Chapter 12 Static Equilibrium and Elasticity


 Madeline Wilson
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1 Chapte Static Equilibium ad Elaticity Coceptual Poblem [SSM] Tue o fale: (a) i 0 i ufficiet fo tatic equilibium to eit. i (b) i 0 i eceay fo tatic equilibium to eit. i (c) I tatic equilibium, the et toque about ay poit i zeo. (d) A object i equilibium caot be movig. (a) ale. The coditio 0 ad τ 0 mut be atified. (b) Tue. The eceay ad ufficiet coditio fo tatic equilibium ae 0 ad τ 0. (c) Tue. The coditio 0 ad τ 0 mut be atified. (d) ale. A object ca be movig with cotat peed (talatioal o otatioal) whe the coditio 0 ad τ 0 ae atified. Tue o fale: (a) The cete of gavity i alway at the geometic cete of a body. (b) The cete of gavity mut be located iide a object. (c) The cete of gavity of a bato i located betwee the two ed. (d) The toque poduced by the foce of gavity about the cete of gavity i alway zeo. (a) ale. The locatio of the cete of gavity deped o how a object ma i ditibuted. (b) ale. A eample of a object fo which the cete of gavity i outide the object i a dout. (c) Tue. The tuctue of a bato ad the defiitio of the cete of gavity guaatee that the cete of gavity of a bato i located betwee the two ed. (d) Tue. Becaue the foce of gavity actig o a object act though the cete of gavity of the object, it leve (o momet) am i alway zeo. 85
2 86 Chapte 3 The hoizotal ba i igue 7 will emai hoizotal if (a) ad R R, (b) M R M R, (c) M R R M, (d) M M, (e) R R. Detemie the Cocept The coditio that the ba i i otatioal equilibium i that the et toque actig o it equal zeo; i.e., R M R M 0. (b) i coect. 4 Sit i a chai with you back taight. Now ty to tad up without leaig fowad. Eplai why you caot do it. Detemie the Cocept You caot tad up becaue, if you ae to tad up, you body cete of gavity mut be above you feet. 5 You have a job diggig hole fo pot to uppot ig fo a ouiiaa etauat (called Moca ). Eplai why the highe above the goud a ig i mouted, the fathe the pot hould eted ito the goud. Detemie the Cocept lat ig of ay kid epeiece ubtatial foce whe the wid blow agait them the lage the uface aea, the lage the foce. I ode to be table, the pot which uppot uch ig mut be buied deeply eough o that the goud ca eet ufficiet foce agait the pot to keep the ig i equilibium ude the toget wid. The pivot poit aoud which the ig might otate i at goud level thu the moe momet am available below goud level, the moe toque may be geeated by the foce of the goud o the pot. Thu the lage the uface aea of the billboad, the geate will be the foce applied above the uface, ad hece the toque applied to the pot will be geate. A uface aea iceae, the pefeed depth of the pot iceae a well o that with the iceaed momet am, the goud ca eet moe toque to balace the toque due to the wid. 6 A fathe (ma M) ad hi o, (ma m) begi walkig out towad oppoite ed of a balaced eeaw. A they walk, the eeaw tay eactly hoizotal. What ca be aid about the elatiohip betwee the fathe peed V ad the o peed v? Detemie the Cocept The quetio i about a ituatio i which a object i i tatic equilibium. Both the fathe ad o ae walkig outwad fom the cete of the eeaw, which alway emai i equilibium. I ode fo thi to happe, at ay time, the et toque about ay poit (let ay, the pivot poit at the cete of the eeaw) mut be zeo. We ca deote the fathe poitio a X, ad the o poitio a, ad chooe the oigi of coodiate to be at the pivot poit. At each momet, the eeaw eet omal foce o the o ad hi fathe equal to thei epective weight, mg ad Mg. By Newto thid law, the fathe eet a dowwad foce equal i magitude to the omal foce, ad the o eet a dowwad foce equal i magitude to the omal foce actig o him.
3 Static Equilibium ad Elaticity 87 Apply τ pivot poit 0 to the eeaw (aume that the fathe walk to the left ad that couteclockwie toque ae poitive): MgX mg 0 () Epe the ditace both the fathe ad hi o walk a a fuctio of time: X VΔt ad vδt Subtitute fo X ad i equatio () to obtai: m MgVΔ t mgvδt 0 V v M Remak: The fathe peed i le tha the o peed by a facto of m/m. 7 Tavel mug that people might et o the dahboad of thei ca ae ofte made with boad bae ad elatively aow mouth. Why would tavel mug be deiged with thi hape, athe tha have the oughly cylidical hape that mug omally have? Detemie the Cocept The mai eao thi i doe i to lowe the cete of gavity of the mug a a whole. o a give volume, it i poible to make a mug with getly lopig ide that ha a igificatly lowe cete of gavity tha the taditioal cylide. Thi i impotat, becaue a the cete of gavity of a object get lowe (ad a it bae boade) the object i hade to tip. Whe ca ae tavelig at cotat velocity, the deig of the mug i ot impotat but whe ca ae toppig ad goig acceleatig ad deceleatig the highe cete of gavity of the uual deig make it much moe poe to tippig. 8 The ailo i the photo ae uig a techique called hikig out. What pupoe doe poitioig themelve i thi way eve? If the wid wee toge, what would they eed to do i ode to keep thei caft table? Detemie the Cocept Dyamically the boat ae i equilibium alog thei lie of motio, but i the plae of thei ail ad the ailo, they ae i tatic equilibium. The toque o the boat, applied by the wid actig o the ail, ha a tedecy to tip the boat. The udde couteact that tedecy to ome degee, but i paticulaly tog wid, whe the boat i ailig at paticula agle with epect to the wid, the ailo eed to hike out to apply ome toque (due to the gavitatioal foce of the Eath o the ailo) by leaig outwad o the beam of the boat. If the wid tegthe, they eed to eted thei bodie futhe ove the ide ad may eed to get ito a cotaptio called a tapeze that eable the ailo to have hi o he etie body outide the boat.
4 88 Chapte 9 [SSM] A alumium wie ad a teel wie of the ame legth ad diamete D ae joied edtoed to fom a wie of legth. Oe ed of the wie i the fateed to the ceilig ad a object of ma M i attached to the othe ed. Neglectig the ma of the wie, which of the followig tatemet i tue? (a) The alumium potio will tetch by the ame amout a the teel potio. (b) The teio i the alumium potio ad the teel potio ae equal. (c) The teio i the alumium potio i geate tha that i the teel potio. (d) Noe of the above Detemie the Cocept We kow that equal legth of alumium ad teel wie of the ame diamete will tetch diffeet amout whe ubjected to the ame teio. Alo, becaue we ae eglectig the ma of the wie, the teio i them i idepedet of which oe i cloe to the oof ad deped oly o Mg. (b) i coect. Etimatio ad Appoimatio 0 A lage cate weighig 4500 N et o fou cmhigh block o a hoizotal uface (igue 8). The cate i.0 m log,. m high ad. m deep. You ae aked to lift oe ed of the cate uig a log teel py ba. The fulcum o the py ba i 0 cm fom the ed that lift the cate. Etimate the legth of the ba you will eed to lift the ed of the cate. Pictue the Poblem The diagam to the ight how the foce actig o the cate a it i beig lifted at it left ed. Note that whe the cowba lift the cate, oly half the weight of the cate i uppoted by the ba. Chooe the coodiate ytem how ad let the ubcipt pb efe to the py ba. The diagam below how the foce actig o the py ba a it i beig ued to lift the ed of the cate. y pb pb B y W W w ' A l B l pb
5 Static Equilibium ad Elaticity 89 Aume that the maimum foce you ca apply i 500 N (about 0 lb). et be the ditace betwee the poit of cotact of the teel ba with the floo ad the cate, ad let be the total legth of the ba. ackig ifomatio egadig the bed i py ba at the fulcum, we ll aume that it i mall eough to be egligible. We ca apply the coditio fo otatioal equilibium to the py ba ad a coditio fo talatioal equilibium to the cate whe it left ed i o the vege of liftig. Apply y 0 to the cate: pb W + 0 () Apply τ 0 to the cate about a ai though poit B ad pepedicula to the plae of the page to obtai: Solve equatio () fo pb ad ubtitute fo to obtai: Apply τ 0 to the py ba about a ai though poit A ad pepedicula to the plae of the page to obtai: Subtitute fo pb to obtai: w ww 0 W a oted i Pictue the Poblem. W W W pb pb + pb ( ) 0 W + Subtitute umeical value ad evaluate : 4500 N ( 0.0m) + ( 500 N) 55cm [SSM] Coide a atomic model fo Youg modulu. Aume that a lage umbe of atom ae aaged i a cubic aay, with each atom at a coe of a cube ad each atom at a ditace a fom it i eaet eighbo. Imagie that each atom i attached to it 6 eaet eighbo by little pig each with foce cotat k. (a) Show that thi mateial, if tetched, will have a Youg modulu Y k/a. (b) Uig Table  ad aumig that a.0 m, etimate a typical value fo the atomic foce cotat k i a metal. Pictue the Poblem We ca deive thi epeio by imagiig that we pull o a aea A of the give mateial, epeig the foce each pig will epeiece, fidig the factioal chage i legth of the pig, ad ubtitutig i the defiitio of Youg modulu.
6 90 Chapte (a) The defiitio of Youg modulu i: Y A () Δ Epe the elogatio Δ of each pig: The foce each pig will epeiece a a eult of a foce actig o the aea A i: Δ () k N Epe the umbe of pig N A i the aea A: N a Subtitutig fo N yield: Subtitute i equatio () to obtai, fo the eteio of oe pig: Δ a A a ka Aumig that the pig eted/compe liealy, the factioal eteio of the pig i: Δ Δ a a tot a ka a ka Subtitute i equatio () ad implify to obtai: (b) om ou eult i Pat (a): Y A a ka k Ya k a om Table : Y 00 GN/m.00 0 N/m Subtitute umeical value ad evaluate k: k 9 (.00 0 N/m )(.0 0 m).0 N/cm By coideig the toque about the cete of the ball joit i you houlde, etimate the foce you deltoid mucle (thoe mucle o top of you houlde) mut eet o you uppe am, i ode to keep you am held out ad eteded at houlde level. The, etimate the foce they mut eet whe you hold a 0lb weight out to the ide at am legth. Pictue the Poblem A model of you am i how i the pictoial epeetatio. You houlde joit i at poit P ad the foce the deltoid mucle eet o you
7 Static Equilibium ad Elaticity 9 eteded am deltoid i how actig at a agle with the hoizotal. The weight of you am i the gavitatioal foce g mg eeted by Eath though the cete of gavity of you am. We ca ue the coditio fo otatioal equilibium to etimate the foce eeted by you deltoid mucle. Note that, becaue it momet am i zeo, the toque due to houlde about a ai though poit P ad pepedicula to the page i zeo. P deltoid houlde mg g P Apply τ 0 to you eteded am: i mg 0 () deltoid Solvig fo deltoid yield: Aumig that 0 cm, 60 cm, mg 0 lb, ad 0, ubtitute umeical value ad evaluate deltoid : mg deltoid i deltoid ( 0 lb)( 60 cm) ( 0 cm) i0 86 lb If you hold a 0lb weight at the ed of you am, equatio () become: ' deltoid i mg m'g 0 whee m i the ma of the 0lb weight. Solvig fo deltoid yield: Subtitute umeical value ad evaluate deltoid : ' deltoid deltoid mg + m'g i ( 0 lb)( 60 cm) + ( 0 lb)( 60 cm) ( 0 cm) i0 60 lb Coditio fo Equilibium 3 You cutch i peed agait the idewalk with a foce c alog it ow diectio, a how i igue 9. Thi foce i balaced by the omal
8 9 Chapte foce ad a fictioal foce f. (a) Show that whe the foce of fictio i at it maimum value, the coefficiet of fictio i elated to the agle by μ ta. (b) Eplai how thi eult applie to the foce o you foot whe you ae ot uig a cutch. (c) Why i it advatageou to take hot tep whe walkig o lippey uface? Pictue the Poblem Chooe a coodiate ytem i which upwad i the poitive y diectio ad to the ight i the poitive diectio ad ue the coditio fo talatioal equilibium. (a) Apply 0 to the foce actig o the tip of the cutch: Solve equatio () fo ad aumig that f f,ma, obtai: Subtitute i equatio () ad olve fo µ : f + c i 0 () ad y c co 0 () f f μ μ,ma μ ta c co (b) Takig log tide equie a lage coefficiet of tatic fictio becaue i lage fo log tide. (c) If μ i mall (the uface i lippey), mut be mall to avoid lippig. 4 A thi od of ma M i upeded hoizotally by two vetical wie. Oe wie i at the left ed of the od, ad the othe wie i /3 of the way fom the left ed. (a) Detemie the teio i each wie. (b) A object i ow hug by a tig attached to the ight ed of the od. Whe thi happe, it i oticed that the wie emai hoizotal but the teio i the wie o the left vaihe. Detemie the ma of the object. Pictue the Poblem The pictoial epeetatio how the thi od with the foce decibed i Pat (a) actig o it. We ca apply τ 0 0 to the od to fid the foce T ad T R. The implet way to detemie the ma m of the object upeded fom the od i (b) i to apply the coditio fo otatioal equilibium a ecod time, but thi time with epect to a ai pepedicula to the page ad though the poit at which T R act.
9 Static Equilibium ad Elaticity 93 T 0 Mg T R 3 (a) Apply τ 0 0 to the od: 3 T Mg 0 Mg Apply vetical 0 to the od: T Mg + T R 0 3 R T R 4 Subtitute fo T R to obtai: 3 T Mg + Mg 0 Mg 4 T 4 (b) With a object of ma m upeded fom the ight ed of the od ad T 0, applyig τ 0 about a ai pepedicula to the page ad though the poit at which T act yield: R Solvig fo m yield: ( ) Mg ( 3 ) mg 0 3 m M The Cete of Gavity 5 A automobile ha 58 pecet of it weight o the fot wheel. The fot ad back wheel o each ide ae epaated by.0 m. Whee i the cete of gavity located? Pictue the Poblem et the weight of the automobile be w. Chooe a coodiate ytem i which the oigi i at the poit of cotact of the fot wheel with the goud ad the poitive ai iclude the poit of cotact of the ea wheel with the goud. Apply the defiitio of the cete of gavity to fid it locatio. Ue the defiitio of the cete of gavity to obtai: cg W wi i 0.58w ( 0.84m)w i ( 0) + 0.4w(.0m) Becaue W w: cg w ( 0.84m)w cg 84cm
10 94 Chapte Static Equilibium 6 igue 30 how a leve of egligible ma with a vetical foce app beig applied to lift a load. The mechaical advatage of the leve i defied a M app, mi, whee app, mi i the mallet foce eceay to lift the load. Show that fo thi imple leve ytem, M /X, whee i the momet am (ditace to the pivot) fo the applied foce ad X i the momet am fo the load. Pictue the Poblem We ca ue the give defiitio of the mechaical advatage of a leve ad the coditio fo otatioal equilibium to how that M /X. Epe the defiitio of mechaical advatage fo a leve: Apply the coditio fo otatioal equilibium to the leve: M () τ app, mi fulcum app, mi X 0 Solve fo the atio of to to obtai: app, mi app, mi X Subtitute fo to obtai: app, mi i equatio () M X 7 [SSM] igue 3 how a 5foot ailboat. The mat i a uifom 0kg pole that i uppoted o the deck ad held foe ad aft by wie a how. The teio i the foetay (wie leadig to the bow) i 000 N. Detemie the teio i the backtay (wie leadig aft) ad the omal foce that the deck eet o the mat. (Aume that the fictioal foce the deck eet o the mat to be egligible.) Pictue the Poblem The foce diagam how the foce actig o the mat. et the oigi of the coodiate ytem be at the foot of the mat with the + diectio to the ight ad the +y diectio upwad. Becaue the mat i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to fid the teio i the backtay, T B, ad the omal foce, D, that the deck eet o the mat. T y 45 mg P D T B
11 Static Equilibium ad Elaticity 95 Apply τ 0to the mat about a ai though poit P: ( 4.88m)( 000 N) i ( 4.88m) T i B Solve fo T B to obtai: ( N) 000 i T B () i 45.0 id, the agle of the foetay with the vetical:.74m ta m Subtitute umeical value i equatio () ad evaluate T B : T ( 000 N) i 9.3 i 45.0 B 69 N Apply the coditio fo talatioal equilibium i the y diectio to the mat: y D T co TB co45 mg 0 Solvig fo D yield: T co + T co 45 + mg D B Subtitute umeical value ad evaluate D : ( 000 N) co ( 69 N) co 45 + ( 0 kg)( 9.8 m/ ).54 kn D 8 A uifom 0.0m beam of ma 300 kg eted ove a ledge a i igue 3. The beam i ot attached, but imply et o the uface. A 60.0kg tudet ited to poitio the beam o that he ca walk to the ed of it. What i the maimum ditace the beam ca eted pat ed of the ledge ad till allow him to pefom thi feat? Pictue the Poblem The diagam how M g, the weight of the beam, m g, the weight of the tudet, ad the foce the ledge eet, actig o the beam. Becaue the beam i i equilibium, we ca apply the coditio fo otatioal equilibium to the beam to fid the locatio of the pivot poit P that will allow the tudet to walk to the ed of the beam. 5.0 m Mg P mg Apply τ 0 about a ai though the pivot poit P: Mg ( 5.0m ) mg 0
12 96 Chapte Solvig fo yield: 5.0M M + m Subtitute umeical value ad evaluate : ( 5.0 m)( 300kg) 300kg kg 4.m 9 [SSM] A gavity boad i a coveiet ad quick way to detemie the locatio of the cete of gavity of a peo. It coit of a hoizotal boad uppoted by a fulcum at oe ed ad a cale at the othe ed. To demotate thi i cla, you phyic pofeo call o you to lie hoizotally o the boad with the top of you head diectly above the fulcum poit a how i igue 33. The cale i.00 m fom the fulcum. I pepaatio fo thi epeimet, you had accuately weighed youelf ad detemied you ma to be 70.0 kg. Whe you ae at et o the gavity boad, the cale advace 50 N beyod it eadig whe the boad i thee by itelf. Ue thi data to detemie the locatio of you cete of gavity elative to you feet. Pictue the Poblem The diagam how w, the weight of the tudet, P, the foce eeted by the boad at the pivot, ad, the foce eeted by the cale, actig o the tudet. Becaue the tudet i i equilibium, we ca apply the coditio fo otatioal equilibium to the tudet to fid the locatio of hi cete of gavity. P P.00 m w mg S Apply τ 0 about a ai though the pivot poit P: (.00m) w 0 Solvig fo yield: (.00m) w Subtitute umeical value ad evaluate : (.00m)( 50 N) ( 70.0 kg)( 9.8m/ ) 0.78m 0 A tatioay 3.0m boad of ma 5.0 kg i higed at oe ed. A foce i applied vetically at the othe ed, ad the boad make at 30 agle with the hoizotal. A 60kg block et o the boad 80 cm fom the hige a how i igue 34. (a) id the magitude of the foce. (b) id the foce eeted by
13 Static Equilibium ad Elaticity 97 the hige. (c) id the magitude of the foce, a well a the foce eeted by the hige, if i eeted, itead, at ight agle to the boad. Pictue the Poblem The diagam how m g, the weight of the boad, hige, the foce eeted by the hige, M g, the weight of the block, ad, the foce actig vetically at the ight ed of the boad. Becaue the boad i i equilibium, we ca apply the coditio fo otatioal equilibium to it to fid the magitude of. P y hige 0.80 m m Mg mg.50 m (a) Apply τ 0 about a ai though the hige to obtai: [( 3.0m) co30 ] mg[ (.50 m) co30 ] Mg[ ( 0.80 m) co30 ] 0 m.50m + M 0.80m 3.0m Solvig fo yield: ( ) ( ) g Subtitute umeical value ad evaluate : ( 5.0kg)(.50m) + ( 60kg)( 0.80m ) ( 9.8m/ ) 8N 0.8kN 3.0m y Apply 0 to the boad to obtai: hige Mg mg + 0 Solvig fo hige yield: Mg + mg ( M + m) g Subtitute umeical value ad evaluate hige : hige hige ( 60kg + 5.0kg)( 9.8m/ ) 0.46 kn 8N
14 98 Chapte (c) The foce diagam howig the foce actig at ight agle to the boad i how to the ight: Apply τ 0 about the hige: P y hige 0.80 m m Mg mg.50 m 30 ( 3.0m) mg[ (.5m) co30 ] Mg[ ( 0.80 m) co30 ] 0 Solvig fo yield: m(.5m) + M( 0.80m) g co30 3.0m Subtitute umeical value ad evaluate : ( 5.0kg)(.5m) + ( 60kg)( 0.80m ) ( 9.8m/ ) co30 57 N 0.6kN 3.0m Apply y 0 to the boad: hige i Mg mg + co30 0 o i M + m g co30 () hige ( ) Apply 0 to the boad: hige co i 30 0 o co i 30 () hige Divide the fit of thee equatio by the ecod to obtai: hige hige i co ( M + m) g co30 i 30 Solvig fo yield: ( M + m) Subtitute umeical value ad evaluate : ta ta ( 65kg)( 9.8m/ ) ( 57 N) ( 57 N) i30 g i 30 co30 8. co30
15 Static Equilibium ad Elaticity 99 Subtitute umeical value i equatio () ad evaluate hige : ( 57 N) i 30 co8. hige 0.5kN A cylide of ma M i uppoted by a fictiole tough fomed by a plae iclied at 30º to the hoizotal o the left ad oe iclied at 60º o the ight a how i igue 35. id the foce eeted by each plae o the cylide. Pictue the Poblem The plae ae fictiole; theefoe, the foce eeted by each plae mut be pepedicula to that plae. et be the foce eeted by the 30 plae, ad let be the foce eeted by the 60 plae. Chooe a coodiate ytem i which the poitive diectio i to the ight ad the poitive y diectio i upwad. Becaue the cylide i i equilibium, we ca ue the coditio fo talatioal equilibium to fid the magitude of ad Mg Apply 0 to the cylide: i 30 i 60 0 () Apply y 0 to the cylide: co30 + co60 Mg 0 () Solve equatio () fo : 3 (3) Subtitute fo i equatio () to obtai: 3 co30 + co60 Mg 0 Solve fo to obtai: Mg Mg 3co30 + co60 Subtitute fo i equatio (3) to obtai: 3 ( ) 3 Mg Mg A uifom 8kg doo that i.0 m high by 0.80 m wide i hug fom two hige that ae 0 cm fom the top ad 0 cm fom the bottom. If each hige uppot half the weight of the doo, fid the magitude ad diectio of the hoizotal compoet of the foce eeted by the two hige o the doo.
16 00 Chapte Pictue the Poblem The dawig how the doo ad it two uppot. The cete of gavity of the doo i 0.80 m above (ad below) the hige, ad 0.40 m fom the hige hoizotally. Chooe a coodiate ytem i which the poitive diectio i to the ight ad the poitive y diectio i upwad. Deote the hoizotal ad vetical compoet of the hige foce by Hh ad Hv. Becaue the doo i i equilibium, we ca ue the coditio fo talatioal ad otatioal equilibium to detemie the hoizotal foce eeted by the hige. Hv Hh.6 m P ' Hv' Hh 0.40 m mg Apply τ 0 about a ai though the lowe hige: (.6 m) mg( 0.40m) 0 Hh Solve fo Hh : mg( 0.40m) Hh.6m Subtitute umeical value ad evaluate Hh : Hh ( 8kg)( 9.8m/ )( 0.40m).6m 44 N Apply 0 to the doo ad olve fo ' Hh : ' ad Hh Hh ' Hh 0 44 N Remak: Note that the uppe hige pull o the doo ad the lowe hige puhe o it. 3 id the foce eeted o the tut by the hige at A fo the aagemet i igue 36 if (a) the tut i weightle, ad (b) the tut weigh 0 N.
17 Static Equilibium ad Elaticity 0 Pictue the Poblem et T be the teio i the lie attached to the wall ad be the legth of the tut. The figue iclude w, the weight of the tut, fo pat (b). Becaue the tut i i equilibium, we ca ue the coditio fo both otatioal ad talatioal equilibium to fid the foce eeted o the tut by the hige. v 0 A h T w 45 W (a) Epe the foce eeted o the tut at the hige: iˆ + h v ˆj () Igoig the weight of the tut, apply τ 0 at the hige: T ( co 45 ) W 0 Solve fo the teio i the lie: T W co45 ( 60 N) Apply N to the tut: h T co45 co45 0 ad y v + T co 45 Mg 0 Solve fo ad evaluate h : T co45 ( 4.4 N) co45 30 N h Solve fo ad evaluate v : v Mg T 60 N co45 ( 4.4 N) co45 30 N Subtitute i equatio () to obtai: ( 30 N) i + ( 30 N)j ˆ ˆ (b) Icludig the weight of the tut, apply τ 0 at the hige: T ( co 45 ) W co45 w 0 Solve fo the teio i the lie: T 45 ( co 45 ) W + co w
18 0 Chapte Subtitute umeical value ad evaluate T: T ( co 45 )( 60 N) + co 45 ( 0 N) 49.5 N to the tut: h T co45 Apply 0 0 ad y v + T co45 W w 0 Solve fo ad evaluate h : T co 45 ( 49.5 N) h 35 N co 45 Solve fo ad evaluate v : v W + w T co N + 0 N 45 N ( 49.5 N) co45 Subtitute fo h ad v to obtai: ( 35 N) i + ( 45 N)j ˆ ˆ 4 Julie ha bee hied to help pait the tim of a buildig, but he i ot coviced of the afety of the appaatu. A 5.0m plak i upeded hoizotally fom the top of the buildig by ope attached at each ed. Julie kow fom peviou epeiece that the ope beig ued will beak if the teio eceed.0 kn. He 80kg bo dimie Julie woie ad begi paitig while tadig.0 m fom the ed of the plak. If Julie ma i 60 kg ad the plak ha a ma of 0 kg, the ove what age of poitio ca Julie tad to joi he bo without cauig the ope to beak? Pictue the Poblem Note that if Julie i at the fa left ed of the plak, T ad T ae le tha.0 kn. et be the ditace of Julie fom T. Becaue the plak i i equilibium, we ca apply the coditio fo otatioal equilibium to elate the ditace to the othe ditace ad foce. T T P m g J.5 m m g p.0 m m g b
19 Apply τ 0 about a ai though the left ed of the plak: Static Equilibium ad Elaticity 03 (.0m) T ( 4.0m) m g (.5m) m g m g 0 5 b p J Solvig fo yield: ( 5.0m) T ( 4.0m) m (.5 ) m J g m b m J m J m p Subtitute umeical value ad implify to obtai: Set T.0 kn ad evaluate : 3 m T N m 3 m N 6.7 m (.0 kn) 6.7 m ad Julie i afe povided <.3m. 5 [SSM] A cylide of ma M ad adiu R oll agait a tep of height h a how i igue 37. Whe a hoizotal foce of magitude i applied to the top of the cylide, the cylide emai at et. (a) id a epeio fo the omal foce eeted by the floo o the cylide. (b) id a epeio fo the hoizotal foce eeted by the edge of the tep o the cylide. (c) id a epeio fo the vetical compoet of the foce eeted by the edge of the tep o the cylide. Pictue the Poblem The figue to the ight how the foce actig o the cylide. Chooe a coodiate ytem i which the poitive diectio i to the ight ad the poitive y diectio i upwad. Becaue the cylide i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to fid ad the hoizotal ad vetical compoet of the foce the coe of the tep eet o the cylide. R h Mg R c,v c,h h (a) Apply τ 0 to the cylide about the tep coe: Mg ( R h) 0
20 04 Chapte Solvig fo yield: ( R h) Mg Epe a a fuctio of R ad h: ( ) R R h Rh h Subtitute fo i the epeio fo ad implify to obtai: Mg ( R h) Rh h R h Mg h (b) Apply 0 to the cylide: c,h + 0 Solve fo c,h : c,h (c) Apply y 0 to the cylide: Mg + c,v 0 c, v Mg Subtitute the eult fom Pat (a) ad implify to obtai: c,v Mg Mg R h h R h h 6 o the cylide i Poblem 5, fid a epeio fo the miimum magitude of the hoizotal foce that will oll the cylide ove the tep if the cylide doe ot lide o the edge. Pictue the Poblem The figue to the ight how the foce actig o the cylide. Becaue the cylide i i equilibium, we ca ue the coditio fo otatioal equilibium to epe i tem of. Becaue, to oll ove the tep, the cylide mut lift off the floo, we ca et 0 i ou epeio elatig ad ad olve fo. R h Mg R c,v c,h h
21 Apply τ 0 coe: about the tep Static Equilibium ad Elaticity 05 Mg ( R h) 0 Solve fo : ( R h) Mg Epe a a fuctio of R ad h: ( ) R R h Rh h Subtitute fo i the epeio fo ad implify to obtai: Mg ( R h) Rh h R h Mg h To oll ove the tep, the cylide mut lift off the floo. That i, 0: 0 Mg R h h Solvig fo yield: Mg h R h 7 igue 38 how a had holdig a epee, a weapo ued i the pot of fecig which you ae takig a a phyical educatio elective. The cete of ma of you epee i 4 cm fom the pommel (the ed of the epee at the gip). You have weighed it o you kow that the epee ma i kg ad it full legth i 0 cm. (a) At the begiig of a match you hold it taight out i tatic equilibium. id the total foce eeted by you had o the epee. (b) id the toque eeted by you had o the epee. (c) You had, beig a eteded object, actually eet it foce alog the legth of the epee gip. Model the total foce eeted by you had a two oppoitely diected foce whoe lie of actio ae epaated by the width of you had (take to be 0.0 cm). id the magitude ad diectio of thee two foce. Pictue the Poblem The diagam how the foce ad that the fece had eet o the epee. We ca ue a coditio fo talatioal equilibium to fid the upwad foce the fece mut eet o the epee whe it i i equilibium ad the defiitio of toque to detemie the total toque eeted. I Pat (c) we ca ue the coditio fo talatioal ad otatioal equilibium to obtai two equatio i ad that we ca olve imultaeouly. I Pat (d) we ca apply Newto d law i otatioal fom ad the coditio fo talatioal equilibium to obtai two equatio i ad that, agai, we ca olve imultaeouly.
22 06 Chapte cm 0 cm 4 cm W (a) ettig the upwad foce eeted by the fece had be, apply 0to the epee to obtai: y Subtitute umeical value ad evaluate : (b) The toque due to the weight about the left ed of the epee i equal i magitude but oppoite i diectio to the toque eeted by you had o the epee: Subtitute umeical value ad evaluate τ : y (c) Apply 0to the epee to obtai: Apply τ 0 0 to obtai: W 0 ad W mg τ w τ ( kg)( 9.8m/ ) 6.87 N ( 0.4m)( 6.87 N).7 N m.65n m N 0 () (.00m) + ( 0.m).65 N m 0 () 0 Solve equatio () ad () imultaeouly to obtai: 8.3N ad 5N. Remak: Note that the foce eaet the butt of the epee i diected dowwad ad the foce eaet the had guad i diected upwad. 8 A lage gate weighig 00 N i uppoted by hige at the top ad bottom ad i futhe uppoted by a wie a how i igue 39. (a) What mut be the teio i the wie fo the foce o the uppe hige to have o hoizotal compoet? (b) What i the hoizotal foce o the lowe hige? (c) What ae the vetical foce o the hige?
23 Static Equilibium ad Elaticity 07 Pictue the Poblem I the foce diagam, the foce eeted by the hige ae y,, y,, ad, whee the ubcipt efe to the lowe hige. Becaue the gate i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to fid the teio i the wie ad the foce at the hige. T y, y,, mg (a) Apply τ 0 about a ai though the lowe hige ad pepedicula to the plae of the page: T i + T co mg 0 Solvig fo T yield: Subtitute umeical value ad evaluate T: T T mg i + co (.5m)( 00 N) (.5m) i 45 + (.5m) 4N 0.4 kn co45 (b) Apply 0 to the gate:, T co45 0 Solve fo ad evaluate, : T co45 ( 4N), 99.7 N.0 0 co45 N (c) Apply y 0 to the gate: y, + y, + T i 45 mg 0 Becaue y, ad y, caot be detemied idepedetly, olve fo ad evaluate thei um: y, + mg T i 45 y, 00 N 99.7 N.0 0 N
24 08 Chapte 9 O a campig tip, you moo you boat at the ed of a dock i a apidly flowig ive. It i achoed to the dock by a chai 5.0 m log, a how i igue 46. A 00N weight i upeded fom the cete i the chai. Thi will allow the teio i the chai to chage a the foce of the cuet which pull the boat away fom the dock ad to the ight vaie. The dag foce by the wate o the boat deped o the peed of the wate. You decide to apply the piciple of tatic you leaed i phyic cla. (Igoe the weight of the chai.) The dag foce o the boat i 50 N. (a) What i the teio i the chai? (b) How fa i the boat fom the dock? (c) The maimum teio the chai ca utai i 500 N. What miimum wate dag foce o the boat would ap the chai? Pictue the Poblem The feebody diagam how to the left below i fo the weight ad the diagam to the ight i fo the boat. Becaue both ae i equilibium ude the ifluece of the foce actig o them, we ca apply a coditio fo talatioal equilibium to fid the teio i the chai. T y T y d 00 N T mg (a) Apply 0 to the boat: T co d 0 T d co y Apply 0 to the weight: T i 00 N 0 () Subtitute fo T to obtai: d ta 00 N 0 Solve fo to obtai: Subtitute fo d ad evaluate : Solve equatio () fo T: ta 00 N d 00 N ta 00 N T i ( 50 N) 45 Subtitute fo ad evaluate T: T 00 N 70.7 N i 45 7N
25 Static Equilibium ad Elaticity 09 (b) Relate the ditace d of the boat fom the dock to the agle the chai make with the hoizotal: Subtitute umeical value ad evaluate d: d d co d co d ( 5.0m) co45 3.5m (c) Relate the eultat teio i the chai to the vetical compoet of the teio v ad the maimum dag foce eeted o the boat by the wate : d, ma v + d, ma ( 500 N) Solve fo : ( ) d, ma d, ma N 500 v Becaue the vetical compoet of the teio i 50 N: d,ma ( 500 N) ( 50 N) 0.50 kn 30 Romeo take a uifom 0m ladde ad lea it agait the mooth (fictiole) wall of the Capulet eidece. The ladde ma i kg ad the bottom et o the goud.8 m fom the wall. Whe Romeo, whoe ma i 70 kg, get 90 pecet of the way to the top, the ladde begi to lip. What i the coefficiet of tatic fictio betwee the goud ad the ladde? Pictue the Poblem The ladde ad the foce actig o it at the citical momet of lippig ae how i the diagam. Ue the coodiate ytem how. Becaue the ladde i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium. y by wall 0.5 mg 0.9 Mg 0 m 0 f,ma.8 m Uig it defiitio, epe µ : f,ma μ ()
26 0 Chapte Apply τ 0 about the bottom of the ladde: [ 0.9 co ] Mg + [ 0.5 co ] mg [ i ] W 0 Solvig fo W yield: ( 0.9M + 0.5m) W i g co id the agle :.8m co m Subtitute umeical value ad evaluate W : [ 0.9( 70kg) + 0.5( kg) ]( 9.8m/ ) co N W i Apply 0 to the ladde ad olve fo f,ma : W f,ma ad f, ma W 0.7 N Apply y 0 to the ladde: Mg mg 0 ( M + m)g Subtitute umeical value ad evaluate : ( 70kg + kg)( 9.8m/ ) 90.5 N Subtitute umeical value i equatio () ad evaluate µ : μ.7 N 90.5 N [SSM] Two 80N foce ae applied to oppoite coe of a ectagula plate a how i igue 4. (a) id the toque poduced by thi couple uig Equatio 6. (b) Show that the eult i the ame a if you detemie the toque about the lowe lefthad coe. Pictue the Poblem The foce how i the figue cotitute a couple ad will caue the plate to epeiece a couteclockwie agula acceleatio. The couple equatio iτ D. The followig diagam how the geometic elatiohip betwee the vaiable i tem of a geealized agle.
27 Static Equilibium ad Elaticity b 80 N D a a 80 N P b (a) The couple equatio i: τ D () om the diagam, D i give by: ( ) D b co () Agai, efeig to the diagam: Subtitutig fo i equatio () ad implifyig yield: Subtitutig fo D i equatio () yield: Subtitute umeical value ad evaluate τ : (b) ettig the couteclockwie diectio be the poitive diectio, apply τ 0 about a ai omal to the plae of the ectagle ad paig though poit P: a ta ( b a ta ) D co bco ai ( bco ai ) τ (3) τ ( 80 N)( bco30 ai 30 ) ( 69 N) b ( 40 N)a ( + D) 0 + Subtitutig fo D yield: ( bco ai ) 0 Solve fo τ to obtai: τ ( bco ai ), i ageemet with equatio (3). 3 A uifom cube of ide a ad ma M et o a hoizotal uface. A hoizotal foce i applied to the top of the cube a i igue 4. Thi foce i ot ufficiet to move o tip the cube. (a) Show that the foce of tatic fictio
28 Chapte eeted by the uface ad the applied foce cotitute a couple, ad fid the toque eeted by the couple. (b) The toque eeted by the couple i balaced by the toque eeted by the couple coitig of the omal foce o the cube ad the gavitatioal foce o the cube. Ue thi fact to fid the effective poit of applicatio of the omal foce whe Mg/3. (c) id the geatet magitude of fo which the cube will ot tip (Aumig the cube doe ot lip.). Pictue the Poblem We ca ue the coditio fo talatioal equilibium ad the defiitio of a couple to how that the foce of tatic fictio eeted by the uface ad the applied foce cotitute a couple. We ca ue the defiitio of toque to fid the toque eeted by the couple. We ca ue ou eult fom (b) to fid the effective poit of applicatio of the omal foce whe Mg/3 ad the coditio fo otatioal equilibium to fid the geatet magitude of fo which the cube will ot tip. (a) Apply 0 to the tatioay cube: + f 0 f Becaue f, thi pai of equal, paallel, ad oppoitely diected foce cotitute a couple. The toque of the couple i: τ couple a (b) et equal the ditace fom the poit of applicatio of to the cete of the cube. Now, Mg, o applyig τ 0 to the cube yield: Subtitutig fo ad implifyig yield: (c) Solve equatio () fo : Mg a 0 Mg a 3 a Mg 3 Mg a a () Mg Notig that ma a/, epe the coditio that the cube will tip: a Mg Mgma > a a Mg
29 Static Equilibium ad Elaticity 3 33 [SSM] A ladde of egligible ma ad of legth lea agait a lick wall makig a agle of with the hoizotal floo. The coefficiet of fictio betwee the ladde ad the floo i μ. A ma climb the ladde. What height h ca he each befoe the ladde lip? Pictue the Poblem et the ma of the ma be M. The ladde ad the foce actig o it ae how i the diagam. Becaue the wall i lick, the foce the wall eet o the ladde mut be hoizotal. Becaue the ladde i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to it. y by wall Mg h 0 f, ma Apply y 0 to the ladde ad olve fo : Apply 0 to the ladde ad olve fo f,ma : Apply τ 0 about the bottom of the ladde to obtai: Solvig fo ad implifyig yield: Refeig to the figue, elate to h: Mg 0 Mg f, ma W 0 f, ma W Mg co W i 0 W i Mg co f,ma ta Mg μ ta μ ta Mg h i Subtitutig fo yield: h μ ta i 34 A uifom ladde of legth ad ma m lea agait a fictiole vetical wall, makig a agle of 60º with the hoizotal. The coefficiet of tatic
30 4 Chapte fictio betwee the ladde ad the goud i If you ma i fou time that of the ladde, how high ca you climb befoe the ladde begi to lip? Pictue the Poblem The ladde ad the foce actig o it ae how i the dawig. Chooe a coodiate ytem i which the poitive diectio i to the ight ad the poitive y diectio i upwad. Becaue the wall i mooth, the foce the wall eet o the ladde mut be hoizotal. Becaue the ladde i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium. 0 y by wall mg f,ma 4mg Apply y 0 to the ladde ad olve fo : Apply 0 to the ladde ad olve fo f, ma : Apply τ 0 about a ai though the bottom of the ladde: mg 4mg 0 5mg W f,ma 0 f, ma W mg co + 4mg co W i 0 Subtitute fo W ad olve fo : Simplify to obtai: f, ma ad μmgi mgco 4mg co 5 5 μ ta 4 8 Subtitute umeical value to obtai: ( 0.45) 5 ta That i, you ca climb about 85% of the way to the top of the ladde befoe it begi to lip. 35 A ladde of ma m ad legth lea agait a fictiole vetical wall, o that it make a agle with the hoizotal. The cete of ma of the ladde i a height h above the floo. A foce diected diectly away fom the wall pull o the ladde at it midpoit. id the miimum coefficiet of tatic fictio μ
31 Static Equilibium ad Elaticity 5 fo which the top ed of the ladde will epaate fom the wall befoe the lowe ed begi to lip. Pictue the Poblem The ladde ad the foce actig o it ae how i the figue. Becaue the ladde i epaatig fom the wall, the foce the wall eet o the ladde i zeo. Becaue the ladde i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium. y f, ma 0 mg h To fid the foce equied to pull the ladde away fom the wall, apply τ 0 about a ai though the bottom of the ladde: Solvig fo yield: ( ) mg( co ) 0 i o, becaue co, ta mgh i 0 ta h mgh () ta i Apply 0 to the ladde: f, ma 0 f,ma μ () Apply y 0 to the ladde: mg 0 mg Equate equatio () ad () ad ubtitute fo to obtai: μ mg mgh ta i Solvig fo µ yield: μ h ta i 36 A 900N ma it o top of a tepladde of egligible ma that et o a fictiole floo a i igue 43. Thee i a co bace halfway up the ladde. The agle at the ape i 30º. (a) What i the foce eeted by the floo o each leg of the ladde? (b) id the teio i the co bace. (c) If the co bace i moved dow towad the bottom of the ladde (maitaiig the ame agle ), will it teio be the ame, geate, o le tha whe it wa i it highe poitio? Eplai you awe.
32 6 Chapte Pictue the Poblem Aume that half the ma weight act o each ide of the ladde. The foce eeted by the fictiole floo mut be vetical. D i the epaatio betwee the leg at the bottom ad i the ditace of the co bace fom the ape. Becaue each leg of the ladde i i equilibium, we ca apply the coditio fo otatioal equilibium to the ight leg to elate the teio i the co bace to it ditace fom the ape. w T m h (a) By ymmety, each leg caie half the total weight, ad the foce o each leg i 450 N. (b) Coide oe of the ladde leg ad apply τ 0 about the ape: D T 0 T D Uig tigoomety, elate h ad though the taget fuctio: D ta D h ta h Subtitute fo D i the epeio fo T ad implify to obtai: T h ta h ta y Apply 0 to the ladde ad olve fo : w 0 w Subtitute fo to obtai: T wh ta () Subtitute umeical value ad evaluate T: T ( 900 N)( 4.0m) (.0m) ta5 0.4 kn (c) om equatio () we ca ee that T i iveely popotioal to. Hece, if the bace i moved lowe, T will deceae. 37 A uifom ladde et agait a fictiole vetical wall. The coefficiet of tatic fictio betwee the ladde ad the floo i What i the
33 Static Equilibium ad Elaticity 7 mallet agle betwee the ladde ad the hoizotal uch that the ladde will ot lip? Pictue the Poblem The figue how the foce actig o the ladde. Becaue the wall i fictiole, the foce the wall eet o the ladde i pepedicula to the wall. Becaue the ladde i o the vege of lippig, the tatic fictio foce i f,ma. Becaue the ladde i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium. 0 y by wall mg f,ma Apply 0 to the ladde: f, ma W 0 W f,ma μ Apply y 0 to the ladde: mg 0 mg Apply τ 0 about a ai though the bottom of the ladde: Subtitute fo W ad ad implify to obtai: mg ( co ) ( i ) 0 W co μ i 0 ta μ Subtitute the umeical value of ta μ ad evaluate : ( 0.30) A uifom log with a ma of 00 kg, a legth of 4.0 m, ad a adiu of cm i held i a iclied poitio, a how i igue 44. The coefficiet of tatic fictio betwee the log ad the hoizotal uface i The log i o the vege of lippig to the ight. id the teio i the uppot wie ad the agle the wie make with the vetical wall. Pictue the Poblem et T the teio i the wie; the omal foce of the uface; ad f,ma µ the maimum foce of tatic fictio. ettig the poit at which the wie i attached to the log be the oigi, the cete of ma of the log i at (.838 m, m) ad the poit of cotact with the floo i at ( m,.594 m). Becaue the log i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium.
34 8 Chapte y T f,ma mg 3 Apply 0 to the log: T i f, ma 0 o T i f μ () Apply y 0 to the log: T co + mg 0 o T co mg (),ma Divide equatio () by equatio () to obtai: Solvig fo yield: Apply τ 0 about a ai though the oigi: T i μ T co mg μ ta (3) mg mg 3μ 0 Solve fo to obtai: mg + μ 3 Subtitute umeical value ad evaluate : ( )( 9.8m/ ) kg ( 0.60) 389 N
35 Static Equilibium ad Elaticity 9 Subtitute umeical value i equatio (3) ad evaluate : 0.60 ta 389 N.5 ( 00kg)( 9.8m/ ) Subtitute umeical value i equatio () ad evaluate T: T ( 0.60)( 389 N) i kn 39 [SSM] A tall, uifom, ectagula block it o a iclied plae a how i igue 45. A cod i attached to the top of the block to pevet it fom fallig dow the iclie. What i the maimum agle fo which the block will ot lide o the iclie? Aume the block ha a heighttowidth atio, b/a, of 4.0 ad the coefficiet of tatic fictio betwee it ad the iclie i μ Pictue the Poblem Coide what happe jut a iceae beyod ma. Becaue the top of the block i fied by the cod, the block will i fact otate with oly the lowe ight edge of the block emaiig i cotact with the plae. It follow that jut pio to thi lippig, ad f µ act at the lowe ight edge of the block. Chooe a coodiate ytem i which up the iclie i the + diectio ad the diectio of i the +y diectio. Becaue the block i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium. a y T b + mg f, ma Apply 0 to the block: T + μ mg i 0 () Apply y 0 to the block: mg co 0 () Apply τ 0 about a ai though the lowe ight edge of the block: ( mg ) + b( mg i ) bt 0 a co (3)
36 0 Chapte Elimiate betwee equatio () ad () ad olve fo T: T mg ( i μ co ) Subtitute fo T i equatio (3): a( mg co ) + b( mg i ) b[ mg( i μ co )] 0 Subtitute 4a fo b: a( mg co ) + ( 4.0a)( mg i ) ( 4.0a) [ mg( i μ co )] 0 Simplify to obtai: ( + 8.0μ ) co 4.0i 0 Solvig fo yield: Subtitute umeical value ad evaluate : Ste ad Stai ta μ + ta ( 8.0)( 0.80) A 50kg ball i upeded fom a teel wie of legth 5.0 m ad adiu.0 mm. By how much doe the wie tetch? Pictue the Poblem i the utetched legth of the wie, i the foce actig o it, ad A i it coectioal aea. The tetch i the wie Δ i elated to Youg modulu by Y ( A) ( Δ ). We ca ue Table  to fid the umeical value of Youg modulu fo teel. id the amout the wie i tetched fom Youg modulu: Y A Δ Δ YA Subtitute fo ad A to obtai: Subtitute umeical value ad evaluate Δ: mg Δ Yπ ( 50 kg)( 9.8m/ )( 5.0 m) Δ 3 ( 00 GN/m )( π)(.0 0 m) 0.98 mm 4 [SSM] Coppe ha a teile tegth of about N/m. (a) What i the maimum load that ca be hug fom a coppe wie of diamete 0.4 mm?
37 Static Equilibium ad Elaticity (b) If half thi maimum load i hug fom the coppe wie, by what pecetage of it legth will it tetch? Pictue the Poblem i the utetched legth of the wie, i the foce actig o it, ad A i it coectioal aea. The tetch i the wie Δ i elated to Y te tai A Δ Youg modulu by ( ) ( ). (a) Epe the maimum load i ma teile tegth A tem of the wie teile tegth: teile tegth π Subtitute umeical value ad evaluate ma : ma 8 3 ( N/m ) π ( 0. 0 m) 4.6 N 4 N (b) Uig the defiitio of Youg modulu, epe the factioal chage i legth of the coppe wie: Δ AY AY ma Δ ( 4.6 N) Δ : Subtitute umeical value ad evaluate π ( 0. mm) (.0 0 N/m ) 0.4% 4 A 4.0kg ma i uppoted by a teel wie of diamete 0.60 mm ad legth. m. How much will the wie tetch ude thi load? Pictue the Poblem i the utetched legth of the wie, i the foce actig o it, ad A i it coectioal aea. The tetch i the wie Δ i elated to Youg modulu by Y ( A) ( Δ ). We ca ue Table  to fid the umeical value of Youg modulu fo teel. Relate the amout the wie i tetched to Youg modulu: Y A Δ Δ YA Subtitute fo ad A to obtai: Subtitute umeical value ad evaluate Δ: mg Δ Yπ π 0 ( 4.0 kg)( 9.8m/ )(. m) Δ 0.83mm N/m 3 ( m)
38 Chapte 43 [SSM] A a ue foot puhe off o the goud, the heaig foce actig o a 8.0mmthick ole i how i igue 46. If the foce of 5 N i ditibuted ove a aea of 5 cm, fid the agle of hea, give that the hea modulu of the ole i N/m. Pictue the Poblem The hea te, defied a the atio of the heaig foce to the aea ove which it i applied, i elated to the hea tai though the defiitio hea te A of the hea modulu; M. hea tai ta Uig the defiitio of hea modulu, elate the agle of hea, to the hea foce ad hea modulu: ta ta M A M A Subtitute umeical value ad evaluate : 5 4 (.9 0 N/m )( 5 0 m ) ta 5 N A teel wie of legth.50 m ad diamete.00 mm i joied to a alumium wie of idetical dimeio to make a compoite wie of legth 3.00 m. id the eultig chage i the legth of thi compoite wie if a object with a ma of 5.00 kg i hug vetically fom oe of it ed. (Neglect ay effect the mae of the two wie have o the chage i thei legth.) Pictue the Poblem The tetch i the wie Δ i elated to Youg modulu by Y ( A) ( Δ ), whee i the utetched legth of the wie, i the foce actig o it, ad A i the coectioal aea of the wie. The chage i legth of the compoite wie i the um of the chage i legth of the teel ad alumium wie. The chage i legth of the compoite wie Δ i the um of the chage i legth of the two wie: Δ Δ + Δ teel Al
39 Static Equilibium ad Elaticity 3 Uig the defiig equatio fo Youg modulu, ubtitute fo Δ teel ad Δ Al i equatio () ad implify to obtai: Δ A Y teel teel A Y teel teel + A + Y Y Al Al Al Al Subtitute umeical value ad evaluate Δ: Δ ( 5.00 kg)( 9.8m/ ) 3 π ( m).8 mm.50 m.00 0 N/m.50 m N/m m 45 [SSM] Equal but oppoite foce of magitude ae applied to both ed of a thi wie of legth ad coectioal aea A. Show that if the wie i modeled a a pig, the foce cotat k i give by k AY/ ad the potetial eegy toed i the wie iu Δ, whee Y i Youg modulu ad Δ i the amout the wie ha tetched. Pictue the Poblem We ca ue Hooke law ad Youg modulu to how that, if the wie i coideed to be a pig, the foce cotat k i give by k AY/. By teatig the wie a a pig we ca how the eegy toed i the wie i U Δ. Epe the elatiohip betwee the tetchig foce, the foce cotat, ad the elogatio of a pig: Uig the defiitio of Youg modulu, epe the atio of the tetchig foce to the elogatio of the wie: kδ k Δ Δ AY () Equate thee two epeio fo /Δ to obtai: k AY Teatig the wie a a pig, epe it toed eegy: U AY k AYΔ Δ ( Δ) ( Δ)
40 4 Chapte Solvig equatio () fo yield: AYΔ Subtitute fo i the epeio fo U to obtai: U Δ 46 The teel E tig of a violi i ude a teio of 53.0 N. The diamete of the tig i 0.00 mm ad the legth ude teio i 35.0 cm. id (a) the utetched legth of thi tig ad (b) the wok eeded to tetch the tig. Pictue the Poblem et epeet the tetched ad the utetched legth of the wie. The tetch i the wie Δ i elated to Youg modulu by Y ( A) ( Δ ), whee i the foce actig o it, ad A i it coectioal aea. I Poblem 45 we howed that the eegy toed i the wie i U Δ, whee Y i Youg modulu ad Δ i the amout the wie ha tetched. (a) Epe the tetched legth of the wie: ' + Δ Uig the defiitio of Youg modulu, epe Δ: Subtitute ad implify: Δ AY ' + + AY AY Solvig fo yield: ' + AY Subtitute umeical value ad evaluate : + π m 53.0 N 3 ( m) (.00 0 N/m ) 34.7 cm (b) om Poblem 45, the wok doe i tetchig the wie i: W ΔU Δ Subtitute umeical value ad evaluate W: W ( 53.0 N)( m m) 0.08J
41 Static Equilibium ad Elaticity 5 47 Duig a mateial ciece epeimet o the Youg modulu of ubbe, you teachig aitat upplie you ad you team with a ubbe tip that i ectagula i co ectio. She tell you to fit meaue the co ectio dimeio ad thei value ae 3.0 mm.5 mm. The lab witeup call fo the ubbe tip to be upeded vetically ad vaiou (kow) mae to attached to it. You team obtai the followig data fo the legth of the tip a a fuctio of the load (ma) o the ed of the tip: oad, kg egth, cm (a) Ue a peadheet o gaphig calculato to fid Youg modulu fo the ubbe tip ove thi age of load. Hit: It i pobably bet to plot /A veu Δ/. Why? (b) id the eegy toed i the tip whe the load i 0.5 kg. (See Poblem 45.) (c) id the eegy toed i the tip whe the load i 0.30 kg. I it twice a much a you awe to Pat (b)? Eplai. Pictue the Poblem We ca ue the defiitio of Youg modulu ad you team data to plot a gaph whoe lope i Youg modulu fo the ubbe tip ove the give age of load. Becaue the ubbe tip tetche liealy fo load le tha o equal to 0.0 kg, we ca ue liea itepolatio i Pat (b) to fid the legth of the ubbe tip fo a load of 0.5 kg. We ca the ue the eult of Poblem 45 to fid the eegy toed i the whe the load i 0.5 kg. I Pat (c) we ca ue the eult of Poblem 45 ad the give legth of the tip whe it load i 0.30 kg to fid the eegy toed i the ubbe tip. (a) The equatio fo Youg modulu ca be witte a: Δ Y A whee Y i the lope of a gaph of /A a a fuctio of Δ/. The followig table ummaize the quatitie, calculated uig you team data, ued to plot the gaph uggeted i the poblem tatemet. oad /A Δ Δ/ U (kg) (N) (N/m ) (m) (J)
42 6 Chapte A peadheetgeeated gaph of /A a a fuctio of Δ/ follow. The peadheet pogam alo plotted the egeio lie o the gaph ad added it equatio to the gaph..e+06.0e+06 /A, N/m^ 8.0E E E+05 y.35e E+04.0E E delta / om the egeio fuctio how o the gaph: Y N/m (b) om Poblem 45: U Δ o, becaue mg, U m Δ ( ) mg Itepolatig fom the data table we ee that the legth of the tip whe the load o it i 0.5 kg i 5.9 cm. Subtitute umeical value ad evaluate U(0.5 kg): U ( 0.30 kg) ( 5.9 cm 5.0 cm)( 0.5 kg)( 9.8 m/ ) 7 mj (c) Evaluate U(0.30 kg) to obtai: U ( 0.30 kg) ( 6.9 cm 5.0 cm)( 0.30 kg)( 9.8 m/ ) 8 mj The eegy toed i the tip whe the load i 0.30 kg i fou time a much a the eegy toed whe the load i 0.5 kg. Although the ubbe tip doe ot tetch liealy (a cocluio you ca cofim eithe gaphically o by eamiig the data table), it tetch i ufficietly liea that, to a good appoimatio, the eegy toed i quadupled whe the load i doubled.
43 Static Equilibium ad Elaticity 7 48 A lage mio i hug fom a ail a how i igue 47. The uppotig teel wie ha a diamete of 0.0 mm ad a utetched legth of.7 m. The ditace betwee the poit of uppot at the top of the mio fame i.5 m. The ma of the mio i.4 kg. How much will the ditace betwee the ail ad the mio iceae due to the tetchig of the wie a the mio i hug? Pictue the Poblem The figue how the foce actig o the wie whee it pae ove the ail. m epeet the ma of the mio ad T i the teio i the uppotig wie. The figue alo how the geomety of the ight tiagle defied by the uppot wie ad the top of the mio fame. The ditace a i fied by the geomety while h ad will chage a the mio i upeded fom the ail. Uig the Pythagoea theoem, epe the elatiohip betwee the ide of the ight tiagle i the diagam: a + h T ' y mg h by ail ail T 0.85 m a 0.75 m Epe the diffeetial of thi equatio ad appoimate diffeetial chage with mall chage: Multiplyig the umeato ad deomiato by yield: Solve the equatio defiig Youg modulu fo Δ/ to obtai: Subtitute fo Δ/ i equatio () to obtai: Notig that T T', apply y 0 to the wie whee it pae ove the uppotig ail: aδa + hδh Δ o, becaue Δa 0, Δ hδh Δ Δ h h Δ Δh () h Δ T AY T T Δh () h AY a π Y whee i the adiu of the wie. mg T co 0 T mg co
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