvis nd evlute Ms : Ms 43 N 5.3 k 3.74 /s (d) he fee-fll cceletion on Pluto is iven b: he fee-fll cceletion on the oon is iven b: won Pluto Pluto () won oon oon () Divide eqution () b eqution () nd siplif to obtin: Pluto oon w w on Pluto on oon w w on Pluto on oon Substitutin nueicl vlues ields: o oon 58 N 9 N Pluto Pluto oon As pnk, ou fiends hve kidnpped ou in ou sleep, nd tnspoted ou out onto the ice covein locl pond. When ou wke up ou e 30.0 fo the neest shoe. he ice is so slippe (i.e. fictionless) tht ou cn not see to et ouself ovin. You elize tht ou cn use Newton s thid lw to ou dvnte, nd choose to thow the heviest thin ou hve, one boot, in ode to et ouself ovin. ke ou weiht to be 595 N. () Wht diection should ou thow ou boot so tht ou will ost quickl ech the shoe? (b) If ou thow ou.0-k boot with n vee foce of 40 N, nd the thow tkes s (the tie intevl ove which ou ppl the foce), wht is the nitude of the vee foce tht the boot eets on ou? (Assue constnt cceletion.) (c) How lon does it tke ou to ech shoe, includin the shot tie in which ou wee thowin the boot? Pictue the Poble he di shown below suizes the infotion bout ou tip to the shoe nd will be helpful in solvin Pt (c) of the poble.

87 Newton s Lws t0 0 t s t? v0 0 v vcostin v vcostin () You should thow ou boot in the diection w fo the closest shoe. (b) he nitude of the vee foce ou eet on the boot equls the nitude of the vee foce the boot eets on ou: v, on ou 40 N (c) he tie equied fo ou to ech the shoe is the su of ou tvel tie while cceletin nd ou tvel tie while costin: Use constnt-cceletion eqution to elte ou displceent Δ 0 to ou cceletion tie Δt 0 : Δt Δt + Δt totl 0 o, becuse Δt s, Δt s + t () totl Δ ( t ) Δ () 0 0 Δ 0 Appl Newton s second lw to epess ou cceletion duin this tie intevl: 0 net v w/ v w evlute 0 : Substitute nueicl vlues in eqution () nd evlute Δ 0 : You costin tie is the tio of ou displceent while costin to ou speed while costin: ( 40 N)( 9.8 /s ) /s 595 N Δ 0 ( 6.95 /s )( s).46 Δ Δt v o, becuse Δ 30.0 Δ0, 30.0 Δ0 Δt (3) v

88 38 Chpte 4 Use constnt-cceletion eqution to elte ou teinl speed (ou speed fte the intevl of cceletion) to ou cceletion nd displceent duin this intevl: v v0 + 0Δt0 o, becuse v 0 0, v 0Δt0 Substitute fo v in eqution (3) to obtin: Δt 30.0 Δ Δt Substitutin fo Δt costin in eqution () ields: Δt totl 30.0 Δ s + Δt evlute Δt totl : Δt s + ( 6.95 /s )( s) totl 7.5 s 98 he pulle of n idel Atwood s chine is iven n upwd cceletion, s shown in iue ind the cceletion of ech ss nd the tension in the stin tht connects the. Pictue the Poble Becuse constnt-upwd cceletion hs the se effect s n incese in the cceletion due to vit, we cn use the esult of Poble 8 (fo the tension) with eplced b +. he ppliction of Newton s second lw to the object whose ss is will connect the cceletion of this bod to tension fo Poble 8., In Poble 8 it is iven tht, when the suppot pulle is not cceletin, the tension in the ope nd the cceletion of the sses e elted ccodin to: + Replce with + : + + ( ) Appl whose ss is : to the object

89 Substitute fo nd siplif to obtin: he epession fo is the se s fo with ll subscipts intechned (note tht positive vlue fo epesents cceletion upwd): ( ) + + ( ) Newton s Lws You e wokin fo n utootive zine nd puttin cetin new utoobile (ss 650 k) thouh its pces. While cceletin fo est, its onbod copute ecods its speed s function of tie s follows: v (/s): t (s): () Usin spedsheet, find the vee cceletion ove ech.8-s tie intevl, nd ph the velocit vesus tie nd cceletion vesus tie fo this c. (b) Whee on the ph of velocit vesus tie is the net foce on the c hihest nd lowest? Eplin ou esonin. (c) Wht is the vee net foce on the c ove the whole tip? (d) o the ph of velocit vesus tie, estite the totl distnce coveed b the c. Pictue the Poble () A spedsheet po is shown below. he fouls used to clculte the quntities in the coluns e s follows: Cell Content/oul Alebic o C 650 D5 (B5 B4)/ Δt E5 (C5 C4)/D5 Δv Δt 5 $D$*E5 0 ( )/5 A B C D E 650 k 3 t (s) v (/s) t idpt (s) (/s ) (N) ve

90 384 Chpte A ph of velocit s function of tie follows: v, /s t, s A ph of cceletion s function of tie is shown below: 4 0, /s t, s (b) Becuse the net foce is lowest whee the cceletion is lowest, we cn see fo the ph of velocit vesus tie tht its slope (the cceletion) is sllest in the intevl fo 0 to.8 s. Becuse the net foce is hihest whee the cceletion is hihest, we cn see fo the ph of velocit vesus tie tht its slope (the cceletion) is etest in the intevl fo.8 s to 3.6 s.

91 Newton s Lws 385 (c) o the tble we see tht: (d) he distnce coveed b the c is the e unde the ph of velocit vesus tie. Becuse the ph of speed vesus tie is ppoitel line, we cn estite the totl distnce coveed b the c b findin the e of the tinul eion unde it. ve A tinle 5500 N bh 60 ( 6.5 s)( 50 /s)

92 386 Chpte 4