Chapter 12 Static Equilibrium and Elasticity

Transcription

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 ee-aw. A they walk, the ee-aw 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 ee-aw, 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 ee-aw) 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 ee-aw 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 ee-aw (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 ed-to-ed 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 -cm-high 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 0-lb 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 0-lb weight at the ed of you am, equatio () become: ' deltoid i mg m'g 0 whee m i the ma of the 0-lb 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 5-foot ailboat. The mat i a uifom 0-kg 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 0.0-m 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 60.0-kg 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 3.0-m 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 60-kg 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 8-kg 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 5.0-m 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 80-kg 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 00-N 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 fee-body 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 0-m 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 80-N 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 left-had 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 900-N 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 height-to-width 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 50-kg 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 co-ectioal 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 co-ectioal 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 4.0-kg 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 co-ectioal 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 8.0-mm-thick 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 co-ectioal 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 co-ectioal 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 co-ectioal 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 wite-up 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 peadheet-geeated 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

44 8 Chapte Subtitutig fo T i equatio () yield: Δ h a mg π Y co Becaue h a co : Δh a mg π Y π Y 3 ( a ) mg a Subtitute umeical value ad evaluate Δh: Δh π (.4 kg)( 9.8 m/ )( 0.85 m) [ ] 3 ( m) (.00 0 N/m )( 0.85 m) ( 0.75 m) 3 7. mm 49 Two mae, M ad M, ae uppoted by wie that have equal legth whe utetched. The wie uppotig M i a alumium wie 0.70 mm i diamete, ad the oe uppotig M i a teel wie 0.50 mm i diamete. What i the atio M /M if the two wie tetch by the ame amout? Pictue the Poblem et the umeal deote the alumium wie ad the umeal the teel wie. Becaue thei iitial legth ad amout they tetch ae the ame, we ca ue the defiitio of Youg modulu to epe the chage i the legth of each wie ad the equate thee epeio to obtai a equatio olvable fo the atio M /M. Uig the defiitio of Youg modulu, epe the chage i legth of the alumium wie: Uig the defiitio of Youg modulu, epe the chage i legth of the teel wie: Becaue the two wie tetch by the ame amout, equate Δ ad Δ ad implify: Mg Δ AY Al M g Δ A Y teel M M M AY Al Al AY teel M AY teel AY

45 Static Equilibium ad Elaticity 9 Subtitute umeical value ad evaluate M /M : M M π 4 π ( 0.70 mm) ( N/m ) ( 0.50 mm) (.00 0 N/m ) 50 A 0.50-kg ball i attached to oe ed of a alumium wie havig a diamete of.6 mm ad a utetched legth of 0.70 m. The othe ed of the wie i fied to the top of a pot. The ball otate about the pot i a hoizotal plae at a otatioal peed uch that the agle betwee the wie ad the hoizotal i 5.0º. id the teio i the wie ad the iceae i it legth due to the teio i the wie. Pictue the Poblem The fee-body diagam how the foce actig o the ball a it otate aoud the pot i a hoizotal plae. We ca apply Newto d law to fid the teio i the wie ad ue the defiitio of Youg modulu to fid the amout by which the alumium wie tetche. T y m mg Apply y 0 to the ball: Subtitute umeical value ad evaluate T: T i mg 0 T T ( 0.50 kg)( 9.8m/ ) 56 N i5.0 mg i 56.3 N Uig the defiitio of Youg modulu, epe Δ: Subtitute umeical value ad evaluate Δ: Δ Δ AY π 4 ( 56.3N)( 0.70m) 3 (.6 0 m) ( N/m ) 0.8mm 5 [SSM] A elevato cable i to be made of a ew type of compoite developed by Acme aboatoie. I the lab, a ample of the cable that i.00 m log ad ha a co-ectioal aea of 0.00 mm fail ude a load of 000 N. The actual cable ued to uppot the elevato will be 0.0 m log ad have a co-

46 30 Chapte ectioal aea of.0 mm. It will eed to uppot a load of 0,000 N afely. Will it? Pictue the Poblem We ca ue the defiitio of te to calculate the failig te of the cable ad the te o the elevato cable. Note that the failig te of the compoite cable i the ame a the failig te of the tet ample. The te o the elevato cable i: The failig te of the ample i: Ste Ste A cable A 0.0 kn.0 0 m 0 N/m failig N 0. 0 m N/m Becaue Ste < Ste, the cable will ot uppot the elevato. failig cable 5 If a mateial deity emai cotat whe it i tetched i oe diectio, the (becaue it total volume emai cotat), it legth mut deceae i oe o both of the othe diectio. Take a ectagula block of legth, width y, ad depth z, ad pull o it o that it ew legth ' + Δ. If Δ << ad Δ y y Δz z, how that Δ y y Δ. Pictue the Poblem et the legth of the ide of the ectagle be, y ad z. The the volume of the ectagle will be V yz ad we ca epe the ew volume V eultig fom the pullig i the diectio ad the chage i volume ΔV i tem of Δ, Δy, ad Δz. Dicadig the highe ode tem i ΔV ad dividig ou equatio by V ad uig the give coditio that Δy/y Δz/z will lead u to the give epeio fo Δy/y. Epe the ew volume of the ectagula bo whe it ide chage i legth by Δ, Δy, ad Δz: V' ( + Δ)( y + Δy)( z + Δz) yz + Δ( yz) + Δy( z) + Δz( y) + { zδδy + yδδz + ΔyΔz + ΔΔyΔz} whee the tem i backet ae vey mall (i.e., ecod ode o highe). Dicad the ecod ode ad highe tem to obtai: V' V + Δ o Δ V V' V Δ ( yz) + Δy( z) + Δz( y) ( yz) + Δy( z) + Δz( y) Becaue ΔV 0: Δ ( yz) [ Δy( z) + Δz( y) ]

47 Static Equilibium ad Elaticity 3 Divide both ide of thi equatio by V yz to obtai: Δ Δy Δz + y z Becaue Δy/y Δz/z, ou equatio become: Δ Δy y Δy y Δ 53 [SSM] You ae give a wie with a cicula co-ectio of adiu ad a legth. If the wie i made fom a mateial whoe deity emai cotat whe it i tetched i oe diectio, the how that Δ Δ, aumig that Δ <<. (See Poblem 5.) Pictue the Poblem We ca evaluate the diffeetial of the volume of the wie ad, uig the aumptio that the volume of the wie doe ot chage ude tetchig ad that the chage i it legth i mall compaed to it legth, how that Δ. Δ Epe the volume of the wie: V π Evaluate the diffeetial of V to obtai: dv π d + π d Becaue dv 0: 0 d + d d d Becaue Δ <<, we ca appoimate the diffeetial chage d ad d with mall chage Δ ad Δ to obtai: Δ Δ 54 o mot mateial lited i Table -, the teile tegth i two to thee ode of magitude lowe tha Youg modulu. Coequetly, mot of thee mateial will beak befoe thei tai eceed pecet. Of ma-made mateial, ylo ha about the geatet eteibility it ca take tai of about 0. befoe beakig. But pide ilk beat aythig ma-made. Cetai fom of pide ilk ca take tai o the ode of 0 befoe beakig! (a) If uch a thead ha a cicula co-ectio of adiu 0 ad utetched legth 0, fid it ew adiu whe tetched to a legth 0 0. (Aume that the deity of the thead emai cotat a it tetche.) (b) If the Youg modulu of the pide thead i Y, calculate the teio eeded to beak the thead i tem of Y ad 0. Pictue the Poblem Becaue the deity of the thead emai cotat duig the tetchig poce, we ca equate the iitial ad fial volume to epe 0 i tem of. We ca alo ue Youg modulu to epe the teio eeded to beak the thead i tem of Y ad 0.

48 3 Chapte (a) Becaue the volume of the thead i cotat duig the tetchig of the pide ilk: π π Subtitute fo ad implify to 0 obtai: (b) Epe Youg modulu i tem of the beakig teio T: Y T A Δ T π Δ 0T π Δ 0 Solvig fo T yield: T Δ π 0 Y 0 Becaue Δ/ 9: T 9π 0 Y 0 Geeal Poblem 55 [SSM] A tadad bowlig ball weigh 6 poud. You wih to hold a bowlig ball i fot of you, with the elbow bet at a ight agle. Aume that you bicep attache to you foeam at.5 cm out fom the elbow joit, ad that you bicep mucle pull vetically upwad, that i, it act at ight agle to the foeam. Alo aume that the ball i held 38 cm out fom the elbow joit. et the ma of you foeam be 5.0 kg ad aume it cete of gavity i located 9 cm out fom the elbow joit. How much foce mut you bicep mucle apply to foeam i ode to hold out the bowlig ball at the deied agle? Pictue the Poblem We ca model the foeam a a cylide of legth 38 cm with the foce how i the pictoial epeetatio actig o it. Becaue the foeam i i both talatioal ad otatioal equilibium ude the ifluece of thee foce, the foce i the diagam mut add (vectoially) to zeo ad the et toque with epect to ay ai mut alo be zeo. bicep 0 elbow elbow joit m foeam g m ball g

49 Static Equilibium ad Elaticity 33 Apply τ 0about a ai though the elbow ad pepedicula to the plae of the diagam: m m bicep foeam ball g g 0 Solvig fo bicep ad implifyig yield: bicep m g + m ( m + m ) foeam foeam ball ball g g Subtitute umeical value ad evaluate bicep : kg ( 5.0 kg) + 6 lb ( 38 cm)( 9.8 m/ ) bicep.05 lb.5 cm.5 kn 56 A biology laboatoy at you uiveity i tudyig the locatio of a peo cete of gavity a a fuctio of thei body weight. They pay well, ad you decide to volutee. The locatio of you cete of gavity whe tadig eect i to be detemied by havig you lie o a uifom boad (ma of 5.00 kg, legth.00 m) uppoted by two cale a how i igue -54. If you height i 88 cm ad the left cale ead 470 N while the ight cale ead 430 N, whee i you cete of gavity elative to you feet? Aume the cale ae both eactly the ame ditace fom the two ed of the boad, ae epaated by 78 cm, ad ae et to each ead zeo befoe you get o the platfom. Pictue the Poblem Becaue the you-boad ytem i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to elate the foce eeted by the cale to the ditace d, meaued fom you feet, to you cete of ma ad the ditace to the cete of gavity of the boad. The followig pictoial epeetatio how the foce actig o the boad. mg epeet you weight. 470 N 78 cm d 89 cm 430 N P ( 5.00 kg)g mg

50 34 Chapte The foce epoible fo a couteclockwie toque about a ai though you feet (poit P) ad pepedicula to the page ae you weight ad the weight of the boad. The oly foce cauig a clockwie toque about thi ai i the 470 N foce eeted by the cale ude you head. Apply τ 0 about a ai though you feet ad pepedicula to the page: m whee m i you ma. Solve fo d to obtai: ( 9.8 m/ ) d ( 5.0 kg)( 0.89 m) (.78 m)( 470 N) 0 d ( 5.00 kg)( 0.89 m) + (.78 m)( 470 N) () m ( 9.8 m/ ) y et upwad be the poitive y diectio ad apply 0 to the plak to obtai: 470 N N m ( 9.8 m/ ) ( 5.00 kg)( 9.8 m/ ) 0 Solvig fo m yield: m kg Subtitute umeical value i equatio () ad evaluate d: d ( 5.0 kg)( 0.89 m) + (.78 m)( 470 N) ( kg)( 9.8 m/ ) 99 cm 57 igue -49 how a mobile coitig of fou object hagig o thee od of egligible ma. id the value of the ukow mae of the object if the mobile i to balace. Hit: id the ma m fit. Pictue the Poblem We ca apply the balace coditio τ 0 ucceively, tatig with the lowet pat of the mobile, to fid the value of each of the ukow weight. Apply τ 0 about a ai though the poit of upeio of the lowet pat of the mobile: ( 3.0cm)(.0 N) ( 4.0cm) m g 0 Solvig fo m yield: m 0.59 kg 0.5kg

51 Static Equilibium ad Elaticity 35 Apply τ 0 about a ai though the poit of upeio of the middle pat of the mobile:.0 N g (.0cm) m g ( 4.0cm) kg g 0 Solvig fo m yield: m kg 0.7kg Apply τ 0 about a ai though the poit of upeio of the top pat of the mobile: (.0cm) (.0 N + ( kg) g + ( 0.59) g) ( 6.0cm) m3 g 0 Solvig fo m 3 yield: m kg kg 58 Steel cotuctio beam, with a iduty deigatio of W, have a weight of poud pe foot. A ew buie i tow ha hied you to place it ig o a 4.0 m log teel beam of thi type. The deig call fo the beam to eted outwad hoizotally fom the fot bick wall (igue -50). It i to be held i place by a 5.0 m-log teel cable. The cable i attached to oe ed of the beam ad to the wall above the poit at which the beam i i cotact with the wall. Duig the iitial tage of cotuctio, the beam i ot to be bolted to the wall, but to be held i place olely by fictio. (a) What i the miimum coefficiet of fictio betwee the beam ad the wall fo the beam to emai i tatic equilibium? (b)what i the teio i the cable i thi cae? Pictue the Poblem Becaue the beam i i both talatioal ad otatioal equilibium ude the ifluece of the foce how below i the pictoial epeetatio, we ca apply 0 ad τ 0 to it to fid the coefficiet of tatic fictio ad the teio i the uppotig cable. 0 f m beam g T

52 36 Chapte (a) Apply 0 obtai: to the beam to The ma of the beam m beam i the poduct of it liea deity λ ad legth : Subtitutig fo yield: m beam i equatio () T co 0 () ad y f m g + T i 0 () mbeam λ beam f λg + T i 0 (3) Relate the foce of tatic fictio to the omal foce eeted by the wall: f μ Subtitutig fo f i equatio (3) yield: Solve fo μ to obtai: Solvig equatio () fo yield: Subtitute fo i equatio (4) ad implify to obtai: Apply τ 0 to the beam about a ai though the oigi ad omal to the page to obtai: Solvig fo T yield: μ λg + T i λg T i μ (4) T co λg T i λg μ ta (5) T co T co ( T ) m g( ) 0 i o T i λg 0 beam ( ) ( ) 0 λg T (6) i Subtitute fo T i equatio (5) ad implify to obtai: om igue -50 we ee that: μ λg ta λg co i 3.0 m ta 4.0 m 3 4 ta (7)

53 Static Equilibium ad Elaticity 37 Subtitutig fo ta i equatio (7) yield: μ 0.75 (b) Subtitute umeical value i equatio (6) ad evaluate T: T lb ft kg.05 lb 3.8ft m 3 i ta 4 ( 4.0 m)( 9.8 m/ ).kn Remak:. kn i appoimately 40 lb. 59 [SSM] Coide a igid.5-m-log beam (igue -5) that i uppoted by a fied.5-m-high pot though it cete ad pivot o a fictiole beaig at it cete atop the vetical.5-m-high pot. Oe ed of the beam i coected to the floo by a pig that ha a foce cotat k 50 N/m. Whe the beam i hoizotal, the pig i vetical ad uteed. If a object i hug fom the oppoite ed of the beam, the beam ettle ito a equilibium poitio whee it make a agle of 7.5 o with the hoizotal. What i the ma of the object? Pictue the Poblem Becaue the beam i i otatioal equilibium, we ca apply τ 0 to it to detemie the ma of the object upeded fom it left ed. by beaig ϕ α co 0 by pig i ( ) i mg α β δ The pictoial epeetatio diectly above how the foce actig o the beam whe it i i tatic equilibium. The pictoial epeetatio to the ight i a elaged view of the ight ed of the beam. We ll ue thi diagam to detemie the legth of the tetched pig. tetched ϕ

54 38 Chapte Apply τ 0 to the beam about a ai though the beaig poit to obtai: ( ) ( iϕ ) 0 mg co by pig o, becaue k, by pig Δ pig mg co kδ pig iϕ 0 () Ue the ight-had diagam above to elate the agle ϕ ad : ϕ ta ta co + i co + i Subtitute fo ϕ i equatio () to obtai: mg co kδ i ta co + i pig 0 Solve fo m to obtai: m kδ pig co i ta + i g co () Δ pig i give by: To fid the value of tetched, efe to the ight-had diagam ad ote that: Δ pig tetched utetched o, becaue, utetched Δ (3) pig tetched π α + π α Agai, efeig to the diagam, π β α + elate β to α: Subtitutig fo α yield: π π β + π Apply the law of coie to the tiagle defied with bold ide: tetched ( ) + ( i ) ( )( i ) co( π ) Ue the fomula fo the coie of the diffeece of two agle to obtai: co ( ) co π

55 Static Equilibium ad Elaticity 39 Subtitutig fo ( π ) co yield: tetched ( ) + ( i ) + ( )( i ) + i + ( i co ) 4 co Ue the tigoometic idetitie obtai: i i co ad co i to tetched 4 co + + ( i ) Simplifyig yield: [ 3 ( co i )] o Subtitutig fo tetched i equatio (3) yield: tetched tetched 4 3 ( co i ) ( ) ( co i ) 3 ( co i ) Δ pig 3 Subtitute fo Δ pig i equatio () to obtai: m k ( 3 ( co i ) ) Subtitute umeical value ad evaluate m: i ta g co co + i m ( ) ( 50 N/m)(.5 m) 3 ( co7.5 i7.5 ).8 kg ( 9.8 m/ ) co7.5 i ta co7.5 + i A ope ad pulley ytem, called a block ad tackle, i ued to aie a object of ma M (igue -5) at cotat peed. Whe the ed of the ope move dowwad though a ditace, the height of the lowe pulley i iceaed by h.

56 40 Chapte (a) What i the atio /h? (b) Aume that the ma of the block ad tackle i egligible ad that the pulley beaig ae fictiole. Show that mgh by applyig the wok eegy piciple to the block tackle object. Pictue the Poblem We ca detemie the atio of to h by otig the umbe of ope uppotig the load whoe ma i M. (a) Notig that thee ope uppot the pulley to which the object whoe ma i M i fateed we ca coclude that: h 3 (b) Apply the wok-eegy piciple to the block-tackle object to obtai: W et ΔE o mgh ytem ΔU block-tackle 6 A plate of ma M i the hape of a equilateal tiagle i upeded fom oe coe ad a ma m i upeded fom aothe of it coe. If the bae of the tiagle make a agle of 6.0º with the hoizotal, what i the atio m/m? Pictue the Poblem The figue how the equilateal tiagle without the ma m, ad the the ame tiagle with the ma m ad otated though a agle. et the ide legth of the tiagle to be a. The the cete of ma of the tiagle i at a ditace of a fom each vete. A the tiagle 3 otate, it cete of ma hift a by, fo <<. Alo, the vete 3 to which m i attached move towad the plumb lie by the ditace d a co30 3 a (ee the dawig). Apply τ 0 about a ai though the poit of upeio: mg a ( a 3 a ) Mg 0 3 Solvig fo m/m yield: m M ( 3 ) 3

57 Static Equilibium ad Elaticity 4 Subtitute umeical value ad evaluate m/m: m M 3 ( 6.0 ) 3 π ad 80 π ad 80 ( 6.0 ) A tadad i-ided pecil i placed o a pad of pape (igue -53) id the miimum coefficiet of tatic fictio μ uch that, if the pad i iclied, the pecil oll dow the iclie athe tha lidig. Pictue the Poblem If the heago i to oll athe tha lide, the iclie agle mut be uch that the cete of ma fall jut beyod the uppot bae. om the geomety of the heago, the citical agle i 30. The fee-body diagam how the foce actig o the heagoal pecil whe it i o the vege of lidig. We ca ue Newto d law to elate the coefficiet of tatic fictio to the agle of the iclie fo which ollig athe tha lidig occu. f,ma mg y Apply 0 to the pecil: mg i f 0 (), ma ad y mg co 0 () Subtitute f,ma µ i equatio (): Divide equatio (3) by equatio () to obtai: mg i μ 0 (3) ta μ Thu, if the pecil i to oll athe tha lide whe the pad i iclied: μ ta A 8.0-kg bo that ha a uifom deity ad i twice a tall a it i wide et o the floo of a tuck. What i the maimum coefficiet of tatic fictio betwee the bo ad floo o that the bo will lide towad the ea of the tuck athe tha tip whe the tuck acceleate fowad o a level oad? Pictue the Poblem The bo ad the foce actig o it ae how i the figue. The foce acceleatig the bo i the tatic fictio foce. Whe the bo i about to

58 4 Chapte tip, act at it edge, a idicated i the dawig. We ca ue the defiitio of µ ad apply the coditio fo otatioal equilibium i a acceleated fame to elate f to the weight of the bo ad, hece, to the omal foce. w w f mg Uig it defiitio, epe µ : μ f Apply τ 0 about a ai though the bo cete of ma: f Subtitute fo to obtai the coditio fo tippig: wf w μ 0.50 f 0 Theefoe, if the bo i to lide: μ A balace cale ha uequal am. The cale i balaced with a.50-kg block o the left pa ad a.95 kg block o the ight pa (igue -54). If the.95-kg block i emoved fom the ight pa ad the.50-kg block i the moved to the ight pa, what ma o the left pa will balace the cale? Pictue the Poblem Becaue the balace i i equilibium, we ca ue the coditio fo otatioal equilibium to elate the mae of the block to the leve am of the balace i the two cofiguatio decibed i the poblem tatemet. < Apply τ 0 about a ai though the fulcum: (.50kg) (.95kg) 0

59 Solvig fo yield:. 30 Static Equilibium ad Elaticity 43 Apply τ 0 about a ai though the fulcum with.50 kg at : M (.50 kg) 0 Solvig fo M yield: (.50 kg).50kg M Subtitute fo / ad evaluate M: M.50kg.30.5kg 65 [SSM] A cube lea agait a fictiole wall makig a agle of with the floo a how i igue -55. id the miimum coefficiet of tatic fictio μ betwee the cube ad the floo that i eeded to keep the cube fom lippig. Pictue the Poblem et the ma of the cube be M. The figue how the locatio of the cube cete of ma ad the foce actig o the cube. The oppoig couple i fomed by the fictio foce f,ma ad the foce eeted by the wall. Becaue the cube i i equilibium, we ca ue the coditio fo talatioal equilibium to etablih that f, ma W ad Mg ad the coditio fo otatioal equilibium to elate the oppoig couple. f, ma P y d Mg a W a i Apply 0 to the cube: y Mg 0 Mg ad f W 0 W f Notig that f, ma ad W fom a couple (thei magitude ae equal), a do ad M g, apply τ 0 about a ai though poit P to obtai: f ai Mgd, ma 0

60 44 Chapte Refeig to the diagam to the ight, a ote that d i( 45 + ). Mg a Subtitute fo d ad f,ma to obtai: μ i i( 45 + ) 0 Solve fo µ ad implify to obtai: 45 d Mga Mg o μ i i a ( 45 + ) 0 μ i i ( 45 + ) ( i 45 co + co45 i ) i co + i i ( cot + ) 66 igue -56 how a 5.00-kg od higed to a vetical wall ad uppoted by a thi wie. The wie ad od each make agle of 45º with the vetical. Whe a 0.0-kg block i upeded fom the midpoit of the od, the teio T i the uppotig wie i 5.0 N. If the wie will beak whe the teio eceed 75 N, what i the maimum ditace fom the hige at which the block ca be upeded? Pictue the Poblem Becaue the od i i equilibium, we ca apply the coditio fo otatioal equilibium to fid the maimum ditace fom the hige at which the block ca be upeded. Apply τ 0 about a ai though the hige to obtai: (.00 m)( 75 N) ( 0.50m)( 5.00 kg)( 9.8m/ ) d( 0.0 kg)( 9.8m/ ) co 45 0 co 45 Solvig fo d yield: d 83cm

61 Static Equilibium ad Elaticity [SSM] igue -57 how a 0.0-kg ladde leaig agait a fictiole wall ad etig o a fictiole hoizotal uface. To keep the ladde fom lippig, the bottom of the ladde i tied to the wall with a thi wie. Whe o oe i o the ladde, the teio i the wie i 9.4 N. (The wie will beak if the teio eceed 00 N.) (a) If a 80.0-kg peo climb halfway up the ladde, what foce will be eeted by the ladde agait the wall? (b) How fa fom the bottom ed of the ladde ca a 80.0-kg peo climb? Pictue the Poblem et m epeet the ma of the ladde ad M the ma of the peo. The foce diagam how the foce actig o the ladde fo Pat (b). om the coditio fo talatioal equilibium, we ca coclude that T by wall, a eult we ll eed i Pat (b). Becaue the ladde i alo i otatioal equilibium, ummig the toque about the bottom of the ladde will elimiate both ad T. 0 by wall T mg Mg (a) Apply τ 0 about a ai though the bottom of the ladde: ( ) mg( co ) Mg( co ) 0 i by wall Solve fo by wall ad implify to obtai: Refe to igue -57 to detemie : Subtitute umeical value ad evaluate by wall: by wall ( m + M ) g co ( m + M ) i 5.0 m ta m by wall g ta ( 0 kg + 80 kg)( 9.8 m/ ) 0.5 kn ta 73.3 (b) Apply 0 to the ladde to obtai: T by wall 0 T by wall

62 46 Chapte Apply τ 0 about a ai though the bottom of the ladde ubject to the coditio that T : by wall ma ( ) mg( co ) Mg( co ) 0 Tma i whee i the maimum ditace alog the ladde that the peo ca climb without eceedig the maimum teio i the wie. Solvig fo ad implifyig yield: Subtitute umeical value ad evaluate : T ma i mg co Mg co ( 00 N)( 5.0 m) ( 0 kg)( 9.8 m/ )(.5 m) ( 80 kg)( 9.8 m/ ) co m 68 A 360-kg object i uppoted o a wie attached to a 5-m-log teel ba that i pivoted at a vetical wall ad uppoted by a cable a how i igue -58. The ma of the ba i 85 kg. With the cable attached to the ba 5.0 m fom the lowe ed a how, fid the teio i the cable ad the foce eeted by the wall o the teel ba. Pictue the Poblem et m epeet the ma of the ba, M the ma of the upeded object, v the vetical compoet of the foce the wall eet o the ba, h the hoizotal compoet of the foce the wall eet o the ba, ad T the teio i the cable. The foce diagam how thee foce ad thei poit of applicatio o the ba. Becaue the ba i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to elate the vaiou foce ad ditace. Apply τ 0 to the ba about a ai though the hige: y v 0 T T h 3 mg co Mg 3 co mg 0 Mg Solvig fo T yield: ( m + M ) T 3 g co

63 Static Equilibium ad Elaticity 47 Subtitute umeical value ad evaluate T: T ( ( )( ) ( )( ) )( 85 kg 7.5 m kg 5 m 9.8 m/ ) co kn 0 kn 5.0 m The magitude ad diectio of the foce eeted by the wall ae give by: + () by wall v h ad v φ ta () h Apply 0 to the ba to obtai: co30 y + T mg Mg 0 v ad T i 30 0 h v m M g T co Solvig the y equatio fo v yield: ( + ) 30 Solve the equatio fo h to obtai: h T i 30 Subtitutig fo v ad h i equatio () yield: (( m + M ) g T co 30 ) + ( i ) by wall T 30 Subtitute umeical value ad evaluate by wall : by wall ( 85 kg kg)( 9.8 m/ ) ( 0.3 kn) co30 ) + ( 0.3 kn) 6.9 kn ( i 30 ) Subtitutig fo v ad h i equatio () yield: φ ta ( m + M ) g T T i30 co30 Subtitute umeical value ad evaluate φ: φ ta ( 85 kg kg)( 9.8 m/ ) ( 0.3 kn) ( 0.3 kn) i 30 4 That i, 4 below the hoizotal. co

64 48 Chapte 69 Repeat Poblem 63 if the tuck acceleate up a hill that make a agle of 9.0º with the hoizotal. Pictue the Poblem The bo ad the foce actig o it ae how i the figue. Whe the bo i about to tip, act at it edge, a idicated i the dawig. We ca ue the defiitio of µ ad apply the coditio fo otatioal equilibium i a acceleated fame to elate f to the weight of the bo ad, hece, to the omal foce. y f w + mg w Uig it defiitio, epe µ : μ f Apply τ 0 about a ai though the bo cete of ma: f Subtitute fo to obtai the coditio fo tippig: wf w μ 0.50 f 0 Theefoe, if the bo i to lide: μ 0. 50, a i Poblem 63. < Remak: The diffeece betwee poblem 63 ad 69 i that i 63 the maimum acceleatio befoe lippig i 0.5g, wheea i 69 it i (0.5 co9.0 i9.0 )g 0.337g. 70 A thi uifom od 60 cm log i balaced 0 cm fom oe ed whe a object whoe ma i (m +.0 gam) i at the ed eaet the pivot ad a object of ma m i at the oppoite ed (igue -59a). Balace i agai achieved if the object whoe ma i (m +.0 gam) i eplaced by the object of ma m ad o object i placed at the othe ed (igue -59b). Detemie the ma of the od. Pictue the Poblem et the ma of the od be epeeted by M. Becaue the od i i equilibium, we ca apply the coditio fo otatioal equilibium to elate the mae of the object placed o the od to it ma.

65 Static Equilibium ad Elaticity 49 Apply τ 0 about a ai though the pivot fo the iitial coditio: ( 0cm)( m +.0g) ( 40cm) ( 0cm) M 0 m Simplifyig yield: ( m +.0g) 4m M 0 Solve fo M to obtai: M 4.0 g 7 [SSM] Thee ae a lage umbe of idetical uifom bick, each of legth. If they ae tacked oe o top of aothe legthwie (ee igue -60), the maimum offet that will allow the top bick to et o the bottom bick i /. (a) Show that if thi two-bick tack i placed o top of a thid bick, the maimum offet of the ecod bick o the thid bick i /4. (b) Show that, i geeal, if you build a tack of N bick, the maimum ovehag of the ( )th bick (coutig dow fom the top) o the th bick i /. (c) Wite a peadheet pogam to calculate total offet (the um of the idividual offet) fo a tack of N bick, ad calculate thi fo 0 cm ad N 5, 0, ad 00. (d) Doe the um of the idividual offet appoach a fiite limit a N? If o, what i that limit? Pictue the Poblem et the weight of each uifom bick be w. The dowwad foce of all the bick above the th bick mut act at it coe, becaue the upwad eactio foce poit though the cete of ma of all the bick above the th oe. Becaue thee i o vetical acceleatio, the upwad foce eeted by the ( + )th bick o the th bick mut equal the total weight of the bick above it. Thu thi foce i jut w. Note that it i coveiet to develop the geeal elatiohip of Pat (b) iitially ad the etact the awe fo Pat (a) fom thi geeal eult. dowwad foce fom block above; thei cete of ma i diectly above thi edge i + w of block d P upwad foce fom coe of block + w N

66 50 Chapte (a) ad (b) Notig that the lie of actio of the dowwad foce eeted by the block above the th block pae though the poit P, eultig i a leve am of zeo, apply 0 P to the th bick to obtai: τ Solvig fo d yield: w ( ) wd 0 whee d i the ovehag of the th bick beyod the edge of the ( + )th bick. d whee,,3, o block umbe : d 4 (c) A peadheet pogam to calculate the um of the offet a a fuctio of i how below. The fomula ued to calculate the quatitie i the colum ae a follow: Cell omula/cotet Algebaic om B5 B4+ + C5 C4+$B$/(*B5) d + A B C D 0.0 m 3 offet

67 Static Equilibium ad Elaticity 5 om the table we ee that d 5 5 cm, d 0 6 cm, ad d cm. (d) The um of the idividual offet S i give by: S N d N Becaue thi eie i a hamoic eie, S appoache ifiity a the umbe of block N gow without boud. The followig gaph, plotted uig a peadheet pogam, ugget that S ha o limit. Offet a a fuctio of fo 0 cm Offet, m A uifom phee of adiu R ad ma M i held at et o a iclied plae of agle by a hoizotal tig, a how i igue -6. et R 0 cm, M 3.0 kg, ad 30º. (a) id the teio i the tig. (b) What i the omal foce eeted o the phee by the iclied plae? (c) What i the fictioal foce actig o the phee?

68 5 Chapte Pictue the Poblem The fou foce actig o the phee: it weight, mg; the omal foce of the plae, ; the fictioal foce, f, actig paallel to the plae; ad the teio i the tig, T, ae how i the figue. Becaue the phee i i equilibium, we ca apply the coditio fo talatioal ad otatioal equilibium to fid f,, ad T. y Mg T f (a) Apply τ 0 about a ai though the cete of the phee: fr TR 0 T f Apply 0 to the phee: f + T co Mg i 0 Subtitutig fo f ad olvig fo T yield: Subtitute umeical value ad evaluate T: i T Mg + co T ( 3.0kg)( 9.8m/ ) 7.9 N i N + co30 (b) Apply y 0 to the phee: T i Mg co 0 Solve fo : T i + Mg co Subtitute umeical value ad evaluate : ( 7.89 N) + 9 N i30 ( 3.0kg)( 9.8m/ ) co30 (c) I Pat (a) we howed that f T: f 7.9 N 73 The leg of a tipod make equal agle of 90º with each othe at the ape, whee they joi togethe. A 00-kg block hag fom the ape. What ae the compeioal foce i the thee leg?

69 Static Equilibium ad Elaticity 53 Pictue the Poblem et be the legth of each leg of the tipod. Applyig the Pythagoea theoem lead u to coclude that the ditace a how i the figue i 3 ad the ditace b, the ditace to the cetoid of the tiagle ABC i 3 3, ad the ditace c i 3. Thee eult allow u to coclude that co 3. Becaue the tipod i i equilibium, we ca apply the coditio fo talatioal equilibium to fid the compeioal foce i each leg. ettig C epeet the compeioal foce i a leg of the tipod, apply 0 to the ape of the tipod: mg 3 C co mg 0 C 3co Subtitute fo co ad implify to obtai: mg 3 3 C 3 3 mg Subtitute umeical value ad evaluate C : 3 C 3 ( 00kg)( 9.8m/ ) 566 N 74 igue -63 how a 0-cm-log uifom beam etig o a cylide that ha a adiu of 4.0-cm. The ma of the beam i 5.0 kg ad that of the cylide i 8.0 kg. The coefficiet of tatic fictio betwee beam ad cylide i zeo, wheea the coefficiet of tatic fictio betwee the cylide ad the floo, ad betwee the beam ad the floo, ae ot zeo. Ae thee ay value fo thee coefficiet of tatic fictio uch that the ytem i i tatic equilibium? If o, what ae thee value? If ot, eplai why oe eit. Pictue the Poblem The foce that act o the beam ae it weight, mg; the foce of the cylide, c, actig alog the adiu of the cylide; the omal foce of the goud, ; ad the fictio foce f µ. The foce actig o the cylide ae it weight, Mg; the foce of the beam o the cylide, cb c i magitude, actig adially iwad; the omal foce of the goud o the cylide, c ; ad the foce of fictio, f c µ c c. Chooe the coodiate ytem how i the figue ad apply the coditio fo otatioal ad talatioal equilibium.

70 54 Chapte y c 90 ο cb Mg mg c f c f Epe : μ, beam floo i tem of f ad f μ,beam floo () Epe μ ad c :, cylide floo i tem of f c Apply τ 0 about a ai though the ight ed of the beam: f c μ,cylide floo () c [( 0cm) co ] mg ( 5cm) c 0 Solve fo c to obtai: [( 0cm) co ] c 5cm mg Subtitute umeical value ad evaluate c : c [ 0co30 ]( 5.0 kg)( 9.8m/ ) 5 8.3N Apply y 0 to the beam: + c co mg 0 Solvig fo yield: mg c co Subtitute umeical value ad evaluate : ( 5.0kg)( 9.8m/ ) ( 8.3N) 4.5 N co30 Apply 0 to the beam: f + co( 90 ) 0 c Solve fo f to obtai: f co( 90 ) c

71 Static Equilibium ad Elaticity 55 Subtitute umeical value ad evaluate f : f 4. N ( 90 ) ( 8.3N) co c co60 cb i the eactio foce to c : cb c 8.3 N adially iwad. Apply y 0 to the cylide: c cb co Mg 0 Solve fo c to obtai: co + Mg c cb Subtitute umeical value ad evaluate c : c ( 8.3N) co30 + ( 8.0kg)( 9.8m/ ) 03N Apply 0 to the cylide: f co( 90 ) 0 c cb Solve fo ad evaluate f c : f co( 90 ) ( 8.3N) c cb 4. N co60 Subtitute umeical value i equatio () ad () ad evaluate µ,beam-floo ad µ,cylide-floo : μ 4. N 4.5N, beam floo ad μ 4. N 03N, cylide floo [SSM] Two olid mooth (fictiole) phee of adiu ae placed iide a cylide of adiu R, a i igue -6. The ma of each phee i m. id the foce eeted by the bottom of the cylide o the bottom phee, the foce eeted by the wall of the cylide o each phee, ad the foce eeted by oe phee o the othe. All foce hould be epeed i tem of m, R, ad.

72 56 Chapte Pictue the Poblem The geomety of the ytem i how i the dawig. et upwad be the poitive y diectio ad to the ight be the poitive diectio. et the agle betwee the vetical cete lie ad the lie joiig the two cete be. The R R i ad ta. R( R) The foce eeted by the bottom of the cylide i jut mg. et be the foce that the top phee eet o the lowe phee. Becaue the phee ae i equilibium, we ca apply the coditio fo talatioal equilibium. W mg R mg R ' W Apply y 0 to the phee: mg mg 0 mg Becaue the cylide wall i mooth, co mg, ad: co R( R) mg mg Epe the compoet of : i mg ta Epe the foce that the wall of the R W cylide eet: R( R) mg Remak: Note that a appoache R/, w. 76 A olid cube of ide legth a balaced atop a cylide of diamete d i i utable equilibium if d << a (igue -64) ad i i table equilibium if d >> a. Detemie the miimum value of the atio d/a fo which the cube i i table equilibium. Pictue the Poblem Coide a mall otatioal diplacemet, δ of the cube fom equilibium. Thi hift the poit of cotact betwee cube ad cylide by Rδ, whee R d/. A a eult of that motio, the cube itelf i otated though the ame agle δ, ad o it cete i hifted i the ame diectio by the amout (a/) δ, eglectig highe ode tem i δ.

73 Static Equilibium ad Elaticity 57 + a + δ R d If the diplacemet of the cube cete of ma i le tha that of the poit of cotact, the toque about the poit of cotact i a etoig toque, ad the cube will etu to it equilibium poitio. If, o the othe had, (a/)δ > (d/) δ, the the toque about the poit of cotact due to mg i i the diectio of δ, ad will caue the diplacemet fom equilibium to iceae. We ee that the miimum value of d/a fo table equilibium i d/a.

74 58 Chapte