Chapter 5 Additional Applications of Newton s Laws

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1 Chapte 5 Additioal Applicatio of Newto Law Coceptual Poble [SSM] Vaiou object lie o the bed of a tuc that i oi alo a taiht hoizotal oad. If the tuc aduall peed up, what foce act o the object to caue the to peed up too? Eplai wh oe of the object iht ta tatioa o the floo while othe iht lip bacwad o the floo. Deteie the Cocept Static ad ietic fictioal foce ae epoible fo the acceleatio. If the coefficiet of tatic fictio betwee the tuc bed ad the object i ufficietl lae, the the object will ot lip o the tuc bed. The lae the acceleatio of the tuc, the lae the coefficiet of tatic fictio that i eeded to peet lippi. Bloc ade of the ae ateial but diffei i ize lie o the bed of a tuc that i oi alo a taiht hoizotal oad. All of the bloc will lide if the tuc acceleatio i ufficietl eat. How doe the iiu acceleatio at which a all bloc lip copae with the iiu acceleatio at which a uch heaie bloc lip? Deteie the Cocept The foce acti o a object ae the oal foce eeted b the floo of the tuc, the aitatioal foce eeted b Eath, ad the fictio foce; alo eeted b the floo of the tuc. Of thee foce, the ol oe that act i the diectio of the acceleatio (choe to be to the iht) i the tatic fictio foce. Appl Newto ecod law to the object to deteie how the citical acceleatio deped o it weiht. f Tai the + diectio to be to the iht, appl Σ a to the object: f μ μ a a μ Becaue a i idepedet of ad, the citical acceleatio ae the ae. 3 A bloc of a et o a plae that i iclied at a ale θ with the hoizotal. It follow that the coefficiet of tatic fictio betwee the bloc ad plae i (a) μ, (b) μ ta θ, (c) μ ta θ, (d) μ ta θ. 387

2 388 Chapte 5 Deteie the Cocept The foce acti o the bloc ae the oal foce eeted b the iclie, the weiht of eeted b Eath, ad the the bloc tatic fictio foce f alo eeted b the iclie. We ca ue the defiitio of μ ad the coditio fo equilibiu to deteie the elatiohip betwee μ ad θ. θ f Appl Appl a to the bloc: f iθ 0 o, becaue, f iθ 0 () a i the diectio: coθ 0 () Diide equatio () b equatio () to obtai: taθ f Subtitute fo f ( μ ) ad iplif to obtai: ta μ θ μ ad (d) i coect. 4 A bloc of a i at et o a plae that i iclied at a ale of 30º with the hoizotal, a how i iue Which of the followi tateet about the aitude of the tatic fictioal foce f i eceail tue? (a) f >. (b) f > co 30º. (c) f co 30º. (d) f i 30. (e) Noe of thee tateet i tue. Deteie the Cocept The bloc i i equilibiu ude the ifluece of,, ad f ; that i, + + f 0 We ca appl Newto ecod law i the diectio to deteie the elatiohip betwee f ad. f 30

3 Additioal Applicatio of Newto Law 389 Appl 0 to the bloc to obtai: f iθ 0 f iθ ad (d) i coect. 5 O a ic wite da, the coefficiet of fictio betwee the tie of a ca ad a oadwa i educed to oe-quate of it alue o a d da. A a eult, the aiu peed a d at which the ca ca afel eotiate a cue of adiu R i educed. The ew alue fo thi peed i (a) a d, (b) 0.7 a d, (c) 0.50 a d, (d) 0.5 a d, (e) educed b a uow aout depedi o the ca a. Pictue the Poble The foce acti o the ca a it oud a cue of adiu R at aiu peed ae how o the fee-bod diaa to the iht. The cetipetal foce i the tatic fictio foce eeted b the oadwa o the tie. We ca appl Newto ecod law to the ca to deie a epeio fo it aiu peed ad the copae the peed ude the two fictio coditio decibed. Appl a to the ca: o the equatio we hae: Epe f,a i te of i the equatio ad ole fo a to obtai: f, a R ad 0 μr a a f, a () Whe μ μ' : ' ' a μ R () Diidi equatio () b equatio () ield: ' a a μ' R μ R μ' μ Sole fo ' a to obtai: μ' ' a a μ

4 390 Chapte 5 Ealuate ' a fo ' 4 μ μ : ' a 4a 0.5a 50% ad (c) i coect. a 6 If it i tated popel o the fictiole iide uface of a coe (iue 5-57), a bloc i capable of aitaii uifo cicula otio. Daw the fee-bod diaa of the bloc ad idetif cleal which foce (o foce o foce copoet) i epoible fo the cetipetal acceleatio of the bloc. Deteie the Cocept The foce acti o the bloc ae the oal foce eeted b the uface of the coe ad the aitatioal foce eeted b Eath. The hoizotal copoet of i epoible fo the cetipetal foce o the bloc. 7 Hee i a iteeti epeiet that ou ca pefo at hoe: tae a woode bloc ad et it o the floo o oe othe flat uface. Attach a ubbe bad to the bloc ad pull etl ad teadil o the ubbe bad i the hoizotal diectio. Keep ou had oi at cotat peed. At oe poit, the bloc will tat oi, but it will ot oe oothl. Itead, it will tat oi, top aai, tat oi aai, top aai, ad o o. Eplai wh the bloc oe thi wa. (The tat-top otio i oetie called tic-lip otio.) Deteie the Cocept A the pi i eteded, the foce eeted b the pi o the bloc iceae. Oce that foce eceed the aiu alue of the foce of tatic fictio, the bloc will lip. A it doe, it will hote the leth of the pi, deceai the foce that the pi eet. The foce of ietic fictio the low the bloc to a top, which tat the ccle oe aai. 8 Viewed fo a ietial efeece fae, a object i ee to be oi i a cicle. Which, if a, of the followi tateet ut be tue. (a) A ozeo et foce act o the object. (b) The object caot hae a adiall outwad foce acti o it. (c) At leat oe of the foce acti o the object ut poit diectl towad the cete of the cicle. (a) Tue. The elocit of a object oi i a cicle i cotiuall chai idepedetl of whethe the object peed i chai. The chae i the elocit ecto ad the acceleatio ecto ad the et foce acti o the object all poit towad the cete of cicle. Thi cete-poiti foce i called a cetipetal foce.

5 Additioal Applicatio of Newto Law 39 (b) ale. The ol coditio that ut be atified i ode that the object oe alo a cicula path i that the et foce acti o it be adiall iwad. (c) ale. The ol coditio that ut be atified i ode that the object oe alo a cicula path i that the et foce acti o it be adiall iwad. 9 A paticle i taeli i a etical cicle at cotat peed. Oe ca coclude that the aitude of it i(ae) cotat. (a) elocit, (b) acceleatio, (c) et foce, (d) appaet weiht. Deteie the Cocept A paticle taeli i a etical cicle epeiece a dowwad aitatioal foce plu a additioal foce that cotai it to oe alo a cicula path. Becaue the peed of the paticle i cotat, the aitude of it elocit i cotat. Becaue the aitude of it elocit i cotat, it acceleatio ut be cotat. Becaue the aitude of it acceleatio i cotat, the aitude of the et foce acti o it ut be cotat. Theefoe, (a), (b), ad (c) ae coect. 0 You place a lihtweiht piece of io o a table ad hold a all itche aet aboe the io at a ditace of.00 c. You fid that the aet caot lift the io, ee thouh thee i obioul a foce betwee the io ad the aet. Net, aai holdi the aet.00 c aboe the io, ou dop the fo a leth, eleai the fo et iultaeoul. A the fall, the aet ad the piece of io ba ito each othe befoe hitti the floo. (a) Daw fee-bod diaa illutati all of the foce o the aet ad the io fo each deotatio. (b) Eplai wh the aet ad io oe cloe toethe while the ae falli, ee thouh the aet caot lift the piece of io whe it i itti o the table. Deteie the Cocept We ca aalze thee deotatio b dawi foce diaa fo each ituatio. I both diaa, h deote had, deote aitatioal, deote aetic, ad deote oal. (a) Deotatio : Deotatio : (b) Becaue the aet doe t lift the io i the fit deotatio, the foce eeted o the io ut be le tha it (the io ) weiht. Thi i till tue whe the two ae falli, but the otio of the io i ot etaied b the table, ad the

6 39 Chapte 5 otio of the aet i ot etaied b the had. Looi at the ecod diaa, the et foce pulli the aet dow i eate tha it weiht, ipli that it acceleatio i eate tha. The oppoite i tue fo the io: the aetic foce act upwad, lowi it dow, o it acceleatio will be le tha. Becaue of thi, the aet will catch up to the io piece a the fall. [SSM] The followi quetio i a ecellet baitwite ieted b Boi Kou. Two idetical bloc ae attached b a ale ti ui oe a pulle a how i iue The ope iitiall u oe the pulle at the ope idpoit, ad the uface that bloc et o i fictiole. Bloc ad ae iitiall at et whe bloc i eleaed with the ti taut ad hoizotal. Will bloc hit the pulle befoe o afte bloc hit the wall? (Aue that the iitial ditace fo bloc to the pulle i the ae a the iitial ditace fo bloc to the wall.) Thee i a e iple olutio. Pictue the Poble The followi fee-bod diaa how the foce acti o the two object oe tie afte bloc i dopped. Note that the teio i the ti i the ae o both ide of the pulle. The ol foce pulli bloc to the left i the hoizotal copoet of the teio. Becaue thi foce i alle tha the T ad bloc ad hae the ae a, the acceleatio of bloc to the iht will alwa be eate tha the acceleatio of bloc to the left., T T Becaue the iitial ditace fo bloc to the pulle i the ae a the iitial ditace of bloc to the wall, bloc will hit the pulle befoe bloc hit the wall. I cla, ot pofeo do the followi epeiet while dicui the coditio ude which ai da ca be elected while aalzi fee-fall. it, a flat piece of pape ad a all lead weiht ae dopped et to each othe, ad cleal the pape acceleatio i le tha that of the lead weiht. The, the pape i cupled ito a all wad ad the epeiet epeated. Oe the ditace of a ete o two, it i clea the acceleatio of the pape i ow e cloe to that of the lead weiht. To ou dia, the pofeo call o ou to eplai wh the pape acceleatio chaed o daaticall. Repeat ou eplaatio hee!

7 Additioal Applicatio of Newto Law 393 Deteie the Cocept Ai da deped o the fotal aea peeted. Reduci it b cupli the pape ae the foce of ai da a lot le o that ait i the ot ipotat foce. The pape will thu acceleate at appoiatel (util peed ae hih eouh fo da foce to coe bac ito pla i pite of the educed aea). 3 [SSM] Ji decide to attept to et a ecod fo teial peed i dii. Ui the owlede he ha aied fo a phic coue, he ae the followi pla. He will be dopped fo a hih a altitude a poible (equippi hielf with oe), o a wa da ad o ito a ife poitio i which hi bod i poited eticall dow ad hi had ae poited ahead. He will outfit hielf with a pecial lee helet ad ouded potectie clothi. Eplai how each of thee facto help Ji attai the ecod. Deteie the Cocept Ai da i popotioal to the deit of ai ad to the co-ectioal aea of the object. O a wa da the ai i le dee. The ai i alo le dee at hih altitude. Poiti hi had eult i le aea bei peeted to ai da foce ad, hece, educe the. Rouded ad lee clothi ha the ae effect a poiti hi had. 4 You ae itti i the paee eat i a ca dii aoud a cicula, hoizotal, flat acetac at a hih peed. A ou it thee, ou feel a foce puhi ou towad the outide of the tac. What i the tue diectio of the foce acti o ou, ad whee doe it coe fo? (Aue that ou do ot lide aco the eat.) Eplai the eatio of a outwad foce o ou i te of the Newtoia pepectie. Deteie the Cocept I ou fae of efeece (the acceleati efeece fae of the ca), the diectio of the foce ut poit towad the cete of the cicula path alo which ou ae taeli; that i, i the diectio of the cetipetal foce that eep ou oi i a cicle. The fictio betwee ou ad the eat ou ae itti o upplie thi foce. The eao ou ee to be "puhed" to the outide of the cue i that ou bod ietia "wat", i accodace with Newto fit law (the law of ietia), to eep it oi i a taiht lie that i, taet to the cue. 5 [SSM] The a of the oo i ol about % of that of Eath. Theefoe, the foce that eep the oo i it obit aoud Eath (a) i uch alle tha the aitatioal foce eeted o the oo b Eath, (b) i uch eate tha the aitatioal foce eeted o the oo b Eath, (c) i the aitatioal foce eeted o the oo b Eath, (d) caot be aweed et, becaue we hae ot et tudied Newto law of ait. Deteie the Cocept The cetipetal foce that eep the oo i it obit aoud Eath i poided b the aitatioal foce Eath eet o the oo. A decibed b Newto 3 d law, thi foce i equal i aitude to the foce the

8 394 Chapte 5 oo eet o Eath. (c) i coect. 6 A bloc i lidi o a fictiole uface alo a loop-the-loop, a i iue The bloc i oi fat eouh o that it ee loe cotact with the tac. Match the poit alo the tac to the appopiate fee-bod diaa i the fiue. Deteie the Cocept The ol foce acti o the bloc ae it weiht ad the foce the uface eet o it. Becaue the loop-the-loop uface i fictiole, the foce it eet o the bloc ut be pepedicula to it uface. At poit A the weiht i dowwad ad the oal foce i to the iht. The oal foce i the cetipetal foce. ee-bod diaa 3 atche thee foce. At poit B the weiht i dowwad, the oal foce i upwad, ad the oal foce i eate tha the weiht o that thei diffeece i the cetipetal foce. ee-bod diaa 4 atche thee foce. At poit C the weiht i dowwad ad the oal foce i to the left. The oal foce i the cetipetal foce. ee-bod diaa 5 atche thee foce. At poit D both the weiht ad the oal foce ae dowwad. Thei u i the cetipetal foce. ee-bod diaa atche thee foce. 7 [SSM] (a) A oc ad a feathe held at the ae heiht aboe the oud ae iultaeoul dopped. Dui the fit few illiecod followi eleae, the da foce o the oc i alle tha the da foce o the feathe, but late o dui the fall the oppoite i tue. Eplai. (b) I liht of thi eult, eplai how the oc acceleatio ca be o obioul lae tha that of the feathe. Hit: Daw a fee-bod diaa of each object. Deteie the Cocept The da foce

9 Additioal Applicatio of Newto Law 395 acti o the object i ie b d CAρ, whee A i the pojected uface aea, i the object peed, ρ i the deit of ai, ad C i a dieiole coefficiet. We ll aue that, oe the heiht of the fall, the deit of ai ρ i cotat. The fee-bod diaa fo a feathe ad a oc eeal illiecod ito thei fall ae how to the iht. The foce acti o both object ae the dowwad aitatioal foce ad a upwad da foce. d, feathe, feathe d, oc, oc (a) The da foce i popotioal to the aea peeted ad oe powe of the peed. The da foce o the feathe i lae becaue the feathe peet a lae aea tha doe the oc. A the oc ai peed, the da foce o it iceae. The da foce o the oc eetuall eceed the da foce o the feathe becaue the da foce o the feathe caot eceed the aitatioal foce o the feathe. A hot tie afte bei eleaed the feathe eache teial peed. Dui the et of it fall the da foce o it i equal to the aitatioal foce acti o it. Howee, the aitatioal foce o the oc i uch eate tha the da foce o the feathe, o the oc cotiue to ai peed lo afte the feathe eache teial peed. A the oc cotiue to ai peed, the da foce o it cotiue to iceae. A a eult, the da foce o the oc eetuall eceed the da foce o the feathe. (b) Teial peed i uch hihe fo the oc tha fo the feathe. The acceleatio of the oc will eai hih util it peed appoache it teial peed. 8 Two puc of ae ad ae li o a fictiole table ad ae coected b a ale pi of foce cotat. A hoizotal foce diected awa fo i the eeted o. What i the aitude of the eulti acceleatio of the cete of a of the two-puc te? (a) /. (b) /( + ). (c) ( + )/( + ), whee i the aout the pi i tetched. (d) ( + ) /. Deteie the Cocept The acceleatio of the cete of a of a te of paticle i decibed b Ma, whee M i the total a of the te. et, et i i,et c

10 396 Chapte 5 Epe the acceleatio of the et,et ac cete of a of the two puc: M + becaue the pi foce i a iteal foce. (b) i coect. 9 The two puc i Poble 8 lie ucoected o a fictiole table. A hoizotal foce diected awa fo i the eeted o. How doe the aitude of the eulti acceleatio of the cete of a of the two-puc te copae to the aitude of the acceleatio of? Eplai ou eaoi. Deteie the Cocept The aitude of the acceleatio of the puc whoe a i i elated to the et foce acti o it thouh Newto ecod law. Becaue the puc ae o loe coected, the aitude of the acceleatio of the cete of a i: a CM, dicoected o Poble 8: a CM, coected + Becaue + > the aitude of the acceleatio of i eate. 0 If ol eteal foce ca caue the cete of a of a te of paticle to acceleate, how ca a ca o leel oud ee acceleate? We oall thi of the ca eie a uppli the foce eeded to acceleate the ca, but i thi tue? Whee doe the eteal foce that acceleate the ca coe fo? Deteie the Cocept Thee i ol oe foce that ca caue the ca to oe fowad the fictio of the oad! The ca eie caue the tie to otate, but if the oad wee fictiole (a i cloel appoiated b ic coditio) the wheel would ipl pi without the ca oi awhee. Becaue of fictio, the ca tie puhe bacwad aait the oad fo Newto thid law, the fictioal foce acti o the tie ut the puh it fowad. Thi a ee odd, a we ted to thi of fictio a bei a etadi foce ol, but it i tue. Whe we puh o the bae pedal to low dow a ca, a bae pad i peed aait the oto o that the fictio of the pad low the wheel otatio. Howee, the fictio of the pad aait the oto caot be the foce that low the ca dow, becaue it i a iteal foce (both the oto ad the wheel ae pat of the ca, o a foce betwee the ae puel iteal to the te). What i

11 Additioal Applicatio of Newto Law 397 the eteal foce that low dow the ca? Gie a detailed eplaatio of how thi foce opeate. Deteie the Cocept The fictio of the tie aait the oad caue the ca to low dow. Thi i athe ubtle, a the tie i i cotact with the oud without lippi at all tie, ad o a ou puh o the bae hade, the foce of tatic fictio of the oad aait the tie ut iceae. Gie a eaple of each of the followi. (a) A thee-dieioal object that ha o atte at it cete of a. (b) A olid object whoe cete of a i outide of it. (c) A olid phee whoe cete of a doe ot lie at it eoetical cete. (d) A lo woode tic whoe cete of a doe ot lie at it iddle. (a) A olid pheical hell, o dout, o tie. (b) A olid heipheical hell. (c) A phee with oe ide a diffeet deit tha the othe, o a deit aiatio that i t adiall etic. (d) A tic with a o-uifo ad o-etic deit aiatio. A baeball bat i a ood eaple of uch a tic. 3 [SSM] Whe ou ae tadi upiht, ou cete of a i located withi the olue of ou bod. Howee, a ou bed oe (a to pic up a pacae), it locatio chae. Appoiatel whee i it whe ou ae bet oe at iht ale ad what chae i ou bod caued the cete of a locatio to chae? Eplai. Deteie the Cocept Relatie to the oud, ou cete of a oe dowwad. Thi i becaue oe of ou a (hip) oed bacwad, oe of ou a (ou head ad houlde) oed fowad, ad the top half of ou bod oed dowwad. 4 Eal o thei thee-da (oe-wa) tip to the oo, the Apollo tea (late 960 to eal 970) would eploiel epaate the lua hip fo the thidtae boote (that poided the fial boot ) while till fail cloe to Eath. Dui the eploio, how did the elocit of each of the two piece of the te chae? How did the elocit of the cete of a of the te chae? Deteie the Cocept The pacecaft peed iceaed towad the oo. The peed of the thid-tae boote deceaed, but the boote cotiued to oe awa fo Eath ad towad the oo. Riht afte the eploio, the cete of

12 398 Chapte 5 a elocit wa the ae a befoe. Afte a while, howee, the bacwad pull of ait of Eath will caue it to deceae becaue the peed of both the lua hip ad the boote deceae. 5 You thow a booea ad fo a while it flie hoizotall i a taiht lie at a cotat peed, while pii apidl. Daw a eie of pictue, a iewed eticall dow fo oehead, of the booea i diffeet otatioal poitio a it oe paallel to the uface of Eath. O each pictue, idicate the locatio of the booea cete of a ad coect the dot to tace the tajecto of it cete of a. What i the cete of a acceleatio dui thi pat of the fliht? Deteie the Cocept The diaa how a pii booea with it cete of a at the locatio of the cicle. A iewed fo aboe, the cete of a oe i a taiht lie a the booea pi about it. The acceleatio of the cete of a i zeo Etiatio ad Appoiatio 6 To deteie the aeodaic da o a ca, autootie eiee ofte ue the coat-dow ethod. The ca i die o a lo, flat oad at oe coeiet peed (60 i/h i tpical), hifted ito eutal, ad allowed to coat to a top. The tie that it tae fo the peed to dop b ucceie 5-i/h iteal i eaued ad ued to copute the et foce lowi the ca dow. (a) Oe da, a oup eaued that a Toota Tecel with a a of 00 coated dow fo 60.0 i/h to 55.0 i/h i 3.9. Etiate the aeae et foce lowi the ca dow i thi peed ae. (b) If the coefficiet of olli fictio fo thi ca i ow to be 0.00, what i the foce of olli fictio that i acti to low it dow? Aui that the ol two foce acti o the ca ae olli fictio ad aeodaic da, what i the aeae da foce acti o the ca? (c) The da foce ha the fo Cρ A, whee A i the co-ectioal aea of the ca faci ito the ai, i the ca peed, ρ i the deit of ai, ad C i a dieiole cotat of ode. If the co-ectioal aea of the ca i.9, deteie C fo the data ie. (The deit of ai i. / 3 ; ue 57.5 i/h fo the peed of the ca i thi coputatio.) Pictue the Poble The foce acti o the Tecel a it low fo 60 to 55 i/h ae a olli-fictio foce eeted b the oadwa, a ai-da foce eeted b the ai, the oal foce eeted b the oadwa, ad the aitatioal foce eeted b Eath. The ca i oi i the poitie diectio. We ca ue Newto ecod law to calculate the aeae foce fo the ate at which the

13 Additioal Applicatio of Newto Law 399 ca peed deceae ad the olli foce fo it defiitio. The da foce ca be ifeed fo the aeae- ad olli-fictio foce ad the da coefficiet fo the defii equatio fo the da foce. d f olli (a) Appl a to the ca to elate the aeae foce acti o it to it aeae acceleatio: a aa Δ Δt Subtitute ueical alue ad ealuate a : i h i 3.9 h ( 00) 58N 0.58N a (b) The olli-fictio foce i the poduct of the coefficiet of olli fictio ad the oal foce: Subtitute ueical alue ad ealuate f olli : f f olli olli μ μ olli olli ( 0.00)( 00 )( 9.8/ ) 0.0N Aui that ol two foce ae acti o the ca oppoite to the diectio of it otio ie: + a d f olli d a folli Subtitute ueical alue ad ealuate d : 58N 00 N d 0.38N (c) Ui the defiitio of the da foce ad it calculated alue fo (b) ad the aeae peed of the ca dui thi 5 ph iteal, ole fo C: d ρ C A C d ρ A

14 400 Chapte 5 Subtitute ueical alue ad ealuate C: C 3 (./ )(.9 ) ( 38N) i h i h [SSM] Ui dieioal aali, deteie the uit ad dieio of the cotat b i the etadi foce b if (a) ad (b). (c) Newto howed that the ai eitace of a falli object with a cicula co ectio hould be appoiatel ρπ, whee ρ.0 / 3, the deit of ai. Show that thi i coitet with ou dieioal aali fo Pat (b). (d) id the teial peed fo a die; appoiate hi coectioal aea a a di of adiu The deit of ai ea the uface of Eath i.0 / 3. (e) The deit of the atophee deceae with heiht aboe the uface of Eath; at a heiht of 8.0, the deit i ol 0.54 / 3. What i the teial elocit at thi heiht? Pictue the Poble We ca ue the dieio of foce ad peed to deteie the dieio of the cotat b ad the dieio of ρ,, ad to how that, fo, Newto epeio i coitet dieioall with ou eult fo pat (b). I Pat (d) ad (e), we ca appl Newto ecod law ude teial peed coditio to fid the teial peed of the die ea the uface of Eath ad at a heiht of 8. Aue that 9.8 / eai cotat. (Note: At 8, 9.78 /. Howee, it will ot affect the eult i Pat (e).) (a) Sole the da foce equatio fo b with : b d Subtitute the dieio of d ad ad iplif to obtai: [] b ML T L T M T ad the uit of b ae / (b) Sole the da foce equatio fo b with : b d Subtitute the dieio of d ad ML ad iplif to obtai: [] b T L T M L ad the uit of b ae /

15 (c) Epe the dieio of Additioal Applicatio of Newto Law 40 [ ] [ ] ( ) Newto epeio: d ρπ L 3 ML T M L L T o Pat (b) we hae: (d) Letti the dowwad diectio be the + diectio, appl a to the die: M L L T [ d ] [ b ] ML T ρπ t 0 t ρπ Subtitute ueical alue ad ealuate t : π ( 56)( 9.8/ ) 3 (. / )( 0.30) t 57/ (e) Ealuate t at a heiht of 8 : ( 56)( 9.8/ ) t π 3 ( 0.54 / )( 0.30) 87/ 8 Etiate the teial elocit of a aeae ized aidop ad a olfball- ized hailtoe. (Hit: See Poble 6 ad 7.) Pictue the Poble o Newto ecod law, the equatio decibi the otio of falli aidop ad lae hailtoe i d a whee ρπ b i the da foce. Ude teial peed coditio (a 0), d the da foce i equal to the weiht of the falli object. Tae the adiu of a aidop to be 0.50 ad the adiu of a olf-ball ized hailtoe to be.0 c. Epe the elatiohip betwee t ad the weiht of a falli object ude teial peed: b t t () b Ui b πρ, ealuate b : 3 3 b π (. / )( ) / Ealuati b h ield: 3 b π (. / )(.0 0 ) h /

16 40 Chapte 5 Epe the a of a phee i 4π ρv te of it olue ad deit: 3 3 ρ Ui ρ / 3, ealuate : 4 π ( ) (.0 0 / ) Ui ρ h 90 / 3, ealuate h : (.0 0 ) ( 90 / ) h π Subtitute ueical alue i equatio () ad ealuate t, : Subtitute ueical alue i equatio () ad ealuate t,h : t, t,h 7 ( )( 9.8/ ) 7 3.3/ / ( )( 9.8/ ) 4 0/ / 9 Etiate the iiu coefficiet of tatic fictio eeded betwee a ca tie ad the paeet i ode to coplete a left tu at a cit teet iteectio at the poted taiht-ahead peed liit of 5 ph ad o aow ie-cit teet. Coet o the wido of attepti uch a tu at that peed. Pictue the Poble I ode to pefo thi etiate, we eed to deteie a ouh adiu of cuatue fo the ca tu i a oal cit iteectio. Aui the ca oe fo iht-had lae to iht-had lae, ad aui fail oal dieio of 40 feet fo the width of the teet, the cete of the ca path tael alo a cicle of, a, 30 feet i adiu. The et cetipetal foce i poided b the foce of tatic fictio ad the acceleatio of the ca i equal to thi et foce diided b the a of the ca. iall, we ole fo the coefficiet of tatic fictio.

17 Additioal Applicatio of Newto Law 403 A diaa howi the foce acti o the ca a it oud the cue i how to the iht. Appl a to the ca tie: f, a a o, becaue f μ, a f, a μ μ () Appl a to the ca tie: 0 o, becaue, 0 Subtituti fo i equatio () ield: Subtitute ueical alue ad ealuate μ : μ i h h 3600 i μ 30 ft ft.4 ( 9.8 / ) Thi i pobabl ot uch a ood idea. Tie o aphalt o cocete hae a aiu coefficiet of tatic fictio of about. 30 Etiate the widet tace ou ca tae whe tadi o a d, ic uface. That i, how wide ca ou afel place ou feet ad ot lip ito a udeied plit? Let the coefficiet of tatic fictio of ubbe o ice be ouhl 0.5. Pictue the Poble We eed to etiate the foce actie at the place of each foot. Aui a etical tace, with the defii ale bei the ale betwee each le ad the oud,θ, we ca the daw a foce diaa ad appl Newto ecod law to ou foot. The fee-bod diaa how the oal

18 404 Chapte 5 foce, eeted b the ic uface, the aiu tatic fictio foce, alo eeted b the ic uface, ad the foce due to ait eeted o ou foot. A fee-bod diaa howi the foce acti o oe foot i how to the iht. θ f, a Appl a to oe of ou feet: f coθ 0 () ad, a iθ 0 () Subtituti f, a μ i equatio () ie: μ coθ 0 o μ coθ (3) Soli equatio () fo ield: iθ (4) Diide equatio (4) b equatio (3) to obtai: Subtitute the ueical alue of ad ealuate θ: ictio μ ta θ μ θ ta μ θ ta Thi ale coepod to a ale betwee ou le of about 8. 3 [SSM] A bloc of a lide at cotat peed dow a plae iclied at a ale of θ with the hoizotal. It follow that (a) μ i θ, (b) μ ta θ, (c) μ co θ, (d) μ co θ i θ. Pictue the Poble The bloc i i equilibiu ude the ifluece of,, ad f ; that i, + + f 0. We ca appl Newto ecod law to deteie the elatiohip betwee f, θ, ad.

19 Additioal Applicatio of Newto Law 405 A pictoial epeetatio howi the foce acti o the lidi bloc i how to the iht. Ui it defiitio, epe the coefficiet of ietic fictio: f θ f μ () Appl a to the bloc: f iθ a o, becaue a 0, f iθ Appl a to the bloc: coθ a o, becaue a 0, coθ Subtitute fo f ad i equatio () ad iplif to obtai: iθ μ coθ taθ ad (b) i coect. 3 A bloc of wood i pulled at cotat elocit b a hoizotal ti aco a hoizotal uface with a foce of 0 N. The coefficiet of ietic fictio betwee the uface i 0.3. The foce of fictio i (a) ipoible to deteie without owi the a of the bloc, (b) ipoible to deteie without owi the peed of the bloc, (c) 0.30 N, (d) 6.0 N, o (e) 0 N. Pictue the Poble The bloc i i equilibiu ude the ifluece of,, app, ad f ; that i + + app + f 0 We ca appl Newto ecod law to deteie f. f app

20 406 Chapte 5 Appl a to the bloc: app f a o, becaue a 0, f 0 N ad (e) i coect. app 33 [SSM] A bloc weihi 0-N et o a hoizotal uface. The coefficiet of tatic ad ietic fictio betwee the uface ad the bloc ae μ 0.80 ad μ A hoizotal ti i the attached to the bloc ad a cotat teio T i aitaied i the ti. What i the ubequet foce of fictio acti o the bloc if (a) T 5 N o (b) T 0 N? Pictue the Poble Whethe the fictio foce i that due to tatic fictio o ietic fictio deped o whethe the applied teio i eate tha the aiu tatic fictio foce. We ca appl the defiitio of the aiu tatic fictio to decide whethe f,a o T i eate. f T Noti that, calculate the aiu tatic fictio foce: f,a μ μ 6 N ( 0.80)( 0 N) (a) Becaue f,a > T: f f T 5 N (b) Becaue T > f,a : f f μ μ ( 0.60)( 0 N) N 34 A bloc of a i pulled at a cotat elocit aco a hoizotal uface b a ti a how i iue The aitude of the fictioal foce i (a) μ, (b) T co θ, (c) μ (T ), (d) μ T i θ, o (e) μ ( T i θ). Pictue the Poble The bloc i i equilibiu ude the ifluece of the foce T, f,, ad ; that i T + f We ca appl Newto ecod law to deteie the elatiohip betwee T ad f.

21 Additioal Applicatio of Newto Law 407 A fee-bod diaa howi the foce acti o the bloc i how to the iht. T θ f Appl a to the bloc: T coθ + f a Becaue a 0: f T coθ ad (b) i coect. 35 [SSM] A 00- cate et o a thic-pile capet. A wea woe the puhe o the cate with a hoizotal foce of 500 N. The coefficiet of tatic ad ietic fictio betwee the cate ad the capet ae ad 0.400, epectiel. id the ubequet fictioal foce eeted b the capet o the cate. Pictue the Poble Whethe the fictio foce i that due to tatic fictio o ietic fictio deped o whethe the applied teio i eate tha the aiu tatic fictio foce. If it i, the the bo oe ad the fictio foce i the foce of ietic fictio. If it i le, the bo doe ot oe. f app The aiu tatic fictio foce i ie b: f, a μ o, becaue, f, a μ Subtitute ueical alue ad ealuate f,a : f,a ( 0.600)( 00)( 9.8/ ) 589 N Becaue ot oe ad : f >, the bo doe, a app app f 500 N 36 A bo weihi 600 N i puhed alo a hoizotal floo at cotat elocit with a foce of 50 N paallel to the floo. What i the coefficiet of ietic fictio betwee the bo ad the floo?

22 408 Chapte 5 Pictue the Poble Becaue the bo i oi with cotat elocit, it acceleatio i zeo ad it i i equilibiu ude the ifluece of app,,, ad f ; that i, app f 0. We ca appl Newto ecod law to deteie the elatiohip betwee f ad. f app The defiitio of μ i: f μ () Appl a to the bo: a o, becaue a 0, N Appl a to the bo: app f a o, becaue a 0, f 50 N app Subtitute ueical alue i equatio () ad ealuate μ : μ 50 N 600 N [SSM] The coefficiet of tatic fictio betwee the tie of a ca ad a hoizotal oad i Nelecti ai eitace ad olli fictio, (a) what i the aitude of the aiu acceleatio of the ca whe it i baed? (b) What i the hotet ditace i which the ca ca top if it i iitiall taeli at 30 /? Pictue the Poble Aue that the ca i taeli to the iht ad let the poitie diectio alo be to the iht. We ca ue Newto ecod law of otio ad the defiitio of μ to deteie the aiu acceleatio of the ca. Oce we ow the ca aiu acceleatio, we ca ue a cotatacceleatio equatio to deteie the leat toppi ditace.

23 Additioal Applicatio of Newto Law 409 (a) A diaa howi the foce acti o the ca i how to the iht. f Appl a to the ca: f, a μ a () Appl a to the ca ad ole fo : a w a 0 o, becaue a 0 ad, () Subtitute fo i equatio () to obtai: f a, a μ Soli fo a, a ield: a a μ, Subtitute ueical alue ad a,a (0.60)(9.8/ ealuate a, a : 5.9 / ) 5.89/ (b) Ui a cotat-acceleatio equatio, elate the toppi ditace of the ca to it iitial peed ad it acceleatio: Ui a 5.89 / becaue the acceleatio of the ca i to the left, ubtitute ueical alue ad ealuate Δ: 0 + a Δ o, becaue 0, aδ Δ a ( 30/) Δ ( 5.89 / ) The foce that acceleate a ca alo a flat oad i the fictioal foce eeted b the oad o the ca tie. (a) Eplai wh the acceleatio ca be eate whe the wheel do ot lip. (b) If a ca i to acceleate fo 0 to 90 /h i, what i the iiu coefficiet of fictio eeded betwee the oad ad tie? Aue that the die wheel uppot eactl half the weiht of the ca.

24 40 Chapte 5 Pictue the Poble We ca ue the defiitio of acceleatio ad appl Newto ecod law to the hoizotal ad etical copoet of the foce to deteie the iiu coefficiet of fictio betwee the oad ad the tie. (a) The fee-bod diaa how the foce acti o the tie o the die wheel, the tie we e aui uppot half the weiht of the ca. f Becaue μ > μ, f will be eate if the wheel do ot lip. (b) Appl a to the ca: f μ a () Appl a to the ca ad ole fo : Subtituti fo i equatio () ield: Subtitute fo a to obtai: Subtitute ueical alue ad ealuate μ : a o, becaue a 0 ad, μ a Δ μ Δt a μ h μ ( ) h A bloc i held at et aait a etical wall b a hoizotal foce of 00 N. (a) What i the fictioal foce eeted b the wall o the bloc? (b) What i the iiu hoizotal foce eeded to peet the bloc fo falli if the tatic coefficiet of fictio betwee the wall ad the bloc i 0.400?

25 Additioal Applicatio of Newto Law 4 Pictue the Poble The bloc i i equilibiu ude the ifluece of the foce how o the foce diaa. We ca ue Newto ecod law ad the defiitio of μ to ole fo f ad. f 00 N (a) Appl a to the bloc: Subtitute ueical alue ad ealuate f : f a o, becaue a 0, f 0 f f ( 5.00 )( 9.8/ ) 49.N (b) Ue the defiitio of μ to epe : f,a μ Subtitute ueical alue ad ealuate : 49.N N 40 A tied ad oeloaded tudet i attepti to hold a lae phic tetboo weded ude hi a, a how i iue 5-6. The tetboo ha a a of 3., while the coefficiet of tatic fictio of the tetboo aait the tudet udea i 0.30 ad the coefficiet of tatic fictio of the boo aait the tudet hit i (a) What i the iiu hoizotal foce that the tudet ut appl to the tetboo to peet it fo falli? (b) If the tudet ca ol eet a foce of 6 N, what i the acceleatio of the tetboo a it lide fo ude hi a? The coefficiet of ietic fictio of a aait tetboo i 0.00, while that of hit aait tetboo i Pictue the Poble We ca appl Newto ecod law to elate the iiu foce equied to hold the boo i place to it a ad to the coefficiet of tatic fictio. I Pat (b), we ca poceed iilal to elate the acceleatio of the boo to the coefficiet of ietic fictio.

26 4 Chapte 5 (a) The foce diaa how the foce acti o the boo. The oal foce i the et foce the tudet eet i queezi the boo. Let the hoizotal diectio be the + diectio ad upwad the + diectio. Note that the oal foce i the ae o eithe ide of the boo becaue it i ot acceleati i the hoizotal diectio. The boo could be acceleati dowwad. Appl a to the boo: ad,i μ, μ,i μ,i,i,i Tiple/Moca,i,i 0 + μ,,i 0 Noti that,i,i, ole the i equatio fo i : μ, + μ, Subtitute ueical alue ad ealuate i : (b) Appl a with the boo acceleati dowwad, to obtai: ( 3.)( 9.8/ ) 65N i , μ + μ a, Soli fo a ield: a μ, + μ, Subtitute ueical alue ad ealuate a : ( 6N) 4.3/, dowwad 9.8/ 4 You ae aci i a all o a ow da whe the tepeatue i ea the feezi poit. The coefficiet of tatic fictio betwee a ca tie ad a ic oad i You cew bo i coceed about oe of the hill o the coue ad wat ou to thi about witchi to tudded tie. To adde the iue, he wat to copae the actual hill ale o the coue to ee which of the ou ca ca eotiate. (a) What i the ale of the teepet iclie that a ehicle with fou-wheel die ca clib at cotat peed? (b) Gie that the hill ae ic, what i the teepet poible hill ale fo the ae fou-wheel die ca to deced at cotat peed? Pictue the Poble We ca ue the defiitio of the coefficiet of tatic fictio a

27 Additioal Applicatio of Newto Law 43 ad Newto ecod law to elate the ale of the iclie to the foce acti o the ca. (a) The fee-bod diaa how the foce acti o the ca whe it i eithe oi up the hill o dow the hill without acceleatio. The fictio foce that the oud eet o the tie i the foce f how acti up the iclie. θ f Appl a to the ca: f iθ 0 () ad coθ 0 () Becaue, equatio () ad () becoe: Soli equatio (3) fo f ad equatio (4) fo ield: Ue the defiitio of μ to elate f ad : f iθ 0 (3) ad coθ 0 (4) f iθ ad coθ f μ iθ coθ taθ Soli foθ ield: θ ( μ ) ( ) ta ta (b) Poceed eactl a i (a) to obtai: θ ta ( 0.080) A 50- bo that i eti o a leel floo ut be oed. The coefficiet of tatic fictio betwee the bo ad the floo i Oe wa to oe the bo i to puh dow o the bo at a ale θ below the hoizotal. Aothe ethod i to pull up o the bo at a ale θ aboe the hoizotal. (a) Eplai wh oe ethod equie le foce tha the othe. (b) Calculate the iiu foce eeded to oe the bo b each ethod if θ 30º ad copae the awe with the eult whe θ 0. Pictue the Poble The fee-bod

28 44 Chapte 5 diaa fo the two ethod ae how to the iht. Method eult i the bo bei puhed ito the floo, iceai the oal foce ad the tatic fictio foce. Method patiall lift the bo,, educi the oal foce ad the tatic fictio foce. We ca appl Newto ecod law to obtai epeio that elate the aiu tatic fictio foce to the applied foce. Method Method f θ f θ (a) Method i pefeable becaue it educe ad, theefoe, f. (b) Appl a to the bo: coθ f coθ μ 0 Method : Appl a to the bloc ad ole fo : iθ ad + iθ 0 Relate f to : f μ μ ( + iθ ), a (), a Method : Appl a to the foce i the diectio ad ole fo : + iθ ad iθ 0 Relate f to : f μ μ ( iθ ), a (), a Epe the coditio that ut be atified to oe the bo b eithe ethod: Method : Subtitute () i (3) ad ole fo : Method : Subtitute () i (3) ad ole fo : f coθ (3), a μ (4) coθ μ iθ μ (5) coθ + μ iθ

29 Subtitute ueical alue ad ealuate equatio (4) ad (5) with θ 30 : Additioal Applicatio of Newto Law 45 ad ( 30 ) ( 30 ) ( 0.60)( 50 )( 9.8 / ) co N ( 0.60) i 30 ( 0.60)( 50 )( 9.8 / ) co N ( 0.60) i 30 Ealuate equatio (4) ad (5) with θ 0 : ad ( 0 ) ( 0 ) ( 0.60)( 50 )( 9.8 / ) co0 0.9 N ( 0.60) i 0 ( 0.60)( 50 )( 9.8 / ) co N ( 0.60) i 0 43 [SSM] A bloc of a 50 i at et o a plae that ae a ale of θ 30 with the hoizotal. The coefficiet of ietic fictio betwee the bloc ad the plae i The bloc i attached to a ecod bloc of a 00 that ha feel b a ti that pae oe a fictiole, ale pulle (iue 5-6). Whe the ecod bloc ha falle 30.0 c, what will be it peed? Pictue the Poble Chooe a coodiate te i which the + diectio i up the iclie fo the bloc whoe a i ad dowwad fo the bloc whoe a i. We ca fid the peed of the te whe it ha oed a ie ditace b ui a cotat-acceleatio equatio. We ll aue that the ti i ale ad that it doe ot tetch. Ude the ifluece of the foce how i the fee-bod diaa, the bloc will hae a coo acceleatio a. The applicatio of Newto ecod law to each bloc, followed b the eliiatio of the teio T ad the ue of the defiitio of f, will allow u to deteie the acceleatio of the te., T T f 30,, 30

30 46 Chapte 5 Ui a cotat-acceleatio equatio, elate the peed of the te to it acceleatio ad diplaceet: 0 + a Δ ad, becaue 0 0, aδ aδ () Appl a i : a to the bloc whoe T f, i 30 a () ad,, co30 0 (3) Becaue,, equatio () ad (3) ca be witte a: T f i 30 a (4) ad co30 (5), Ui f μ,, ubtitute equatio T μ co30 i30 (5) i equatio (4) to obtai: a (6) Appli a to the bloc whoe a i ie: Add equatio (6) ad (7) to eliiate T ad the ole fo a to obtai:, T a o, becaue,, T (7) a ( μ co30 i ) a + 30 Subtituti fo a i equatio () ad iplifi ield: [ ( μ co30 + i30 )] + Δ

31 Additioal Applicatio of Newto Law 47 Subtitute ueical alue ad ealuate : [ ( 0.50 )( ( 0.00) co30 + i 30 )]( 9.8 / )( ) c/ 44 I iue ad the coefficiet of tatic fictio betwee the bloc ad the iclie i (a) id the ae of poible alue fo fo which the te will be i tatic equilibiu. (b) id the fictioal foce o the 4.0- bloc if.0? Pictue the Poble Chooe a coodiate te i which the + diectio i up the iclie fo the bloc whoe a i ad dowwad fo the bloc whoe a i. We ll aue that the ti i ale ad that it doe ot tetch. Ude the ifluece of the foce how i the fee-bod diaa, the bloc ae i tatic equilibiu. While f ca be eithe up o dow the iclie, the fee-bod diaa how the ituatio i which otio i ipedi up the iclie. The applicatio of Newto ecod law to each bloc, followed b the eliiatio of the teio T ad the ue of the defiitio of f, will allow u to deteie the ae of alue fo. T, f 30, T, (a) Noti that,, appl a to the bloc whoe a i : Ui f, a μ, ubtitute equatio () i equatio () to obtai: Noti that,, appl a to the bloc whoe a i : T ± f i 30 0 (), a ad co30 0 (), T ± μ co30 i 30 0 (3) T 0 (4)

32 48 Chapte 5 Add equatio (3) ad (4) to eliiate T ad ole fo : ( ± co30 + i ) 30 (5) μ Subtitute ueical alue to obtai: ( 4.0) [ ± ( 0.40) co30 + i ] 30 Deoti the alue of with a plu i a,+ ad the alue of with the iu i a,- deteie the ae of alue of fo which the te i i tatic equilibiu: (b) With, the ipedi otio i dow the iclie ad the tatic fictio foce i up the iclie. Appl a to the bloc whoe a i : Appl a to the bloc whoe a i : Add equatio (6) ad (7) ad ole fo f to obtai: Subtitute ueical alue ad ealuate f : 3.4 ad,-, + ad T + f i 30 0 (6) T 0 (7) ( i30 ) f f [( 4.0) i 30.0]( 9.8/ ) 9.8 N 45 I iue 5-6, 4.0, 5.0, ad the coefficiet of ietic fictio betwee the iclied plae ad the 4.0- bloc i μ 0.4. id the aitude of the acceleatio of the ae ad the teio i the cod. Pictue the Poble Chooe a coodiate te i which the + diectio i up the iclie fo the bloc whoe a i ad dowwad fo the bloc whoe a i. We ll aue that the ti i ale ad that it doe ot tetch. Ude the ifluece of the foce how i the fee-bod diaa, the bloc will hae a coo acceleatio a. The applicatio of Newto ecod law to each bloc, followed b the eliiatio of the teio T ad the ue of the defiitio of f, will allow u to deteie the acceleatio of the te. iall, we ca ubtitute fo the teio i eithe of the otio equatio to deteie the acceleatio of the ae.

33 Additioal Applicatio of Newto Law 49 f, 30, T T, Noti that,, appl a to the bloc whoe a i : T f i 30 a () ad co30 0 (), Ui f μ, ubtitute equatio T μ co30 () i equatio () to obtai: i30 a (3) Appl a to the bloc whoe a i : Add equatio (3) ad (4) to eliiate T ad ole fo a to obtai: T a (4) ( μ co30 i ) a + 30 Subtituti ueical alue ad ealuati a ield: a [ 5.0 ( 0.4)( 4.0 ) co30 ( 4.0 ) i 30 ]( 9.8 / ) / Soli equatio (3) fo T ield: [ co 30 + i ] T a + μ 30 Subtitute ueical alue ad ealuate T: T ( 4.0 )(.36 / ) + [( 0.4) co30 + i 30 ]( 4.0 )( 9.8 / ) 37 N 46 A - tutle et o the bed of a zooeepe tuc, which i taeli dow a cout oad at 55 i/h. The zooeepe pot a dee i the oad, ad low to a top i. Aui cotat acceleatio, what i the iiu coefficiet of tatic fictio betwee the tutle ad the tuc bed uface that i eeded to peet the tutle fo lidi?

34 40 Chapte 5 Pictue the Poble We ca deteie the acceleatio ecea fo the tuc ad tutle b coidei the diplaceet of both dui the ie tie iteal. The tatic fictio foce ut poide the ecea acceleatio fo the tutle. The tutle, if it i ot to lip, ut hae thi acceleatio which i poduced b the tatic fictio foce acti o it f The equied coefficiet of tatic fictio i ie b: Letti epeet the a of the tutle, appl a to the tutle: Soli equatio () fo f ield: Becaue a 0 ad, equatio (3) becoe: f μ () f a () ad (3) a f a Subtituti fo f ad i equatio a a μ (4) () ad iplifi ield: The acceleatio of the tuc ad tutle i ie b: a Δ f, i, Δt Δt o, becaue f, 0, i, a Δt Subtitute fo a i equatio (4) to obtai: μ i, Δt Subtitute ueical alue ad ealuate μ : i μ h i ( 9.8 / )( ) h

35 Additioal Applicatio of Newto Law 4 47 [SSM] A 50- bloc i pojected up a ap with a iitial peed of 7.0 /. The coefficiet of ietic fictio betwee the ap ad the bloc i 0.3. (a) If the ap i iclied 5 with the hoizotal, how fa alo the uface of the ap doe the bloc lide befoe coi to a top? (b) The bloc the lide bac dow the ap. What i the coefficiet of tatic fictio betwee the bloc ad the ap if the bloc i ot to lide bac dow the ap? Pictue the Poble The foce diaa how the foce acti o the bloc a it lide up the ap. Note that the bloc i acceleated b f ad the copoet of. We ca ue a cotat-acceleatio equatio to epe the diplaceet of the bloc up the ap a a fuctio of it acceleatio ad Newto ecod law to fid the acceleatio of the bloc a it lide up the ap. f θ (a) Ue a cotat-acceleatio equatio to elate the ditace the bloc lide up the iclie to it iitial peed ad acceleatio: Appl 0 + aδ o, becaue 0, aδ Δ () a a to the bloc: f iθ a () ad coθ 0 (3) Subtituti f μ ad i equatio () ad (3) ield: Eliiate betwee equatio (4) ad (5) to obtai: μ iθ a (4) ad coθ 0 (5) μ coθ iθ a Soli fo a ield: ( μ coθ iθ ) a + Subtitute fo a i equatio () to 0 obtai: Δ ( μ coθ + iθ )

36 4 Chapte 5 Subtitute ueical alue ad ealuate Δ: ( 7.0 /) [( 0.3) co5 + i 5 ]( 9.8 / ) Δ (b) At the poit at which the bloc i itataeoul at et, tatic fictio becoe opeatie ad, if the tatic fictio coefficiet i too hih, the bloc will ot eue otio, but will eai at the hih poit. We ca deteie the aiu poible alue of μ fo which the bloc till lide bac dow the iclie b coidei the equalit of the tatic fictio foce ad the copoet of the weiht of the bloc dow the ap. Appl a to the bloc whe it i i equilibiu at the poit at which it i oetail at et: Soli equatio (6) fo ield: Becaue f becoe:, a μ, equatio (5) θ f,a f, a iθ 0 (5) ad coθ 0 (6) coθ μ coθ iθ 0 o μ coθ iθ 0 μ taθ Subtitute the ueical alue of θ ad ealuate μ : μ ta A autoobile i oi up a 5º ade at a peed of 30 /. The coefficiet of tatic fictio betwee the tie ad the oad i (a) What iiu ditace doe it tae to top the ca? (b) What iiu ditace would it tae to top if the ca wee oi dow the ade? Pictue the Poble We ca fid the toppi ditace b appli Newto ecod law to the autoobile ad the ui a cotat-acceleatio equatio. The fictio foce the oad eet o the tie ad the copoet of the ca weiht alo the iclie cobie to poide the et foce that top the ca. The pictoial epeetatio uaize what we ow about the otio of the ca. We ca ue Newto ecod law to deteie the acceleatio of the ca ad a cotatacceleatio equatio to obtai it toppi ditace.

37 Additioal Applicatio of Newto Law 43 θ t / t? i 0 (a) Ui a cotat-acceleatio equatio, elate the fial peed of the ca to it iitial peed, acceleatio, ad diplaceet; ole fo it diplaceet: 0 + aa, i o, becaue 0, 0 i () a a, Daw the fee-bod diaa fo the ca oi up the iclie: f,a θ Noti that, appl a to the ca: Subtitute f ad fo, a μ equatio (3) i equatio () ad ole fo a a, : f, a iθ a () ad coθ 0 (3) ( μ coθ iθ ) aa, + Subtituti fo a a, i equatio () 0 i ield: ( μ coθ + iθ ) Subtitute ueical alue ad ealuate i : 9.8/ ( ) ( ) ( 30 /) ( 0.70 co5 + i5 ) i 49

38 44 Chapte 5 (b) Whe the ca i oi dow the iclie, the tatic fictio foce i up the iclie a how i the fee-bod diaa to the iht. Note the chae i coodiate te fo Pat (a). θ f,a Appl a to the ca: ad iθ f a, a coθ 0 Poceed a i (a) to obtai: ( iθ μ coθ ) Subtituti fo a a, i equatio () aa, 0 i ield: ( iθ μ coθ ) Subtitute ueical alue ad ealuate i : 9.8/ ( 30 /) ( i co5 ) ( ) ( ) i A ea-wheel-die ca uppot 40 pecet of it weiht o it two die wheel ad ha a coefficiet of tatic fictio of 0.70 with a hoizotal taiht oad. (a) id the ehicle aiu acceleatio. (b) What i the hotet poible tie i which thi ca ca achiee a peed of 00 /h? (Aue the eie ha uliited powe.) Pictue the Poble The fictio foce the oad eet o the tie poide the et foce that acceleate the ca. The pictoial epeetatio uaize what we ow about the otio of the ca. We ca ue Newto ecod law to deteie the acceleatio of the ca ad a cotat-acceleatio equatio to calculate how lo it tae it to each 00 /h. 0 t 0 0 t? 0 0? /h

39 Additioal Applicatio of Newto Law 45 (a) Becaue 40% of the ca weiht i o it two die wheel ad the acceleati fictio foce act jut o thee wheel, the fee-bod diaa how jut the foce acti o the die wheel. f 0.4 Appl a to the die wheel: f a a, a, () ad () Ue the defiitio of f, a i aa, 4 0. μ equatio () ad eliiate betwee the two equatio to obtai: Subtitute ueical alue ad ealuate a : a, a a, 0.4 ( 0.70)( 9.8/ ).747 /.7 / (b) Ui a cotat-acceleatio equatio, elate the iitial ad fial elocitie of the ca to it acceleatio ad the elaped tie; ole fo the tie: Subtitute ueical alue ad ealuate t :, 0, + aa, Δt o, becaue 0, 0 ad Δt t,, t a t a, h h / 0 50 You ad ou bet pal ae a fiedl bet that ou ca place a.0- bo aait the ide of a cat, a i iue 5-63, ad that the bo will ot fall to the oud, ee thouh ou uaatee to ue o hoo, ope, fatee, aet, lue, o adheie of a id. Whe ou fied accept the bet, ou bei puhi the cat i the diectio how i the fiue. The coefficiet of tatic fictio betwee the bo ad the cat i (a) id the iiu acceleatio fo which ou will wi the bet. (b) What i the aitude of the fictioal foce i thi cae? (c) id the foce of fictio o the bo if the acceleatio i twice the iiu eeded fo the bo ot to fall. (d) Show that, fo a bo of a a, the bo will ot fall if the aitude of the fowad acceleatio i a /μ, whee μ i the coefficiet of tatic fictio.

40 46 Chapte 5 Pictue the Poble To hold the bo i place, the acceleatio of the cat ad bo ut be eat eouh o that the tatic fictio foce acti o the bo will equal the weiht of the bo. We ca ue Newto ecod law to deteie the iiu acceleatio equied. f,a (a) Noti that, appl a to the bo: Subtituti μ fo f, a i equatio () ield: Subtitute fo fo equatio () to obtai: ai, () ad f 0 () μ μ, a 0 ( a ) 0 i, a i, μ Subtitute ueical alue ad ealuate a i, : a i, 9.8/ / (b) o equatio () we hae: Subtitute ueical alue ad ealuate f :, a f, a f (.0 )( 9.8 / ) 0 N, a (c) If a i twice that equied to hold the bo i place, f will till hae it aiu alue ie b: f, a 0 N (d) Becaue a μ, the bo will ot fall if a. i, μ 5 Two bloc attached b a ti (iue 5-64) lide dow a 0º iclie. Bloc ha a 0.80 ad bloc ha a 0.5. I additio, the ietic coefficiet of fictio betwee the bloc ad the iclie ae 0.30 fo bloc ad 0.0 fo bloc. id (a) the aitude of the acceleatio of the bloc, ad (b) the teio i the ti.

41 Additioal Applicatio of Newto Law 47 Pictue the Poble Aue that the ti i ale ad doe ot tetch. The the bloc hae a coo acceleatio ad the teio i the ti act o both bloc i accodace with Newto thid law of otio. Let dow the iclie be the + diectio. Daw the fee-bod diaa fo each bloc ad appl Newto ecod law of otio ad the defiitio of the ietic fictio foce to each bloc to obtai iultaeou equatio i a ad T. Daw the fee-bod diaa fo the bloc whoe a i :, f, θ T Appl a to the uppe bloc: The elatiohip betwee f, ad, i: f, + T + iθ a () ad coθ 0 () f, μ,,, (3) Eliiate f, ad, betwee (), μ coθ + T + (), ad (3) to obtai: a, i θ (4) Daw the fee-bod diaa fo the bloc whoe a i : T, f, θ

42 48 Chapte 5 Appl a to the bloc: The elatiohip betwee f, ad, i: f, T + iθ a (5) ad coθ 0 (6) f, μ,,, (7) Eliiate f, ad, betwee (5), μ coθ T + (6), ad (7) to obtai: a, i θ (8) Add equatio (4) ad (8) to eliiate T, ad ole fo a : a μ, + μ, iθ coθ + Subtitute ueical alue ad ealuate a : a i / ( 0.0)( 0.5 ) + ( 0.30)( 0.80 ) co0 ( 9.8 / ) (b) Eliiate a betwee equatio (4) ad (8) ad ole fo T T T to obtai: T ( μ μ ),, co + θ Subtitute ueical alue ad ealuate T: T ( 0.5 )( 0.80 )( )( 9.8 / ) co0 0.8 N Two bloc of ae ad ae lidi dow a iclie a how i iue The ae coected b a ale od. The coefficiet of ietic fictio betwee the bloc ad the uface ae μ fo bloc ad μ fo bloc. (a) Deteie the acceleatio of the two bloc. (b) Deteie the foce that the od eet o each of the two bloc. Show that the foce ae both 0 whe μ μ ad ie a iple, oatheatical auet wh thi i tue. Pictue the Poble The fee-bod diaa how the foce acti o the two bloc a the lide dow the iclie. Dow the iclie ha bee choe a the + diectio. T i the foce taitted b the od; it ca be eithe teile (T > 0) o

43 Additioal Applicatio of Newto Law 49 copeie (T < 0). B appli Newto ecod law to thee bloc, we ca obtai equatio i T ad a fo which we ca eliiate eithe b oli the iultaeoul. Oce we hae epeed T, the ole of the od will becoe appaet. f, θ, T f, θ T, (a) Appl a to bloc : Appl a to bloc : Letti T T T, ue the defiitio of the ietic fictio foce to eliiate f, ad, betwee the equatio fo bloc ad f, ad, betwee the equatio fo bloc to obtai: θ ad, coθ 0 T + i f, θ ad, coθ 0 i T f, a iθ + T μ coθ () ad a iθ T μ coθ () a a Add equatio () ad () to eliiate T ad ole fo a : a μ + μ iθ coθ + (b) Rewite equatio () ad () b diidi both ide of () b ad both ide of () b to obtai. T a iθ + μ coθ (3) ad T a iθ μ coθ (4) Subtacti (4) fo (3) ad T eaai ield: ( μ μ ) θ + co