Chapter 4 Newton s Laws

Transcription

1 Chpter 4 Newton s Lws Conceptul Probles While on ver sooth level trnscontinentl plne fliht, our coffee cup sits otionless on our tr. Are there forces ctin on the cup? If so, how do the differ fro the forces tht would be ctin on the cup if it st on our kitchen tble t hoe? Deterine the Concept Yes, there re forces ctin on it. he re the norl force of the tble nd the rvittionl pull of Erth (weiht). Since the cup is not ccelertin reltive to the round, the re the se s bein otionless t hoe (which is lso not ccelertin reltive to the round). Suppose ou re pssin nother cr on hihw nd deterine, fro our reference fre, tht the cr ou pss ppers to hve n ccelertion to the west, in spite of the fct its driver is intinin constnt speed nd direction. Is our fre n inertil one? If not, which w is our fre ccelertin copred to the (pprent) ccelertion of the other cr? Deterine the Concept You re in non-inertil fre tht is ccelertin to the est, opposite the other cr s pprent ccelertion. 3 [SSM] You re ridin in liousine tht hs opque windows which do not llow ou to see outside. he cr cn ccelerte b speedin up, slowin down, or turnin. Equipped with just sll hev object on the end of strin, how cn ou use it to deterine if the liousine is chnin either speed or direction? Cn ou deterine the liousine s velocit? Deterine the Concept he su of the eternl forces on the object is lws proportionl to its ccelertion reltive to n inertil reference fre. An reference fre tht intins zero ccelertion reltive to n inertil reference fre is itself n inertil reference fre, nd vice vers. he round is n inertil reference fre. If the lio does not ccelerte (tht is if it does not chne direction or speed) reltive to the round, the pendulu will dnle striht down so tht the net force on the bob is zero (no ccelertion). In tht cse, the lio is n inertil reference fre. Just with this pprtus nd not lookin outside ou cnnot tell the lio s velocit; ll ou know is tht it is constnt. 4 If onl sinle nonzero force cts on n object, ust the object hve n ccelertion reltive to ll inertil reference fres? Is it possible for such n object to hve zero velocit in soe inertil reference fre nd not in nother? If so, ive specific eple. 89

2 90 Chpter 4 Deterine the Concept An object ccelertes when net force cts on it. he fct tht n object is ccelertin tells us nothin bout its velocit other thn tht it is lws chnin. Yes, the object ust hve n ccelertion reltive to the inertil fre of reference. Accordin to Newton s st nd nd lws, n object ust ccelerte, reltive to n inertil reference fre, in the direction of the net force. If there is onl sinle nonzero force, then this force is the net force. Yes, the object s velocit be oentril zero. Durin the period in which the force is ctin, the object be oentril t rest, but its velocit cnnot rein zero becuse it ust continue to ccelerte. hus, its velocit is lws chnin. 5 A bsebll is cted upon b sinle known force. ro this infortion lone, cn ou tell in which direction the bsebll is ovin? Eplin. Deterine the Concept No. Predictin the direction of the subsequent otion correctl requires knowlede of the initil velocit s well s the ccelertion. While the ccelertion cn be obtined fro the net force throuh Newton s nd lw, the velocit cn onl be obtined b intertin the ccelertion. 6 A truck oves w fro ou t constnt velocit, s observed b ou t rest on the surfce of the Erth. It follows tht () no forces ct on the truck, (b) constnt force cts on the truck in the direction of its velocit, (c) the net force ctin on the truck is zero, (d) the net force ctin on the truck its weiht. Deterine the Concept An object in n inertil reference fre ccelertes if there is net force ctin on it. Becuse the object is ovin t constnt velocit, the net force ctin on it is zero. (c) is correct. 7 Severl spce probes hve been lunched tht re now fr out in spce (for eple, Pioneer 0 in the 970s is well beond our solr sste liits) nd the re still ovin w fro the Sun nd its plnets. How is its ss chnin? Which of the known fundentl forces continue to ct on it? Does it hve net force on it? Deterine the Concept he ss of the probe is constnt. However, the solr sste will eert ttrct the probe with rvittionl. As the distnce between the probe nd the solr sste becoes lrer the nitude of the rvittionl force becoes sller. here is net force on the probe becuse no other forces ct on it.

3 Newton s Lws 9 8 Astronuts in pprent weihtlessness durin their st on the Interntionl Spce Sttion ust crefull onitor their ss becuse such conditions re known to cuse serious edicl probles. Give n eple of how ou iht desin equipent to esure the ss of n stronut in Erth orbit. Deterine the Concept You could use clibrted sprin ( sprin with known stiffness constnt) to pull on ech stronut nd esure their resultin ccelertion. hen ou could use Newton s second lw to clculte their ss. 9 [SSM] You re ridin in n elevtor. Describe two situtions in which our pprent weiht is reter thn our true weiht. Deterine the Concept Your pprent weiht is the redin of scle. If the ccelertion of the elevtor (nd ou) is directed upwrd, the norl force eerted b the scle on ou is reter thn our weiht. You could be ovin down but slowin or ovin up nd speedin up. In both cses our ccelertion is upwrd. 0 Suppose ou re in trin ovin t constnt velocit reltive to the round. You toss bll to our friend severl sets in front of ou. Use Newton s second lw to eplin wh ou cnnot use our observtions of the tossed bll to deterine the trin s velocit reltive to the round. Deterine the Concept Becuse ou re ovin with constnt velocit, our fre of reference is n inertil reference fre. In n inertil reference fre there re no fictitious forces. hus ovin or not ovin, the bll will follow the se trjector in our reference fre. he net force on the bll is the se, so its ccelertion is the se. Eplin wh, of the fundentl interctions, the rvittionl interction is the in concern in our everd lives. One other on this list lso pls n incresinl sinificnt role in our rpidl dvncin technolo. Which one is tht? Wh re the others not obviousl iportnt? Deterine the Concept he stron nucler force ct onl over the diensions of nucleus nd the wek nucler force is wek. he ost sinificnt force in our everd world is rvit. It literll keeps us on or ner the round. he other ost coon force is the electronetic force. It provides the lue to hold solid toether nd ke the riid. It is of ret iportnce in electric circuits. Give n eple of n object tht hs three forces ctin on it, nd () ccelertes, (b) oves t constnt (non-zero) velocit, nd (c) reins t rest. Deterine the Concept () An object for which the vector su of the three forces doesn t dd to zero. or eple, sled on frictionless surfce pulled horizontll. Norl force plus weiht plus the pull don t dd to zero, so it ccelertes.

4 9 Chpter 4 (b) Pullin fish verticll upwrd t constnt velocit while it is still in the wter. he forces ctin on the fish re the pull, the weiht of the fish, nd wter dr forces. hese forces dd up to zero. (c) he three forces need to dd vectorill to zero. An eple is picture hun b two wires. 3 [SSM] Suppose block of ss rests on block of ss nd the cobintion rests on tble s shown in iure ell the ne of the force nd its cteor (contct versus ction-t--distnce) for ech of the followin forces; () force eerted b on, (b) force eerted b on, (c) force eerted b on the tble, (d) force eerted b the tble on, (e) force eerted b the erth on. Which, if n, of these forces constitute Newton s third lw pir of forces? Deterine the Concept () he force eerted b on. (b) he force eerted b on. (c) he force eerted b on the tble. (d) he force eerted b the tble on. (e) he force eerted b the erth on. Norl force, contct tpe. Norl force, contct tpe. Norl force, contct tpe. Norl force, contct tpe. Grvittionl force, ction-t--distnce tpe. he Newton s 3 rd lw force pirs re the two norl forces between the two blocks nd the norl force between the tble nd the botto block. he rvittionl force hs 3 rd lw force pir, tht cts on Erth nd, so, is not in the question set. 4 You nk fish ou hve just cuht on our line upwrd fro rest into our bot. Drw free-bod dir of the fish fter it hs left the wter nd s it ins speed s it rises. In ddition, tell the tpe (tension, sprin, rvit, norl, friction, etc.) nd cteor (contct versus ction-t--distnce) for ech force on our dir. Which, if n, pirs of the forces on our dir constitute Newton s third lw pir? Cn ou tell the reltive nitudes of the forces fro the infortion iven? Eplin.

5 Newton s Lws 93 Deterine the Concept A free-bod dir showin the forces ctin on the fish is shown to the riht. A tble surizin the tpe nd cteor of the forces is shown below. fish strin on fish r Erth on fish r w orce pe Cteor ension Contct strin on fish Grvit Action-t--distnce Erth on fish Becuse the fish ccelertes upwrd, the tension force ust be reter in nitude thn the rvittionl force ctin on the fish. 5 If ou entl set fnc plte on the tble, it will not brek. However if ou drop it fro heiht, it iht ver well brek. Discuss the forces tht ct on the plte (s it contcts the tble) in both these situtions. Wht is different bout the second sitution tht cuses the plte to brek? Deterine the Concept When the plte is sittin on the floor, the norl force n ctin upwrd on it is eerted b the floor nd is the se size s the rvittionl force on the plte. Hence, the plte does not ccelerte. However, to slow the plte down s it hits the floor requires tht n > (or n >> if the floor is hrd nd the plte slows quickl). A lre norl force eerted on delicte chin cn esil brek it. 6 or ech of the followin forces, ive wht produces it, wht object it cts on, its direction, nd the rection force. () he force ou eert on our briefcse to hold it t rest verticll. (b) he norl force on the soles of our feet s ou stnd brefooted on horizontl wood floor. (c) he rvittionl force on ou s ou stnd on horizontl floor. (d) he horizontl force eerted on bsebll b bt s the bll is hit striht up the iddle towrds center field for sinle. Deterine the Concept () he force ou eert on our briefcse to hold it t rest verticll. You produce this force. It cts on the briefcse. It cts upwrd. he rection force is the force the briefcse eerts on our hnd.

6 94 Chpter 4 (b) he norl force on the soles of our feet s ou stnd brefooted on horizontl wood floor. (c) he rvittionl force on ou s ou stnd on horizontl floor. (d) he horizontl force eerted on bsebll b bt s the bll is hit striht up the iddle towrds center field for sinle. he floor produces this force. It cts on our feet. It cts upwrd. he rection force is the force our feet eert on the floor. he erth produces this force. It cts on ou. It cts downwrd. he rection force is the rvittionl force ou eert on the erth. he bt produces this force. It cts on the bll. It cts horizontll. he rection force is the force the bll eerts on the bt. 7 or ech cse, identif the force (includin its direction) tht cuses the ccelertion. () A sprinter t the ver strt of the rce. (b) A hocke puck skiddin freel but slowl coin to rest on the ice. (c) A lon fl bll t the top of its rc. (d) A bunee juper t the ver botto of her descent. Deterine the Concept () A sprinter t the ver strt of the rce: (b) A hocke puck skiddin freel but slowl coin to rest on the ice: (c) A lon fl bll t the top of its rc: (d) A bunee juper t the ver botto of her descent: he block s force on the sprinter ccelertes the sprinter forwrd when the sprinter pushes bckwrd on the blocks. he frictionl force b the ice on the puck cuses the puck to slow down. his force is directed opposite to the velocit of the puck. he downwrd rvittion force b Erth on the bll. he force eerted b the stretched bunee cord ccelertes the juper upwrd. Its direction is upwrd. 8 rue or flse: () If two eternl forces tht re both equl in nitude nd opposite in direction ct on the se object, the two forces cn never be Newton s third lw pir. (b) he two forces of Newton s third lw pir re equl onl if the objects involved re not ccelertin.

7 Newton s Lws 95 () rue. B definition, third lw pirs cnnot ct on the se object. (b) lse. Action nd rection forces re 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 force of 00 N. oether, the ove cross the ice stedil inin speed. () he force eerted b the bo on his fther is () 00 N, () 00 N, (3) 50 N, or (4) 40 N. (b) How do the nitudes of the two ccelertions copre? (c) How do the directions of the two ccelertions copre? Deterine the Concept () () hese forces re Newton 3 rd lw force pir, nd so the force eerted b the bo on his fther is 00 N. (b) Becuse the fther nd son ove toether, their ccelertions will be the se. (c) he directions of their ccelertion re the se. 0 A irl holds stone in her hnd nd cn ove it up or down or keep it still. rue or flse: () he force eerted b her hnd on the rock is lws the se nitude s the weiht of the stone. (b) he force eerted b her hnd on the rock is the rection force to the pull of rvit on the stone. (c) he force eerted b her hnd is lws the se size the force her hnd feels fro the stone but in the opposite direction. (d) If the irl oves her hnd down t constnt speed, then her upwrd force on the stone is less thn the weiht of the stone. (e) If the irl oves her hnd downwrd but slows the stone to rest, the force of the stone on the irl s hnd is the se nitude s the pull of rvit on the stone. () lse. If the rock is ccelertin, the force the irl eerts ust be reter thn the weiht of the stone. (b) lse. he rection force to the pull of rvit is the force the rock eerts on the rock. (c) rue. hese forces constitute Newton s third lw pir. (d) lse. If she oves the stone downwrd t constnt speed, the net force ctin on the stone ust be zero. (e) lse. If she is slowin the stone, it is eperiencin ccelertion nd the net force ctin on it cn not be zero.

8 96 Chpter 4 [SSM] A.5-k object hns t rest fro strin ttched to the ceilin. () Drw free bod dir of the object, indicte the rection force to ech force drwn, nd tell wht object the rection force cts on. (b) Drw free bod dir of the strin, indicte the rection force to ech force drwn, nd tell wht object ech rection force cts on. Do not nelect the ss of the strin. Deterine the Concept he force dirs will need to include forces eerted b the ceilin, on the strin, on the object, nd forces eerted b the erth. () r strin (b) b ceilin.5 k, strin, object, object () he forces ctin on the.5-k object re the rvittionl force, object nd the tension strin in the strin. he rection to strin is the force the object eerts downwrd on the strin. he rection to is, object the force the object eerts upwrd on the erth. (b) he forces ctin on the strin re its weiht, strin, the weiht of the object, object, nd b ceilin, the force eerted b the ceilin. he rection to, strin is the force the strin eerts upwrd on the erth. he rection to b ceilin is downwrd force the strin eerts on the ceilin. he rection to, object is the force the object eerts upwrd on the erth. () Which of the free-bod dirs in iure represents block slidin down frictionless inclined surfce? (b) or the correct fiure, lbel the forces nd tell which re contct forces nd which re ction-t--distnce forces. (c) or ech force in the correct fiure, identif the rection force, the object it cts on nd its direction. Deterine the Concept Identif the objects in the block s environent tht re eertin forces on the block nd then decide in wht directions those forces ust be ctin if the block is slidin down the inclined plne. () ree-bod dir (c) is correct.

9 (b) Becuse the incline is frictionless, the force n the incline eerts on the block ust be norl to the surfce nd is contct force. he second object cpble of eertin force on the block is the erth nd its force; the rvittionl force ctin on the block cts directl downwrd nd is n ction-t--distnce force. he nitude of the norl force is less thn tht of the weiht becuse it supports onl portion of the weiht. Newton s Lws 97 n (c) he rection to the norl force is the force the block eerts perpendiculrl on the surfce of the incline. he rection to the rvittionl force is the upwrd force the block eerts on the Erth. 3 A bo is held inst copressed, horizontl sprin tht is ttched to wll. he horizontl floor beneth the bo is frictionless. Drw the free bod dir of the bo in the followin cses. () he bo is held t rest inst the copressed sprin. (b) he force holdin the bo inst the sprin no loner eists, but the bo is still in contct with the sprin. (c) When the bo no loner hs contct with the sprin. Deterine the Concept In the followin free-bod dirs we ll ssue tht the bo is initill pushed to the left to copress the sprin. () Note tht, in the free-bod dir to the riht, tht n nd b sprin b hnd. b hnd n b sprin (b) Note tht while n, b sprin is now the net force ctin on the bo. As the sprin decopresses, b sprin will becoe sller. n b sprin

10 98 Chpter 4 (c) When the bo seprtes fro the sprin, the force eerted b the sprin on the bo oes to zero. Note tht it is still true tht n. n 4 Iine ourself seted on wheeled desk chir t our desk. Consider n friction forces between the chir nd the floor to be neliible. However, the friction forces between the desk nd the floor re not neliible. When sittin t rest, ou decide ou need nother cup of coffee. You push horizontll inst the desk, nd the chir rolls bckwrd w fro the desk. () Drw our free-bod dir of ourself durin the push nd clerl indicte which force ws responsible for our ccelertion. (b) Wht is the rection force to the force tht cused our ccelertion? (c) Drw the free-bod dir of the desk nd eplin wh it did not ccelerte. Does this violte Newton s third lw? Eplin. Deterine the Concept In the followin free-bod dirs we ll ssue tht the desk is to the left nd tht our otion is to the riht. () Newton s third lw ccounts for this s follows. When ou push with our hnds inst the desk, the desk pushes bck on our hnds with force of the se nitude but opposite direction. his force ccelertes ou bckwrd. n b desk (b) he rection force to the force tht cused our ccelertion is the force tht ou eerted on the desk. (c) When ou pushed on the desk, ou did not ppl sufficient force to overcoe the force of friction between the desk nd the floor. In ters of forces on the desk, ou pplied force, nd the floor pplied friction force tht, when dded s vectors, cncelled. he desk, therefore, did not ccelerte nd Newton s third lw is not violted. b ou n b floor, on desk

11 Newton s Lws 99 5 he se (net) horizontl force is pplied for fied tie intervl Δt to ech of two objects, hvin sses nd, tht sit on flt, frictionless surfce. (Let >.) () Assuin the two objects re initill t rest, wht is the rtio of their ccelertions durin the tie intervl in ters of, nd? (b) Wht is the rtio of their speeds v nd v t the end of the tie intervl? (c) How fr prt re the two objects (nd which is hed) the end of the tie intervl? Picture the Proble We cn ppl Newton s nd Lw to find the rtios of the ccelertions nd speeds of the two objects nd constnt-ccelertion equtions to epress the seprtion of the objects s function of the elpsed tie. () Use Newton s nd Lw to epress the ccelertions of the two objects: nd Dividin the first of these equtions b the second nd siplifin ields: (b) Becuse both objects strted fro rest, their speeds fter tie Δt hs elpsed re: v Δt nd v Δt Dividin the first of these equtions b the second nd siplifin ields: v v Δt Δt (c) he seprtion of the two objects t the end of the tie intervl is iven b: Usin constnt ccelertion eqution, epress the distnces trveled b the two objects when tie Δt hs elpsed: Substitute for Δ nd Δ in eqution () nd siplif to obtin: Δ Δ () Δ nd Δ Δ ( t) Δ Δ ( t) Δ ( Δt) ( Δt) ( ) Δt nd, becuse >, the object whose ss is is hed.

12 300 Chpter 4 Estition nd Approition 6 Most crs hve four sprins ttchin the bod to the fre, one t ech wheel position. Devise n eperientl ethod of estitin the force constnt of one of the sprins usin our known weiht nd the weihts of severl of our friends. Use the ethod to estite the force constnt of our cr s sprins. Picture the Proble Suppose ou put in 800 lbs (or bout 3600 N) of weiht nd the cr ss severl inches (or 6.00 c). hen ech sprin supports bout 900 N nd we cn use the definition of the force constnt k to deterine its vlue. he force constnt is the rtio of the copressin (or stretchin) force to the copression (or stretch): k Δ evlute k: k 4 ( 3600 N) 6.00 c 50 N/c 7 [SSM] Estite the force eerted on the olie s love b the puck when he ctches hrd slp shot for sve. Picture the Proble Suppose the olie s love slows the puck fro 60 /s to zero s it recoils distnce of 0 c. urther, ssue tht the puck s ss is 00. Becuse the force the puck eerts on the olie s love nd the force the olie s love eerts on the puck re ction-nd-rection forces, the re equl in nitude. Hence, if we use constnt-ccelertion eqution to find the puck s ccelertion nd Newton s nd lw to find the force the love eerts on the puck, we ll hve the nitude of the force eerted on the olie s love. Appl Newton s nd lw to the puck s it is slowed b the olie s love to epress the nitude of the force the love eerts on the puck: Use constnt-ccelertion eqution to relte the initil nd finl speeds of the puck to its ccelertion nd stoppin distnce: Solvin for puck ields: () love on puck v v0 + puck puck Δ v v puck puck ( ) puck 0 ( Δ ) puck

13 Newton s Lws 30 Substitute for puck in eqution () to obtin: loveon 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 Rerks: he force on the puck is bout 800 ties its weiht. 8 A bsebll pler slides into second bse durin stel ttept. Assuin resonble vlues for the lenth of the slide, the speed of the pler t the beinnin of the slide, nd the speed of the pler t the end of the slide, estite the vere force of friction ctin on the pler. Picture the Proble Let s ssue tht the pler s ss is 00 k, tht he ets oin firl quickl down the bse pth, nd tht his speed is 8.0 /s when he beins his slide. urther, let s ssue tht he pproches the bse t the end of the slide t 3.0 /s. ro these speeds, nd the lenth of the slide, we cn use Newton s nd lw nd constnt-ccelertion eqution to find the force due to friction (which cuses the slowin down). Appl Newton s nd lw to the slidin runner: Usin constnt-ccelertion eqution, relte the runner s initil nd finl speeds to his ccelertion nd the lenth of his slide: Substitutin for in eqution () ields: Assuin the pler slides, substitute nuericl vlues nd evlute friction : friction () v vf v vi Δ Δ f + friction friction vf v Δ i i ( ) ( ) 3.0 /s ( 8.0 /s 00 k ) ( ) kn where the inus sin indictes tht the force of friction opposes the runner s otion. 9 A rce cr skiddin out of control nes to slow down to 90 k/h before crshin hed on into brick wll. ortuntel, the driver is werin set belt. Usin resonble vlues for the ss of the driver nd the stoppin distnce,

14 30 Chpter 4 estite the vere force eerted on the driver b the set belt, includin its direction. Nelect n effects of frictionl forces on the driver b the set. Picture the Proble Assue crush distnce of bout t 90 k/h (5 /s) nd driver s ss of 55 k. We cn use constnt-ccelertion eqution (the definition of vere ccelertion) to find the ccelertion of the driver nd Newton s nd lw to find the force eerted on the driver b the set belt. Appl Newton s nd lw to the driver s she is brouht to rest b her set belt: Use constnt-ccelertion eqution to relte the initil nd finl speeds of the driver to her ccelertion nd stoppin distnce: () set belt on driverl v v0 + driver driver driver ( Δ ) driver Solvin for driver ields: driver v v 0 ( Δ ) driver Substitute for driver in eqution () to obtin: set belt on driverl driver ( Δ) driver evlute set belt on driver : set belt on driver ( 55 k) v 7 kn v 0 ( 5 /s) ( ) 0.0 where the inus sin indictes tht the force eerted b the set belt is in the opposite direction fro the driver s otion. Rerks: he vere force on the bll is bout 3 ties her weiht. Newton s irst nd Second Lws: Mss, Inerti, nd orce 30 A prticle of ss is trvelin t n initil speed v /s. Suddenl constnt force of 5.0 N cts on it, brinin it to stop in distnce of 6.5. () Wht is the direction of the force? (b) Deterine the tie it tkes for the prticle to coe to stop. (c) Wht is its ss? Picture the Proble he ccelertion of the prticle, its stoppin tie, nd its ss cn be found usin constnt-ccelertion equtions nd Newton s nd lw. A convenient coordinte sste is shown in the followin dir.

15 Newton s Lws 303 v r 0 () Becuse the constnt force slows the prticle, we cn conclude tht, s shown in the dir, its direction is opposite the direction of the prticle s otion. (b) Use constnt-ccelertion eqution to relte the initil nd finl velocities of the prticle to its ccelertion nd stoppin tie: v v0 + Δt or, becuse v 0, 0 v0 + Δt Δ v () 0 t Use constnt-ccelertion eqution to relte the initil nd finl velocities of the prticle to its ccelertion nd stoppin distnce: v v0 + Δ or, becuse v 0, 0 v Δ 0 + v0 Δ Substitutin for in eqution () ields: evlute Δt: Δ Δ t v 0 ( 6.5 ) Δ t 5.0 /s 5.00 s (c) Appl Newton s nd lw to the prticle to obtin: Solvin for ields: net net () Becuse the force is constnt, ou cn use constnt-ccelertion eqution to relte the prticle s initil nd finl speeds, ccelertion, nd stoppin distnce: v v0 + Δ or, becuse v 0, 0 v Δ 0 + v0 Δ Substitute for in eqution () Δ to obtin: v 0 net (6.5 )(5.0 N) evlute : ( 5.0 /s ) 3.00 k

16 304 Chpter 4 3 An object hs n ccelertion of 3.0 /s when the onl force ctin on it is 0. () Wht is its ccelertion when this force is doubled? (b) A second object hs n ccelertion of 9.0 /s under the influence of the force 0. Wht is the rtio of the ss of the second object to tht of the first object? (c) If the two objects re lued toether to for coposite object, wht ccelertion will the force 0 ctin on the coposite object produce? Picture the Proble he ccelertion of the object is relted to its ss nd the net force ctin on it b. net 0 () Use Newton s nd lw of otion to relte the ccelertion of the object to the net force ctin on it: net When net 0 : evlute : 0 0 ( 3.0 /s ) 6.0 /s (b) Let the subscripts nd distinuish the two objects. he rtio of the two sses is found fro Newton s nd lw: /s 9.0 /s 3 (c) he ccelertion of the coposite object is the net force divided b the totl ss + of the coposite object: net Substitute for nd evlute : ( ) /s.3 /s 3 A tubot tows ship with constnt force. he increse in the ship s speed durin 0-s intervl is 4.0 k/h. When second tubot pplies n dditionl constnt force in the se direction, the speed increses b 6 k/h durin 0-s intervl. How do the nitudes of the two tubot forces copre? (Nelect the effects of wter resistnce nd ir resistnce.) Picture the Proble he ccelertion of n object is relted to its ss nd the net force ctin on it b net. Let be the ss of the ship, be the ccelertion of the ship when the net force ctin on it is, nd be its ccelertion when the net force is +. 4

17 Newton s Lws 305 Usin Newton s nd lw, epress the net force ctin on the ship when its ccelertion is : Epress the net force ctin on the ship when its ccelertion is : + Divide the second of these equtions b the first nd solve for the rtio / : + Substitute for the ccelertions to deterine the rtio of the ccelertin forces nd solve for to obtin: 6 k/h 0s k/h 0s 33 A sinle constnt force of N cts on prticle of ss. he prticle strts fro rest nd trvels in striht line distnce of 8 in 6.0 s. ind. Picture the Proble he ss of the prticle is relted to its ccelertion nd the net force ctin on it b Newton s nd lw of otion. Becuse the force is constnt, we cn use constnt-ccelertion foruls to clculte the ccelertion. Choose coordinte sste in which the positive direction is the direction of otion of the prticle. he ss is relted to the net force nd the ccelertion b Newton s nd lw: () Becuse the force is constnt, the ccelertion is constnt. Use constnt-ccelertion eqution to relte the displceent of the prticle to it s ccelertion: ( t) Δ ( t) Δ v0 t + Δ or, becuse v 0 0, Δ Δ ( Δt) Substitute for in eqution () to obtin: ( t) Δ Δ evlute : ( N)( 6.0s) ( 8) k 34 A net force of (6.0 N) ˆ i (3.0 N) j ˆ cts on n object of ss.5 k. ind the ccelertion.

18 306 Chpter 4 Picture the Proble he ccelertion of n object is relted to its ss nd the net force ctin on it ccordin to net. Appl Newton s nd 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 tree stup nd penetrtes 6.00 c into the wood before coin to rest. () Assuin tht the ccelertion of the bullet is constnt, find the force (includin direction) eerted b the wood on the bullet. (b) If the se force cted on the bullet nd it hd the se speed but hlf the ss, how fr would it penetrte into the wood? Picture the Proble Choose coordinte sste in which the + direction is in the direction of the otion of the bullet nd use Newton s nd lw nd constntccelertion eqution to epress the reltionship between stoppin nd the ss of the bullet nd its displceent s it is brouht to rest in the block of wood. () Appl Newton s nd lw to the bullet to obtin: stoppin () Use constnt-ccelertion eqution to relte the bullet s initil nd finl speeds, ccelertion, nd stoppin distnce: vf vi + Δ or, becuse v f 0, 0 v Δ i + vi Δ Substitute for in eqution () to obtin: evlute stoppin stoppin vi () Δ : ( ) ( 500 /s.80 0 k ) 3 stoppin 3.8 kn ( 6.00 c) where the inus sin indictes tht stoppin opposes the otion of the bullet. (b) Solvin eqution () for Δ ields: vi Δ (3) stoppin

19 Newton s Lws 307 or nd Δ Δ : Evlute this epression for ' to obtin: v Δ' ' i stoppin vi Δ' (4) 4 stoppin Dividin eqution (4) b eqution (3) ields: v Δ' 4 Δ or Δ' Δ i stoppin vi stoppin evlute Δ : ( 6.00 c) 3.00 c Δ' 36 A crt on horizontl, liner trck hs fn ttched to it. he crt is positioned t one end of the trck, nd the fn is turned on. Strtin fro rest, the crt tkes 4.55 s to trvel distnce of.50. he ss of the crt plus fn is 355. Assue tht the crt trvels with constnt ccelertion. () Wht is the net force eerted on the crt-fn cobintion? (b) Mss is dded to the crt until the totl ss of the crt-fn cobintion is 7, nd the eperient is repeted. How lon does it tke for the crt to trvel.50 now? Inore the effects due to friction. Picture the Proble Choose the coordinte sste shown in the dir to the riht. he force ctin on the crt-fn cobintion is the consequence of the fn blowin ir to the left. We cn use Newton s nd lw nd constnt-ccelertion eqution to epress the reltionship between nd the ss of the crt-fn cobintion nd the distnce it trvels in iven intervl of tie. n () Appl Newton s nd lw to the crt-fn cobintion to obtin: ()

20 308 Chpter 4 Usin constnt-ccelertion eqution, relte the distnce the crtfn cobintion trvels to its initil speed, ccelertion, nd the elpsed tie: ( t) Δ ( t) Δ v0 Δt + Δ or, becuse v 0 0, Δ Δ ( Δt) Substitute for in eqution () to obtin: ( ) Δ () Δt ( ) (.50 ) evlute : k ( 4.55 s) N N (b) Solve eqution () for Δt to obtin: Δ t Δ evlute Δt: ( 0.7 k)(.50 ) Δ t N 6.49 s 37 A horizontl force 0 cuses n ccelertion of 3.0 /s when it cts on n object of ss slidin on frictionless surfce. ind the ccelertion of the se object in the circustnces shown in iure 4-35 nd 4-35b. Picture the Proble he ccelertion of n object is relted to its ss nd the net force ctin on it ccordin to net. Let be the ss of the object nd choose coordinte sste in which the direction of 0 in (b) is the + nd the direction of the left-ost 0 in () is the + direction. Becuse both force nd ccelertion re vector quntities, find the resultnt force in ech cse nd then find the resultnt ccelertion. () Appl obtin: to the object to Epress the net force ctin on the object: net () net iˆ + ˆj iˆ 0 + ˆj 0 he nitude nd direction of this net force re iven b: net nd θ tn + tn

21 Substitute for net in eqution () nd 0 siplif to obtin: 0 Substitute the nuericl vlue of 0 nd evlute : ( 3.0 /s ) Newton s Lws fro ech force. (b) Proceedin s in (), epress the net force ctin on the object: iˆ + ˆj ( sin 45 ) iˆ + ( + cos 45 )j ˆ net he nitude nd direction of this net force re iven b: nd ( 0 sin 45 ) + ( cos 45 ). 0 net + 80 θ tn tn 4.6 fro cos sin 45 Substitute for net in eqution () nd 0 siplif to obtin: Substitute the nuericl vlue of 0 nd evlute : ( ) /s 8.4 fro 38 Al nd Bert stnd in the iddle of lre frozen lke (frictionless surfce). Al pushes on Bert with force of 0 N for.5 s. Bert s ss is 00 k. Assue tht both re t rest before Al pushes Bert. () Wht is the speed tht Bert reches s he is pushed w fro Al? (b) Wht speed does Al rech if his ss is 80 k? 0 Picture the Proble he speed of either Al or Bert cn be obtined fro their ccelertions; in turn, the cn be obtined fro Newtons nd lw pplied to ech person. he free-bod dirs to the riht show the forces ctin on Al nd Bert. he forces tht Al nd Bert eert on ech other re ction-ndrection forces. Al on Bert n,bert Bert r n, Al Bert on Al r Al

22 30 Chpter 4 () Appl to Bert: Alon Bert Bert Bert Bert Al on Bert Bert evlute Bert : Bert 0 N 00k 0.00/s Usin constnt-ccelertion eqution, relte Bert s speed to his initil speed, speed fter.5 s, nd ccelertion: evlute Bert s speed t the end of.5 s: v v v Bert, Δt ( 0.00 /s )(.5 s) 0.30 /s (b) ro Newton's 3 rd lw, n equl but oppositel directed force cts on Al while he pushes Bert. Becuse the ice is frictionless, Al speeds off in the opposite direction. Appl to Al:, Al Bert on Al, AlAl, Solvin for Al s ccelertion ields: Al, Bert on Al, Al evlute Al, : Al, 0 N 80k 0.50/s Usin constnt-ccelertion eqution, relte Al s speed to his initil speed, speed fter.5 s, nd ccelertion: 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 cross frictionless floor with force of iven nitude, the block hs n ccelertion of /s. If ou push on different block whose ss is with force of nitude 0, its ccelertion is 3.0 /s. (Both forces re pplied horizontll.) () Wht ccelertion will horizontl force of nitude 0 ive to block with ss? (b) Wht ccelertion will horizontl force of nitude 0 ive to block with ss +?

23 Newton s Lws 3 Picture the Proble he free-bod dirs show the forces ctin on the two blocks. We cn ppl Newton s second lw to the forces ctin on the blocks nd eliinte to obtin reltionship between the sses. Additionl pplictions of Newton s nd lw to the su nd difference of the sses will led us to vlues for the ccelertions of these cobintions of ss. () Appl to the two blocks: n, n, r,, nd,, r Eliinte between the two, equtions nd solve for : Substitute nuericl vlues to /s obtin: /s, Epress the ccelertion of n object whose ss is when the net force ctin on it is : 3, 4 3 evlute : 3 ( /s ) 4.0 /s (b) Epress the ccelertion of n object whose ss is + when the net force ctin on it is : 5, evlute : 5 ( /s ).4 /s 40 o dr 75.0-k lo lon the round t constnt velocit, our trctor hs to pull it with horizontl force of 50 N. () Drw the free bod dir of the lo. (b) Use Newton s lws to deterine the force of friction on the lo. (c) Wht is the norl force of the round on the lo? (d) Wht horizontl force ust ou eert if ou wnt to ive the lo n ccelertion of.00 /s ssuin the force of friction does not chne. Redrw the lo s free bod dir for this sitution.

24 3 Chpter 4 Picture the Proble Becuse the velocit is constnt, the net force ctin on the lo ust be zero. Choose coordinte sste in which the positive direction is the direction of otion of the lo nd ppl Newton s nd lw to the lo. () he free-bod dir shows the forces ctin on the lo when it is bein dred in the + direction t constnt velocit. (b) Appl to the lo when it is ovin t constnt speed: b round or b round n pull pull b round pull 0 Substitute for pull nd evlute the force of friction b round : b round pull 50 N (c) Appl to the lo to obtin: Becuse the rvittionl force is iven b : evlute n : (d) he free-bod dir shows the forces ctin on the lo when it is ccelertin in the positive direction. Appl to the lo when it is ccelertin to the riht: or n n n ( 75.0 k)( 9.8 /s ) 736 N n b round n 0 pull pull b round

25 Solvin for pull ields: pull + b round 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 forces, (.0 N) ˆ i + ( 3.0 N) j ˆ nd (4.0 N) ˆ i ( N) j ˆ. he object is t rest t the oriin t tie t 0. () Wht is the object s ccelertion? (b) Wht is its velocit t tie t 3.0 s? (c) Where is the object t tie t 3.0 s? Picture the Proble he ccelertion cn be found fro Newton s nd lw. Becuse both forces re constnt, the net force nd the ccelertion re constnt; hence, we cn use the constnt-ccelertion equtions to nswer questions concernin the otion of the object t vrious ties. () Appl Newton s nd 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-ccelertion eqution, epress 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) Epress the position of the object in ters of its vere velocit: r v v v0 + v t t vt Substitute for v nd evlute this epression t t 3.0 s: r ( 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 34 Chpter 4 Mss nd Weiht 4 On the oon, the ccelertion due to rvit is onl bout /6 of tht on Erth. An stronut, whose weiht on erth is 600 N, trvels to the lunr surfce. His ss, s esured on the oon, will be () 600 k, (b) 00 k, (c) 6. k, (d) 9.8 k, (e) 360 k. Picture the Proble he ss of the stronut is independent of rvittionl fields nd will be the se on the oon or, for tht tter, out in deep spce. Epress the ss of the stronut in ters of his weiht on erth nd the rvittionl field t the surfce of the erth: w 600 N erth 9.8N/k nd (c) is correct. erth 6. k 43 ind the weiht of 54-k student in () newtons nd (b) pounds. Picture the Proble he weiht of n object is relted to its ss nd the rvittionl field throuh. () he weiht of the student is: w ( 54k)( 9.8N/k) 530 N N (b) Convert newtons to pounds: w 530 N 9lb. 0 lb 4.45 N/lb 44 ind the ss of 65-lb enineer in kilors. Picture the Proble he ss of n object is relted to its weiht nd the rvittionl field. Convert the weiht of the n into newtons: ( 65lb)( 4.45 N/lb) 734 N 65 lb Clculte the ss of the n fro his weiht nd the rvittionl field: w 734 N 9.8N/k 74.8k 45 [SSM] o trin stronuts to work on the oon, where the ccelertion due to rvit is onl bout /6 of tht on erth, NASA suberes the in tnk of wter. If n stronut, who is crrin bckpck, ir conditionin unit, oen suppl, nd other equipent, hs totl ss of 50 k, deterine the followin quntities. () her weiht on Erth, (b) her weiht on

27 Newton s Lws 35 the oon, (c) the required upwrd buonc force of the wter durin her trinin for the oon s environent on Erth. Picture the Proble We cn use the reltionship between weiht (rvittionl force) nd ss, toether with the iven infortion bout the ccelertion due to rvit on the oon, to find the stronut s weiht on Erth nd on the oon. () Her weiht on Erth is the product of her ss nd the rvittionl field t the surfce of the erth: evlute w: (b) Her weiht on the oon is the product of her ss nd the rvittionl field t the surfce of the oon: Substitute for her weiht on Erth nd evlute her weiht on the oon: w erth w erth oon ( 50 k)( 9.8 /s ).45 kn w w oon kn w erth (.453 kn) 409 N oon 6 (c) he required upwrd buonc force of the wter equls her weiht of erth: w buonc werth.45 kn 46 It is the er 075 nd spce trvel is coon. A phsics professor brins his fvorite techin deonstrtion with hi to the oon. he pprtus consists of ver sooth horizontl (frictionless) tble nd n object to slide on it. On Erth when the professor ttches sprin (sprin constnt 50 N/) to the object nd pulls horizontll so the sprin stretches.0 c, the object ccelertes t.5 /s. () Drw the free bod dir of the object nd use it nd Newton s lws to deterine its ss. (b) Wht would the block s ccelertion be under identicl conditions on the oon? Picture the Proble he forces ctin on the object re the norl force eerted b the tble, the rvittionl force eerted b the erth, nd the force eerted b the stretched sprin.

28 36 Chpter 4 () he free-bod dir shown to the riht ssues tht the sprin hs been stretched to the riht. Hence the force tht the sprin eerts on the object is to the left. Note tht the + direction hs been chosen to be in the se direction s the force eerted b the sprin. Appl obtin: to the object to he force eerted b the sprin on the object is iven b: s n s s () s kδ where Δ is the ount b which the sprin hs been stretched or copressed nd k is the force constnt. Substitutin for 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 the Erth nd the force eerted b the sprin is the se, its ccelertion on the oon would be the se s on the Erth. ree-bod Dirs: Sttic Equilibriu 47 A trffic liht (ss 35.0 k) is supported b two wires s in iure () Drw the liht s free bod dir nd use it to nswer the followin question qulittivel: Is the tension in wire reter thn or less thn the tension in wire? (b) Prove our nswer b pplin Newton s lws nd solvin for the two tensions. Picture the Proble Becuse the trffic liht is not ccelertin, the net force ctin on it ust be zero; i.e.,

29 Newton s Lws 37 Construct free-bod dir showin the forces ctin on the support point: Appl to the support point: Solve for in ters of : cos30 cos60 0 cos cos60 hus is reter thn. 48 A lp with ss of 4.6 k is hnin fro wires s shown in iure he rin hs neliible ss. he tension in the verticl wire is () 09 N, (b) 48 N, (c) 570 N, (d) 360 N, (e) 730 N. Picture the Proble ro the fiure, it is cler tht supports the full weiht of the lp. Drw free-bod dir showin the forces ctin on the lp nd ppl 0 r. Appl 0 to the lp to obtin: Solve for nd substitute for to obtin: evlute : 0 ( 4.6 k)( 9.8/s ) 48N nd (b) is correct. 49 [SSM] In iure 4-38, k block is suspended t the idpoint of.5--lon strin. he ends of the strin re ttched to the ceilin t points seprted b.00. () Wht nle does the strin ke with the ceilin? (b) Wht is the tension in the strin? (c) he k block is reoved nd two 0.50-k blocks re ttched to the strin such tht the lenths of the three strin seents re equl (iure 4-38b). Wht is the tension in ech seent of the strin?

30 38 Chpter 4 Picture the Proble he free-bod dirs for prts (), (b), nd (c) re shown below. In both cses, the block is in equilibriu under the influence of the forces nd we cn use Newton s nd lw of otion nd eoetr nd trionoetr to obtin reltionships between θ nd the tensions. () nd (b) (c) () Referrin to the free-bod dir for prt (), use trionoetr to deterine θ : (b) Notin tht, ppl to the k block nd solve for the tension : θ cos sinθ 0 becuse 0 nd sinθ 37 evlute : ( k)( 9.8/s ) 4.N sin36.9 (c) he lenth of ech seent is: ind the distnce d: d Epress θ in ters of d nd solve for its vlue: Appl to the 0.50-k block: θ cos cos 0.47 d sinθ 0 3 sinθ

31 evlute 3 : 3 ( 0.50 k)( 9.8/s ) 3.4 N sin45.57 Newton s Lws N Appl to the 0.50-k block nd solve for the tension : evlute : 3 cosθ 0 since 0. nd 3 cosθ ( N) cos N B setr: N 50 A bll weihin 00 N is shown suspended fro sste of cords (iure 4-39). Wht re the tensions in the horizontl nd nled cords? Picture the Proble he suspended bod is in equilibriu under the influence of the forces hor, 45, nd. ht is, hor Drw the free-bod dir of the forces ctin on the knot just bove the 00-N bod. Choose coordinte sste with the positive direction to the riht nd the positive direction upwrd. Appl the conditions for trnsltionl equilibriu to deterine the tension in the horizontl cord. r hor 45 r 45 Appl to the knot: sin Solvin for 45 ields: 45 () sin 45 Appl to the knot: cos hor Solve for hor to obtin: cos 45 hor 45 Substitutin for 45 ields: hor cos 45 sin N Substitutin for in eqution () ields: 00 N sin N

32 30 Chpter 4 5 [SSM] A 0-k object on frictionless tble is subjected to two horizontl forces, nd, with nitudes 0 N nd 30 N, s shown in iure ind the third force 3 tht ust be pplied so tht the object is in sttic equilibriu. Picture the Proble he ccelertion of n object is directl proportionl to the net force ctin on it. Choose coordinte sste in which the positive direction is the se s tht of nd the positive direction is to the riht. Add the two forces to deterine the net force nd then use Newton s nd lw to find the ccelertion of the object. If 3 brins the sste into equilibriu, it ust be true tht Epress 3 in ters of nd : 3 () Epress nd in unit vector nottion: nd iˆ (0 N) {( 30 N)sin 30 } iˆ + {(30 N)cos30 } ˆj ( 5 N)ˆ i (6 N) ˆj + Substitute for 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 or the sstes to be in equilibriu in iure 4-4, iure 4-4b, nd iure 4-4c find the unknown tensions nd sses. Picture the Proble he free-bod dirs for the sstes in equilibriu re shown below. Appl the conditions for trnsltionl equilibriu to find the unknown tensions. () (b) (c)

33 Newton s Lws 3 () Appl 0 nd 0 to the knot bove the suspended ss to obtin: cos60 30 N 0 () nd sin 60 0 () Solvin eqution () for ields: 30 N cos60 60 N Solvin eqution () for ields: sin 60 ( 60 N) 5.96 N 5 N sin 60 Becuse is the weiht of the object whose ss is : evlute : 5.96 N 9.8 /s 5.3 k (b) Appl 0 nd 0 to the knot bove the suspended ss to obtin: nd ( 80 N) cos60 sin 60 0 ( N) sin 60 cos Solvin the first of these equtions for ields: Solvin the second of these equtions for ields: ( 80 N) 46 N 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: nd cos cos60 sin 60 + sin

34 3 Chpter 4 Solvin the first of these equtions for ields: Solvin the second of these equtions for ields: evlute nd 3 : sin 60 sin N ( 6.0 k)( 9.8 /s ) sin N Becuse : ( 6.0 k)( 9.8 /s ) 60 N 58.9 N Becuse the effect of the pulle is to chne the direction cts, is the weiht of the object whose ss is : evlute : N 9.8 /s 3.5 k 53 Your cr is stuck in ud hole. You re lone, but ou hve lon, stron rope. Hvin studied phsics, ou tie the rope tutl to telephone pole nd pull on it sidews, s shown in iure 4-4. () ind the force eerted b the rope on the cr when the nleθ is 3.00º nd ou re pullin with force of 400 N but the cr does not ove. (b) How stron ust the rope be if it tkes force of 600 N to ove the cr whenθ is 4.00º? Picture the Proble Construct the free-bod dir for tht point in the rope t which ou eert the force nd choose the coordinte sste shown in the free-bod dir. We cn ppl Newton s nd lw to the rope to relte the tension to. () Notin tht nd tht the cr s ccelertion is zero, ppl to the cr: r r θ sinθ 0 θ sinθ

35 Newton s Lws 33 evlute : 400 N sin kn (b) Proceed s in Prt () to obtin: 600 N sin kn 54 Blloon rches re often seen t festivls or celebrtions; the re de b ttchin heliu-filled blloons to rope tht is fied to the round t ech end. he lift fro the blloons rises the structure into the rch shpe. iure 4-43 shows the eoetr of such structure: N blloons re ttched t equll spced intervls lon ssless rope of lenth L, which is ttched to two supports t its ends. Ech blloon provides lift force. he horizontl nd verticl coordintes of the point on the rope where the ith blloon is ttched re 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 first blloon, nd seent N is the seent between the lst blloon nd the other point of ttchent). () iure 4-43b shows free-bod dir for the ith blloon. ro this dir, show tht the horizontl coponent of the force i (cll it H ) is the se for ll the strin seents. (b) B considerin the verticl coponent of the forces, use Newton s lws to derive the followin reltionship 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) ro the dir nd the two epressions bove, show tht i i L L tn θ i (N i)/ H nd tht i cosθ N + j, i sin θ N + j. j 0 j 0 Picture the Proble In Prt () we cn ppl Newton s nd lw to obtin the iven epression for. In (b) we cn use setr ruent to find n epression for tn θ 0. In (c) we cn use our results obtined in () nd (b) to epress i nd i. () Appl 0 to the blloon: Solvin for ields: + i sinθi i sinθi 0 i sinθi sinθ i i

36 34 Chpter 4 (b) B setr, ech support ust blnce hlf of the force ctin on the entire rch. herefore, the verticl coponent of the force on the support ust be N/. he horizontl coponent of the tension ust be H. Epress tnθ 0 in ters of N/ nd H : B setr, θ N+ θ 0. herefore, becuse the tnent function is odd: tnθ 0 tnθ N H N tn 0 θ N+ H N H (c) Usin H i cosθ i i cosθ i, divide both sides of the result in () b H nd siplif to obtin: H i i tnθ sinθ cosθ i i i tnθ i sinθi cosθ i i i Usin this result, epress tn θ : tnθ tnθ0 H Substitute for tn θ 0 fro (): N tnθ H H ( N ) H Generlize this result to obtin: tnθ i ( N i) H Epress the lenth of rope between L two blloons: between blloons N + Epress the horizontl coordinte of the point on the rope where the ith blloon is ttched, i, in ters of i nd the lenth of rope between two blloons: i L i + cosθ N + i Su over ll the coordintes to obtin: L i N + θ i cos j j 0 Proceed siilrl for the verticl i coordintes to obtin: L i sinθ j N + j 0

37 Newton s Lws () Consider nuericl solution to Proble 54. Write spredsheet pror to ke rph of the shpe of blloon rch. Use the followin preters: N 0 blloons ech providin lift force.0 N nd ech ttched to rope lenth L, with horizontl coponent of tension H 0 N. How fr prt re the two points of ttchent? How hih is the rch t its hihest point? (b) Note tht we hven t specified the spcin between the supports it is deterined b the other preters. Vr H while keepin the other preters the se until ou crete n rch tht hs spcin of 8.0 between the supports. Wht is H then? As ou increse H, the rch should et fltter nd ore spred out. Does our spredsheet odel show this? Picture the Proble () A spredsheet pror is shown below. he foruls used to clculte the quntities in the coluns re s follows: Cell Content/orul Alebric or C9 ($B$-*B9)/(*$B$4) ( N i) D9 SIN(AAN(C9)) sin( tn θ i ) E9 COS(AAN(C9)) cos( tn θ ) G $B$/($B$+)*E9 L i + cosθi N + G0 G9+$B$/($B$+)*D9 L i + cosθi N + A B C D E G L N 0 3 N 4 H 3.7 N i tnθ i sinθ i cosθ i i i H i