# Chapter 12 Static Equilibrium and Elasticity

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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

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

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

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

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

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.

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

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

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

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

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)

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

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

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