CHAPTER 9: Static Equilibrium; Elasticity and Fracture

Transcription

1 CHPE 9: Static Equilibrium; Elasticit and racture nswers t Questins. If te bject as a net frce n it f zer, ten its center f mass des nt accelerate. ut since it is nt in equilibrium, it must ave a net trque, and terefre ave an angular acceleratin. Sme eamples are: a) cmpact disk in a plaer as it cmes up t speed, after just being put in te plaer. b) ard drive n a cmputer wen te cmputer is first turned n. c) windw fan immediatel after te pwer t it as been sut ff.. e bungee jumper is nt in equilibrium, because te net frce n te jumper is nt zer. If te jumper were at rest and te net frce were zer, ten te jumper wuld sta at rest b ewtn s st law. e jumper as a net upward frce wen at te bttm f te dive, and tat is w te jumper is ten pulled back upwards.. If te fingers are nt te same distance frm te CG, te finger clser t te CG will supprt a larger fractin f te weigt f te meter stick s tat te net trque n te stick is zer. at larger vertical frce means tere will be mre frictin between te stick and tat clser finger, and tus te finger furter frm te CG will be easier t mve. e mre distant finger will slide easier, and terefre mve in clser t te CG. at finger, wen it becmes te ne clsest t te CG, will ten ave mre frictin and will stick. e ter finger will ten slide. Yu ten repeat te prcess. Wicever finger is farter frm te CG will slide clser t it, until te tw fingers eventuall meet at te CG. 4. ike almst an beam balance, te mvable weigts are cnnected t te fulcrum pint b relativel lng lever arms, wile te platfrm n wic u stand is cnnected t te fulcrum pint b a ver srt lever arm. e scale balances wen te trque prvided b ur weigt (large mass, small lever arm) is equal t tat prvided b te sliding weigts (small mass, large lever arm). 5. (a) If we assume tat te pivt pint f rtatin is te lwer left crner f te wall in te picture, ten te gravit frce acting trug te CM prvides te trque t keep te wall uprigt. te tat te gravit frce wuld ave a relativel small lever arm (abut alf te widt f te wall) and s te sidewas frce wuld nt ave t be particularl large t start t mve te wall. (b) Wit te rizntal etensin, tere are factrs tat make te wall less likel t verturn. e mass f te secnd wall is larger, and s te trque caused b gravit (elping t keep te wall uprigt) will be larger fr te secnd wall. e center f gravit f te secnd wall is furter t te rigt f te pivt pint and s gravit eerts a larger trque t cunteract te trque due t. e weigt f te grund abve te new part f te wall prvides a large clckwise trque tat elps t cunteract te trque due t. 6. r rtating te upper alf bd, te pivt pint is near te waist and ips. In tat psitin, te arms ave a relativel small trque, even wen etended, due t teir smaller mass, and te mre massive trunk ead cmbinatin as a ver srt lever arm, and s als as a relativel small trque. us te frce f gravit n te upper bd causes relativel little trque abut te ips tending t rtate u frward, and s te back muscles need t prduce little trque t keep u frm rtating frward. e frce n te upper alf bd due t te back muscles is small, and s te 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 8

2 Giancli Psics: Principles wit pplicatins, 6 t Editin (partiall rigtward) frce at te base f te spinal clumn, t keep te spine in equilibrium, will be small. Wen standing and bending ver, te lever arm fr te upper bd is muc larger tan wile sitting, and s causes a muc larger trque. e CM f te arms is als furter frm te supprt pint, and s causes mre trque. e back muscles, assumed t act at te center f te back, d nt ave a ver lng lever arm. us te back muscles will ave t eert a large frce t cause a cunter-trque tat keeps u frm falling ver. nd accrdingl, tere will ave t be a large frce (mstl t te rigt in te picture) at te base f te spine t keep te spine in equilibrium. 7. Wen te persn stands near te tp, te ladder is mre likel t slip. In te accmpaning diagram, te frce f te persn pusing dwn n te ladder Mg causes a clckwise trque abut te cntact pint wit te grund, wit lever arm d. e nl frce causing a cunterclckwise trque abut tat same pint is te reactin frce f te wall n te ladder, W. Wile te ladder is in equilibrium, W will be te same magnitude as te frictinal frce at te grund,. Since as a maimum G G value, W will ave te same maimum value, and s W will d ave a maimum cunterclckwise trque tat it can eert. s te persn climbs te ladder, teir lever arm gets lnger and s te trque due t teir weigt gets larger. Eventuall, if te trque caused b te persn is larger tan te maimum trque caused b W, te ladder will start t slip it will nt sta in equilibrium. 8. e mass f te meter stick is equal t tat f te rck. r purpses f calculating trques, te meter stick can be treated as if all f its mass were at te 5 cm mark. us te CM f te meter stick is te same distance frm te pivt pint as te rck, and s teir masses must be te same in rder t eert te same trque. 9. If te sum f te frces n an bject are nt zer, ten te CM f te bject will accelerate in te directin f te net frce. If te sum f te trques n te bject are zer, ten te bject as n angular acceleratin. Sme eamples are: a) satellite in a circular rbit arund te Eart. b) blck sliding dwn an inclined plane. c) n bject tat is in prjectile mtin but nt rtating d) e startup mtin f an elevatr, canging frm rest t aving a nn-zer velcit.. d G G W Mg stable unstable neutral equilib. equilib. equilib. 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 9

3 Capter 9 Static Equilibrium; Elasticit and racture. Cnfiguratin (b) is mre likel t be stable. In cnfiguratin (a), te CG f te bttm brick is at te edge f te table, and te CG f te tp brick is t te rigt f te edge f te table. us te CG f te tw-brick sstem is nt abve te base f supprt, and s gravit will eert a trque t rll te bricks clckwise ff te table. nter wa t see tis is tat mre tan 5% f te brick mass is nt abve te base f supprt 5% f te bttm brick and 75% f te tp brick are t te rigt f te edge f te table. It is nt in stable, neutral, r unstable equilibrium. In cnfiguratin (b), eactl alf f te mass (75% f te tp brick and 5% f te bttm brick) is ver te edge f te table. us te CG f te pair is at te edge f te table it is in unstable equilibrium.. Wen walking, u must keep ur CG ver ur feet. If u ave a eav lad in ur arms, ur CG is sifted frward, and s u must lean backwards t realign ur CG ver ur feet.. Wen u rise n ur tiptes, ur CM sifts frward. Since u are alread standing wit ur nse and abdmen against te dr, ur CM cannt sift frward. us gravit eerts a trque n u and u are unable t sta n ur tiptes u will return t being flat-fted n te flr. 4. Wen u start t stand up frm a nrmal sitting psitin, ur CM is nt ver ur pint f supprt (ur feet), and s gravit will eert a trque abut ur feet tat rtates u back dwn int te cair. Yu must lean frward in rder tat ur CM be ver ur feet s tat u can stand up. 5. In te midst f ding a sit-up, te abdmen muscles prvide a trque t rtate u up awa frm te flr, wile te frce f gravit n ur upper alf-bd is tending t pull u back dwn t te flr, prviding te difficult fr ding sit-ups. e frce f gravit n ur lwer alf-bd prvides a trque tat ppses te trque caused b te frce f gravit n ur upper alf-bd, making te sit-up a little easier. Wit te legs bent, te lever arm fr te lwer alf-bd is srter, and s less cunter-trque is available. 6. Psitin is unstable equilibrium, psitin is stable equilibrium, and psitin C is neutral equilibrium. 7. e Yung s mdulus fr a bungee crd is muc smaller tan tat fr rdinar rpe. rm its beavir, we knw tat te bungee crd stretces relativel easil, cmpared t rdinar rpe. rm Eq. 9-4, we ave E. e value f Yung s mdulus is inversel prprtinal t te cange in lengt f a material under a tensin. Since te cange in lengt f a bungee crd is muc larger tan tat f an rdinar rpe if ter cnditins are identical (stressing frce, unstretced lengt, crss-sectinal area f rpe r crd), it must ave a smaller Yung s mdulus. 8. n bject under sear stress as equal and ppsite frces applied acrss it. One blade f te scissrs puses dwn n te cardbard wile te ter arm puses up. ese tw frces cause te cardbard t sear between te tw blades. us te name sears is justified. 9. e left supprt is under tensin, since te frce frm te supprt pulls n te beam. us it wuld nt be wise t use cncrete r stne fr tat supprt. e rigt supprt is under cmpressin, since it puses n te beam. Cncrete r stne wuld be acceptable fr tat supprt. 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

4 Giancli Psics: Principles wit pplicatins, 6 t Editin Slutins t Prblems. If te tree is nt accelerating, ten te net frce in all directins is. cs C C cs 45 cs 64.6 sin C C sin 45 sin 99.4 C C C C tan tan 67.6, C C nd s C is 4, at an angle f clckwise frm.. e trque is te frce times te lever arm. r 58 kg 9.8m s. m.7 m, clckwise. ecause te mass m is statinar, te tensin in te rpe pulling up n te sling must be, and s te frce f te sling n te leg must be, upward. Calculate trques abut te ip jint, wit cunterclckwise trque taken as psitive. See te free-bd diagram fr te leg. te tat te frces n te leg eerted b te ip jint are nt drawn, because te d nt eert a trque abut te ip jint. 5. cm Mg m M 5. kg 6.5 kg 8.5 cm M g 4. e trque is te frce times te lever arm. m r r 58 kg 9.8m s.9 m 5. Write ewtn s nd law fr te junctin, in bt te and directins. cs 45 rm tis, we see tat. us set 55. sin 45 sin sin Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

5 Capter 9 Static Equilibrium; Elasticit and racture 6. (a) et m =. Calculate te net trque abut te left end f te diving bard, wit cunterclckwise trques psitive. Since te bard is in equilibrium, te net trque is zer.. m Mg 4. m. m Mg 4Mg 4 58 kg 9.8 m s 74. Use ewtn s nd law in te vertical directin t find.. m 4. m Mg Mg Mg Mg Mg 4 58 kg 9.8 m s 75.7 (b) epeat te basic prcess, but wit m = 5 kg. e weigt f te bard will add mre clckwise trque.. m. m Mg 4. m Mg kg 5 kg 9.8 m s 96. Mg Mg 4Mg Mg Mg 58 kg 5 kg 9.8 m s e CG f eac beam is at its center. Calculate trques abut te left end f te beam, and take cunterclckwise trques t be psitive. e cnditins f equilibrium fr te beam are used t find te frces tat te supprt eerts n te beam. l Mg l Mg l Mg 94 kg 9.8 m s l 4 l M g l Mg Mg Mg Mg Mg kg 9.8 m s et m be te mass f te beam, and M be te mass f te pian. Calculate trques abut te left end f te beam, wit cunterclckwise trques psitive. e cnditins f equilibrium fr te beam are used t find te frces tat te supprt eerts n te beam. Mg m M g 4 4 kg kg 9.8 m s Mg m M g 46 kg 9.8 m s.47.4 e frces n te supprts are equal in magnitude and ppsite in directin t te abve tw results..5 dwn. dwn /4 M g 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

6 Giancli Psics: Principles wit pplicatins, 6 t Editin 9. e pivt suld be placed s tat te net trque n te bard is zer. We calculate trques abut te pivt pint, wit cunterclckwise trques as psitive. e upward frce at P te pivt pint is swn, but it eerts n trque abut te pivt pint. e mass f te bard is m, and te CG is at te middle f te bard. (a) Ignre te frce m m g. Mg m M (b) Include te frce m g. 5 kg 5 kg 75 kg Mg m g m m 5 kg 7.5 kg M m m 75 kg 5 kg 5 kg Mg 9. m.5 m. m frm adult P / - mg 9. m.54 m.5 m frm adult. Calculate trques abut te left end f te beam, wit cunterclckwise trques psitive. e cnditins f equilibrium fr te beam are used t find te frces tat te supprt eerts n te beam.. m 5. m kg 9.8 m s.5. m 5. m tice tat pints dwn kg 9.8 m s.6. Using te free-bd diagram, write ewtn s secnd law fr bt te rizntal and vertical directins, wit net frces f zer. cs cs sin sin cs cs.6 7 kg 9.8 m s sin tan tan 7 kg 9.8 m s sin sin. 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

7 Capter 9 Static Equilibrium; Elasticit and racture. Draw a free-bd diagram f te junctin f te tree wires. e tensins can be fund frm te cnditins fr frce equilibrium. cs7 cs7 cs5 cs5 sin 7 sin 5 cs 7 cs 5 sin 7 sin 5 kg 9.8m s cs 7 sin 7 sin 5 cs cs 7 cs cs 5 cs e table is smmetric, s te persn can sit near eiter edge and te same distance will result. We assume tat te persn (mass M) is n te rigt side f te table, and tat te table (mass m) is n te verge f tipping, s tat te left leg is n te verge f.6 m lifting ff te flr. ere will ten be n nrmal frce between te left leg f te table and te flr. Calculate trques abut te rigt leg f te table, s tat te nrmal frce between te table and te flr causes n trque. Cunterclckwise trques are taken t be psitive. e cnditins f equilibrium fr te table are used t find te persn s lcatin..6 m Mg.6 m m. kg.6 m M 66. kg.8 m us te distance frm te edge f te table is.5 m.8 m. m M g 4. Draw a frce diagram fr te seet, and write ewtn s secnd law fr te vertical directin. te tat te tensin is te same in bt parts f te cltesline. sin.5 sin.5.6 kg 9.8 m s sin.5 sin.5 48 e 48- tensin is muc iger tan te ~ 6- weigt f te seet because f te angle. Onl te vertical cmpnents f te tensin are supprting te seet, and since te angle is small, te tensin as t be large t ave a large enug vertical cmpnent e beam is in equilibrium, and s bt te net trque and net frce n it must be zer. rm te free-bd diagram, calculate te net trque abut te center f te left supprt, wit cunterclckwise trques as psitive. Calculate te net frce, wit upward as psitive. Use tse tw equatins t find and Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 4

8 Giancli Psics: Principles wit pplicatins, 6 t Editin m 6. m 9. m 5 kg 9.8m s 5. m m 96 5 kg 9.8 m s rm te free-bd diagram, te cnditins f equilibrium are used t find te lcatin f te girl (mass m ). e 5- kg b is represented b m, and te 5-kg girl b C m. Calculate trques abut te center f te see-saw, and take cunterclckwise trques t be psitive. e upward frce f te fulcrum n te see-saw mg causes n trque abut te center. mcg mg m g m g m g C m m C m 5 kg 5 kg 5 kg.6 m. m 7. Since eac alf f te frceps is in equilibrium, te net trque n eac alf f te frceps is zer. Calculate trques wit respect t an ais perpendicular t te plane f te frceps, trug pint P, cunterclckwise being psitive. Cnsider a frce diagram fr ne alf f te frceps. is te frce n te alf-frceps due t te plastic rd, and frce is te frce n P te alf-frceps frm te pin jint. abut pint P. des nt eert an trque P d d d cs cs. 4.6 d 8.5 cm.7 cm e frce tat te frceps eerts n te rd is te ppsite f, and s is als 4.6. d P d 8. e beam is in equilibrium, and s te net frce and net trque n te beam must be zer. rm te free-bd diagram fr te beam, calculate te net trque (cunterclckwise psitive) abut te wall supprt pint t find, and calculate te net frce in bt te and directins t find te cmpnents f W. sin 4 7 kg 9.8 m s sin 4 sin Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 5 W 4

9 Capter 9 Static Equilibrium; Elasticit and racture cs 4 cs cs W W W sin 4 sin 4 7 kg 9.8 m s 5.8 sin 4. W W W W. tan tan W W e persn is in equilibrium, and s bt te net trque and net frce must be zer. rm te free-bd diagram, calculate te net trque abut te center f gravit, wit cunterclckwise trques as psitive. Use tat calculatin t find te lcatin f te center f gravit, a distance frm te feet. m g m 5. kg m g m g m m.6 kg 5. kg e center f gravit is abut 9.5 cm frm te feet. -.7 m 9.5 m. e beam is in equilibrium. Use te cnditins f equilibrium t calculate te tensin in te wire and te frces at te inge. Calculate trques abut te inge, and take cunterclckwise trques t be psitive. sin l m g l m gl m gl l m gl m 45.7 m sin.5 m sin 5. H l l mg l cs cs 78 cs H H sin m g m g H m g m g sin sin dwn H. (a) e ple is in equilibrium, and s te net trque n it must be zer. rm te free-bd diagram, calculate te net trque abut te lwer end f te ple, wit cunterclckwise trques as psitive. Use tat calculatin t find te tensin in te cable. e lengt f te ple is. cs Mg cs m M g cs 6. kg.5 kg 9.8 m s 7.5 m cs m (b) e net frce n te ple is als zer since it is in equilibrium. Write ewtn s nd law in bt 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 6 P P cs M g

10 Giancli Psics: Principles wit pplicatins, 6 t Editin te and directins t slve fr te frces at te pivt. 4.5 P P Mg m M g.5 kg 9.8 m s.8 P P. e man is in equilibrium, s te net frce and te net trque n im are bt zer. rm te frce diagram, write an epressin fr te net trque abut a vertical ais trug is rigt and, wit cunterclckwise trques as psitive. ls write an epressin fr te net frce in te vertical directin. left left.6 m.7 m left rigt 7 kg 9.8 m s rigt.7 m.6 m left rigt left 7 kg 9.8 m s (a) e meter stick is in equilibrium, s te net trque and te net frce are bt zer. rm te frce diagram, write an epressin fr te net trque abut te 9-cm mark, wit cunterclckwise trques as psitive..4 m.9 m kg 9.8 m s (b) Write ewtn s nd law fr te vertical directin wit a net frce f t find te ter tensin kg 9.8 m s m.9 m 9 4. Since te backpack is midwa between te tw trees, te angles in te diagram are equal. Write ewtn s nd law fr te vertical directin fr te pint at wic te backpack is attaced t te crd, wit te weigt f te backpack being te dwnward vertical frce. e angle is determined b te distance between te trees and te amunt f sag at te midpint, as illustrated in te secnd diagram..5 m (a) tan tan.5.8 m (b) sin 9 kg 9.8 m s sin sin m 9 kg 9.8 m s tan tan m sin sin.6 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 7

11 Capter 9 Static Equilibrium; Elasticit and racture 5. e frces n te dr are due t gravit and te inges. Since te dr is in equilibrium, te net trque and net frce must be zer. Write te tree equatins f equilibrium. Calculate trques abut te bttm inge, wit cunterclckwise trques as psitive. rm te statement f te prblem,. w d w. kg 9.8 m s. m d. m.8 m 55. w d d 55.. kg 9.8m s Write te cnditins f equilibrium fr te ladder, wit trques taken abut te bttm f te ladder, and cunterclckwise trques as psitive. l l sin cs W W tan G W G W tan G G l sin r te ladder t nt slip, te frce at te grund must be less tan G r equal t te maimum frce f static frictin. tan tan G G tan G W G l cs us te minimum angle is tan. min 7. e ladder is in equilibrium, s te net trque and net frce must be zer. stating tat te ladder is n te verge f slipping, te static frictinal frce at te grund, is at its maimum value and s C W. Since te persn is standing 7% f te wa up te C s C ladder, te eigt f te ladder is d.7.8 m.7 4. m. e widt f te ladder is d.7. m.7. m. rques are taken abut te pint f cntact f te ladder wit te grund, and cunterclckwise trques are taken as psitive. e tree cnditins f equilibrium are as fllws. C W C W d C Mg C d 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 8

12 Giancli Psics: Principles wit pplicatins, 6 t Editin G Mg G M m g Mgd W 67. kg 9.8 m s Slving te trque equatin gives m Md. kg. m 55. kg. m g W 9.8 m s m e cefficient f frictin ten is fund t be G s G 8. If te lamp is just at te pint f tipping, ten te nrmal frce will be acting at te edge f te base, cm frm te lamp stand ple. We assume te lamp is in equilibrium and just n te verge f tipping, and is being pused sidewas at a cnstant speed. ake trques abut te center f te base, wit cunterclckwise trques psitive. ls write ewtn s nd law fr bt te vertical and rizntal directins. P fr P fr. m. m. m.6 m P 9. irst cnsider te triangle made b te ple and ne f te wires (first diagram). It as a vertical leg f.6 m, and a rizntal leg f. m. e angle tat te tensin (alng te wire) makes wit te vertical is. tan 7.6. e part f te tensin tat is parallel t te grund is.6 terefre sin. w cnsider a tp view f te ple, swing nl P. m frce parallel t te grund (secnd diagram). e rizntal parts f te tensin lie as te sides f an equilateral triangle, and s eac make a angle wit te tensin frce f te net. Write te equilibrium equatin fr te frces alng te directin f te tensin in te net. cs net net sin cs 95 sin 7.6 cs...6 m. m net. e arm is in equilibrium. ake trques abut te elbw jint (te dt in te free-bd diagram), s tat te frce at te elbw jint des nt enter te calculatin. Cunterclckwise trques are psitive. e mass f te lwer arm is m. kg, and te mass f te lad is M. It is given tat 45. M J d M Mg d d Mgd M d d M d d M gd 45.6 m. kg 9.8 m s.5 m 9.8 m s.5 m 7. kg 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 9

13 Capter 9 Static Equilibrium; Elasticit and racture. Calculate te trques abut te elbw jint (te dt in te free bd diagram). e arm is in equilibrium. Cunterclckwise trques are psitive. d D Mg M M md M g d.8 kg. m 7. kg. m 9.8m s m M d D Mg. (a) Calculate te trques abut te elbw jint (te dt in te freebd diagram). e arm is in equilibrium. ake cunterclckwise trques as psitive. sin d D M M D d sin. kg 9.8 m s.4 m. m sin5.5 (b) find te cmpnents f J, write ewtn s nd law fr bt te and directins. en cmbine tem t find te magnitude. cs cs 5 cs5 4 J M J M sin J M J sin 5 sin5. kg 9.8 m s M J J J M J d D. Calculate te trques abut te sulder jint, wic is at te left end f te free-bd diagram f te arm. Since te arm is in equilibrium, te sum f te trques will be zer. ake cunterclckwise trques t be psitive. e frce due t te sulder jint is drawn, but it des nt eert an trque abut te sulder jint. d sin D Mg m m md M g d sin. kg.4 cm 5 kg.5 m 9.8m s.7. m sin5 J d M D M g 4. ere will be a nrmal frce upwards at te ball f te ft, equal t te persn s weigt. Calculate trques abut a pint n te flr directl belw te leg bne (and s in line wit te leg bne frce, ). Since te ft is in equilibrium, te sum f te trques will be zer. ake cunterclckwise trques as psitive. d d d D d 7 kg 9.8 m s.4 e net frce in te directin must be zer. Use tat t find.. 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

14 Giancli Psics: Principles wit pplicatins, 6 t Editin 5. igures 9-4 (b) and (c) are redrawn ere wit te persn 45 frm te rizntal, instead f te riginal. e distances are all te same as in te riginal prblem. We still assume tat te back muscles pull at a angle t te spine. e 8 angle frm te riginal prblem becmes. rques are taken abut te same pint at te base f te spine, wit cunterclckwise trques as psitive..48 m sin.7 m w sin 45 M.48 m w sin 45.6 m w sin 45 s in te riginal prblem, w.7 w, w. w, w.46w. Wit H tis, te trque equatin gives te fllwing result..7 m.7.48 m..6 m.46 wsin 45.94w M.48 m sin ake te sum f te frces in te vertical directin, set equal t zer. sin.7w.w.46w.7w V M V ake te sum f te frces in te rizntal directin, set equal t zer. cs.6w e final result is V M V w.4 V V V is cmpares t.5w fr te mre bent psitin. H M M V 45 w w H w rm Sectin 9-4: n bject wse CG is abve its base f supprt will be stable if a vertical line prjected dwnward frm te CG falls witin te base f supprt. r te twer, te base f supprt is a circle f radius.5 m. If te tp is 4.5 m ff center, ten te CG will be.5 m ff center, and a vertical line dwnward frm te CG will be.5 m frm te center f te base. us te twer is in stable equilibrium. be unstable, te CG as t be mre tan.5 m ff center, and tus te tp must be mre tan 7. m ff center. us te tp will ave t lean.5 m furter t reac te verge f instabilit. 7. (a) e maimum distance fr brick # t remain n brick # will be reaced wen te CM f brick # is directl ver te edge f brick #. us brick # will verang brick # b. # # e maimum distance fr te tp tw bricks t remain n brick # will be reaced wen te center f mass f te tp tw bricks is directl ver te edge f brick #. e CM f te tp tw bricks is (bviusl) at te pint labeled n brick #, a distance f 4 frm te rigt edge f brick #. us 4. # # # e maimum distance fr te tp tree bricks t remain n brick #4 will be reaced wen te center f mass f te tp tree bricks is directl ver te edge f brick #4. e CM f te tp tree bricks is at te pint labeled n brick #, and is fund relative t te center f brick # b # # # # 4 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

15 Capter 9 Static Equilibrium; Elasticit and racture m m CM, r 6 frm te rigt edge f brick #. us 6. m e maimum distance fr te fur bricks t remain n a tabletp will be reaced wen te center f mass f te fur bricks is directl ver te edge f te table. e CM f all fur bricks is at te pint labeled n brick #4, and is fund relative t te center f brick #4 b m m CM 8, r 8 frm te rigt 4m edge f brick #4. us 8. 4 (b) rm te last diagram, te distance frm te edge f te tabletp t te rigt edge f brick # is Since tis distance is greater tan, te answer is es, te first brick is cmpletel bend te edge f te table. (c) rm te wrk in part (a), we see tat te general frmula fr te ttal distance spanned b n bricks is 4 6 n n (d) e arc is t span. m, s te span frm ne side will be.5 m. us we must slve n. m.5 m. Evaluatin f tis epressin fr varius values f n sws tat 5 bricks i i will span a distance f.498 m, and tat 6 bricks will span a distance f.57 m. us it takes 6 bricks fr eac alf-span, plus brick n tp and brick as te base n eac side (as in ig. 9-67(b)), fr a ttal f 5 bricks. 8. e amunt f stretc can be fund using te elastic mdulus in Eq m. m 9 4 E 5 m 5. # # # # 4 n i i 4 9. (a) (b) 5 kg 9.8 m s Stress.4 m. m.m 5 Stress.4 m Strain 4. 9 Yung's Mdulus 5 m e cange in lengt is fund frm te strain. 6 5 Strain Strain 9.6 m 4..9 m 4. (a) (b) kg 9.8 m s Stress.7 m.4 m.5m Stress.7 m Strain Yung's Mdulus m Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

16 Giancli Psics: Principles wit pplicatins, 6 t Editin (c) 7 6 Strain m 6.5 m 4. e cange in vlume is given b Eq We assume te riginal pressure is atmsperic pressure, 5. m. P V V m. m cm.5 cm. m 9 V V V cm.5 cm 997 cm 4. e Yung s Mdulus is te stress divided b te strain. Yung's Mdulus Stress m Strain.7 m 5 m 9.6 m e relatinsip between pressure cange and vlume cange is given b Eq P V V V P. 9 m 9. m V P P atm 9. m 7. m e percentage cange in vlume is fund b multipling te relative cange in vlume b. e cange in pressure is 99 times atmsperic pressure, since it increases frm atmsperic pressure t times atmsperic pressure. 5 V P 99. m % 9 V 9 m e negative sign indicates tat te interir space gt smaller. 46. Elastic ptential energ is given b using as. elastic PE k. e frce is fund frm Eq. 9-4, 6 4 E. m.5 m elastic. m PE.7 J. m 47. (a) e trque due t te sign is te prduct f te weigt f te sign and te distance f te sign frm te wall. d 5. kg 9.8 m s. m. m, clckwise (b) Since te wall is te nl ter bject tat can put frce n te ple wall (ignring te weigt f te ple), ten te wall must put a trque n te ple. e trque due t te sign is clckwise, s te trque due t te wall must be cunterclckwise. See te diagram. ls nte tat te wall must put an upward frce n te ple as well, s tat te net frce n te ple will be zer. wall 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te

17 Capter 9 Static Equilibrium; Elasticit and racture (c) e trque n te rd can be cnsidered as te wall pulling rizntall t te left n te tp left crner f te rd and pusing rizntall t te rigt at te bttm left crner f te rd. e reactin frces t tese put a sear n te wall at te pint f cntact. ls, since te wall is pulling upwards n te rd, te rd is pulling dwn n te wall at te tp surface f cntact, causing tensin. ikewise te rd is pusing dwn n te wall at te bttm surface f cntact, causing cmpressin. us all tree are present. 48. Set te cmpressive strengt f te bne equal t te stress f te bne. ma Cmpressive Strengt 7 m. m 5. ma 49. (a) e maimum tensin can be fund frm te ultimate tensile strengt f te material. ma ensile Strengt ma 6 4 ensile Strengt 5 m 5. m 9 (b) prevent breakage, ticker strings suld be used, wic will increase te crss-sectinal area f te strings, and tus increase te maimum frce. reakage ccurs because wen te strings are it b te ball, te stretc, increasing te tensin. e strings are reasnabl tigt in te nrmal racket cnfiguratin, s wen te tensin is increased b a particularl ard it, te tensin ma eceed te maimum frce. 5. (a) Cmpare te stress n te bne t te cmpressive strengt t see if te bne breaks. Stress m 4 8. m < Cmpressive Strengt f bne e bne will nt break. (b) e cange in lengt is calculated frm Eq m 8. m.5 m 9 E 5 m 5. (a) e area can be fund frm te ultimate tensile strengt f te material. ensile Strengt Safet actr Safet actr 7. kg 9.8 m s 4.4 m 6 ensile Strengt 5 m (b) e cange in lengt can be fund frm te stress-strain relatinsip, equatin (9-5). E E 7.5 m kg 9.8m s 4.4 m m m 5. r eac supprt, t find te minimum crss-sectinal area wit a Strengt safet factr means tat, were eiter te tensile r Safet actr cmpressive strengt is used, as apprpriate fr eac frce. find te frce n eac supprt, use te cnditins f equilibrium fr te beam. ake trques abut te left end f te beam, calling cunterclckwise. m 5. m 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 4

18 Giancli Psics: Principles wit pplicatins, 6 t Editin trques psitive, and als sum te vertical frces, taking upward frces as psitive. 5.. m 5. m tice tat te frces n te supprts are te ppsite f and. S te frce n supprt # is directed upwards, wic means tat supprt # is in tensin. e frce n supprt # is directed dwnwards, s supprt # is in cmpressin. ensile Strengt kg 9.8 m s m ensile Strengt 4 m 6 Cmpressive Strengt kg 9.8 m s m Cmpressive Strengt 5 m 6 5. e diameter can be fund frm te ultimate sear strengt f te material. Sear Strengt Safet actr d d 4 Safet actr 4 6. Sear Strengt 7 m 6. m 54. See te free-bd diagram. e largest tensin will ccur wen te elevatr as an upward acceleratin. Use tat wit te maimum tensin t calculate te diameter f te blt, d. Write ewtn s secnd law fr te elevatr t find te tensin. ma ma m g a m m 9.8m s. m s. 7 d ensile strengt elevatr d 8 ensile strengt 8.. kg 5 m 6.5 m 55. Draw free-bd diagrams similar t igures 9-(a) and 9-(b) fr te frces n te rigt alf f a rund arc and a pinted arc. e lad frce is placed at te same rizntal psitin n eac arc. r eac alf-arc, take trques abut te lwer rigt and crner, wit cunterclckwise as psitive. r te rund arc: ad H H ad rund rund 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 5 H rund ad H rund V

19 Capter 9 Static Equilibrium; Elasticit and racture r te pinted arc: ad H H ad pinted pinted Slve fr, given tat. H pinted 8. m m H rund H H ad ad pinted rund H pinted ad H pinted V 56. Write ewtn s nd law fr te rizntal directin. cs cs us te tw frces are te same size. w write ewtn s nd law fr te vertical directin butress 6 sin sin.5 butress sin sin 5 utress 57. Eac crssbar in te mbile is in equilibrium, and s te net trque abut te suspensin pint fr eac crssbar must be. Cunterclckwise trques will be taken as psitive. e suspensin pint is used s tat te tensin in te suspensin string need nt be knwn initiall. e net vertical frce must als be. e bttm bar: m g m g D D C C 7.5 cm m m m.5 m 5. cm D C D D D C m g m g m m g 4.5m g CD C D CD C D D mdg D CD C mcg e middle bar: m g m g 4.5m g m g CD CD CD D m D m CD.885 kg 5. cm kg cm CD CD CD CD CD mg m C.5m kg.9 kg D m g m g 4.5m m g e tp bar: m g CD CD CD CD D CD CD CD 4.5m m g m 4.5m m g D CD CD D kg.885 kg.94 kg 7.5 cm. cm mg CD CD 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 6

20 Giancli Psics: Principles wit pplicatins, 6 t Editin 58. rm te free-bd diagram (nt t scale), write te frce equilibrium cnditin fr te vertical directin. sin 6. kg 9.8 m s sin. m m. m. m It is nt pssible t increase te tensin s tat tere is n sag. ere must alwas be a vertical cmpnent f te tensin t balance te gravit frce. e larger te tensin gets, te smaller te sag angle will be, wever. 59. (a) If te weel is just lifted ff te lwest level, ten te nl frces n te weel are te rizntal pull, its weigt, and te cntact frce at te crner. ake trques abut te crner pint, fr te weel just barel ff te grund, being eld in equilibrium. e cntact frce at te crner eerts n trque and s des nt enter te calculatin. e pulling frce as a lever arm f, and gravit as a lever arm f, fund frm te triangle swn. Mg Mg Mg Mg Mg. (b) e nl difference is tat nw te pulling frce as a lever arm f. Mg Mg Mg Mg 6. e mass is t be placed smmetricall between tw legs f te table. Wen enug mass is added, te table will rise up ff f te tird leg, and ten te nrmal frce n te table will all be n just tw legs. Mg Since te table legs are equall spaced, te angle marked in te diagram is dwn. ake trques abut a line cnnecting te tw legs tat remain n te flr, s tat te nrmal frces cause n trque. It is seen frm te secnd diagram (a prtin f te first diagram but enlarged) tat te tw frces are equidistant frm te line jining te tw legs n te flr. Since te lever arms are equal, ten te trques will be equal if te frces are equal. us, t be in equilibrium, te tw frces must be te same. If te frce n te edge f te table is an bigger tan te weigt f te table, it will tip. us M 5 kg will cause te table t tip. M g dwn 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 7

21 Capter 9 Static Equilibrium; Elasticit and racture 6. (a) e weigt f te self eerts a dwnward frce and a clckwise trque abut te pint were te self eft. cm tuces te wall. us tere must be an upward frce and a cunterclckwise trque eerted b te slt fr te self t be in equilibrium. Since an frce eerted igt b te slt will ave a srt lever arm relative t te pint were te self tuces te wall, te upward frce.cm must be larger tan te gravit frce. ccrdingl, tere ten must be a dwnward frce eerted b te slt at its left edge, eerting n trque, but balancing te vertical frces. (b) Calculate te values f te tree frces b first taking trques abut te left end f te self, wit te net trque being zer, and ten sum te vertical frces, wit te sum being zer. igt. m 7. m igt eft 7. m 5. kg 9.8 m s m igt igt eft kg 9.8 m s.7 (c) e trque eerted b te supprt abut te left end f te rd is igt. m m 8. m 6. ssume tat te building as just begun t tip, s tat it is essentiall vertical, but tat all f te frce n te building due t cntact wit te Eart is at te lwer left crner, as swn in te figure. ake trques abut tat crner, wit cunterclckwise trques as psitive.. m. m 95 m. m 7. m. m. m. m 7.8 kg 9.8 m s. m E 9. m Since tis is a negative trque, te building will tend t rtate clckwise, wic means it will rtate back dwn t te grund. us te building will nt tpple. E 6. e truck will nt tip as lng as a vertical line dwn frm te CG is between te weels. Wen tat vertical line is at te weel, it is in unstable equilibrium and will tip if te rad is inclined an mre. See te diagram fr te truck at te tipping angle, swing te truck s weigt vectr.. m tan tan tan 9. m 64. Draw a frce diagram fr te cable tat is supprting te rigt-and sectin. e frces will be te tensin at te left end,, te tensin at te rigt end,, and te weigt f te sectin,. e weigt acts at te midpint f te rizntal span f te cable. e sstem is in equilibrium. Write ewtn s nd law in bt te and directins t find te tensins. 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 8

22 Giancli Psics: Principles wit pplicatins, 6 t Editin cs9 sin 6 cs9 sin 6 cs 6 sin9 6 d cs9 cs 6 cs 6 sin 6 sin9 sin9 sin cs9 cs 6 sin9 sin 6 9 cs9 cs sin 6 sin 6 find te eigt f te twer, take trques abut te pint were te radwa meets te grund, at te rigt side f te radwa. te tat ten will eert n trque. ake cunterclckwise trques as psitive. r purpses f calculating te trque due t, split it int and cmpnents. d d cs cs 6.5 d d sin sin 6 58 m d 4 m 65. e radius f te wire can be determined frm te relatinsip between stress and strain, epressed b equatin (9-5). E r r E E Use te free-bd diagram fr te pint f cnnectin f te mass t te wire t determine te tensin frce in te wire. 5 kg 9.8m s sin 589. sin sin e fractinal cange in te lengt f te wire can be fund frm te gemetr f te prblem. cs.4 cs cs us te radius is r 589. E 7 m m / 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 9

23 Capter 9 Static Equilibrium; Elasticit and racture 66. e airplane is in equilibrium, and s te net frce in eac directin and te net trque are all equal t zer. irst write ewtn s nd law fr bt te rizntal and vertical directins, t find te values f te frces D D kg 9.8m s 6.6 Calculate te trques abut te CM, calling cunterclckwise trques psitive. d D 5 5 d 6.6. m 5..6 m 5 D 5. D.6 m d 67. Draw a free-bd diagram fr alf f te cable. Write ewtn s nd law fr bt te vertical and rizntal directins, wit te net frce equal t in eac directin. sin 6.58 sin 6 S te results are: (a).9 cs 6.58 cs (b).58 (c) e directin f te tensin frce is tangent t te cable at all pints n te cable. us te directin f te tensin frce is rizntal at te lwest pint, and is 6 abve te rizntal at te attacment pint. 68. (a) r te etreme case f te beam being read t tip, tere wuld be n nrmal frce at pint frm te supprt. Use te free-bd diagram t write te equatin f rtatinal equilibrium under tat cnditin t find te weigt f te persn, wit. ake trques abut te lcatin f supprt, and call cunterclckwise trques psitive. W is te weigt f te persn. m g 5. m W 5. m W m g 55 (b) Wit te persn standing at pint D, we ave alread assumed tat te vertical directin must als be zer. C D. m 7. m 5. m 5. m m g W m g W mg W. e net frce in 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 4

24 Giancli Psics: Principles wit pplicatins, 6 t Editin (c) w te persn mves t a different spt, s te free-bd diagram canges as swn. gain use te net trque abut supprt and ten use te net vertical frce. m g 5. m W. m. m m g 4 5. m W. m 55. m. m. m m g W m g W 4 96 (d) gain te persn mves t a different spt, s te free-bd diagram canges again as swn. gain use te net trque abut supprt and ten use te net vertical frce. m g 5. m W. m. m m g W 5. m W. m m 55. m. m. m 69 m g W m g W m C D. m 7. m 5. m mg W. m C D. m 5. m 5. m m g 69. If te blck is n te verge f tipping, te nrmal frce will be acting at te lwer rigt crner f te blck, as swn in te free-bd diagram. e blck will begin t rtate wen te trque caused b te pulling frce is larger tan te trque caused b gravit. r te blck t be able t slide, te pulling frce must be as large as te maimum static frictinal frce. Write te equatins f equilibrium fr frces in te and directins and fr trque wit te cnditins as stated abve. l/ fr fr fr s s l l s Slve fr te cefficient f frictin in tis limiting case, t find (a) If l s, ten sliding will appen befre tipping. s l. (b) If l s, ten tipping will appen befre sliding. 7. e limiting cnditin fr te safet f te painter is te tensin in te rpes. e rpes can nl eert an upward tensin n te scaffld. e tensin will be least in te rpe tat is farter frm te painter. e mass f te pail is m, te mass f te scaffld is m, and te mass f te p painter is M. left. m. m. m. m mpg rigt. m M g 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 4

25 Capter 9 Static Equilibrium; Elasticit and racture ind te distance t te rigt tat te painter can walk befre te tensin in te left rpe becmes zer. ake trques abut te pint were te rigt-side tensin is attaced t te scaffld, s tat its value need nt be knwn. ake cunterclckwise trques as psitive.. m m g. m Mg p m. m m. m p 5 kg. m 4. kg. m. m M 6. kg Since te maimum value fr is. m, te painter can walk t te rigt edge f te scaffld safel. w find te distance t te left tat te painter can walk befre te tensin in te rigt rpe becmes zer. ake trques abut te pint were te left-side tensin is attaced t te scaffld, s tat its value need nt be knwn. ake cunterclckwise trques as psitive. Mg m g. m. m p m. m m. m p 5 kg. m 4. kg. m.9 m M 6. kg us te left end is dangerus, and e can get witin. m f te left end safel. 7. (a) e ple will eert a dwnward frce and a clckwise trque abut te wman s rigt and. us tere must be an upward frce eerted b te left and t cause a cunterclckwise trque fr te ple t ave a net trque f zer. e frce eerted b te rigt and is ten f suc a magnitude and directin fr te net vertical frce n te ple t be zer.. m. m eft igt eft eft eft. m igt. kg 9.8 m s. m. (b) We see tat te frce due t te left and is larger tan te frce due t te rigt and, since bt te rigt and and gravit are dwnward. Set te left and frce equal t 5 and calculate te lcatin f te left and, b setting te net trque equal t zer.. m eft 6.7., upward 6.7. kg 9.8 m s 8.7., dwnward 98.. m. m.65 m 5 eft s a ceck, calculate te frce due t te rigt and OK igt eft. m M g igt igt left rigt. m. m. m. m mpg. m. m. m eft eft 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 4

26 Giancli Psics: Principles wit pplicatins, 6 t Editin (c) llw te same prcedure, setting te left and frce equal t m. m. m.5 m. m eft OK igt eft te tat nw te frce due t te rigt and must be pulling upwards. eft 7. e man is in equilibrium, s te net frce and te net trque n im must be zer. We represent te frce n te ands as twice te frce n ne and, and te frce n te feet as twice te frce n ne ft. ake trques abut te pint were is ands tuc te grund, wit cunterclckwise as psitive. d d d f f d 75 kg 9.8m s.4 m 9. d d.5 m d d f f f 75 kg 9.8m s e frce n te spere frm eac plane will be nrmal t te spere, and s perpendicular t te plane at te pint f cntact. Use ewtn s nd law in bt te rizntal and vertical directins t determine te magnitudes f te frces. sin sin 7 sin sin sin sin sin 7 cs cs cs 7 cs sin sin 7 sin 7 cs 7 cs cs 7 cs sin sin sin 7 sin sin sin kg 9.8 m s find te nrmal frce eerted n te rad b te trailer tires, take te trques abut pint, wit cunterclckwise trques as psitive. 5.5 m 8. m m 5.5 m kg 9.8 m s 4,8 8. m 8. m e net frce in te vertical directin must be zer. kg 9.8 m s 4, m 5.5 m 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 4

27 Capter 9 Static Equilibrium; Elasticit and racture 75. ssume a cnstant acceleratin as te persn is brugt t rest, wit up as te psitive directin. Use Eq. -c t find te acceleratin. rm te acceleratin, find te average frce f te snw n te persn, and cmpare te frce per area t te strengt f bd tissue. v v 6 m s 8 m s v v a a. m ma 75 kg 8 m s m issue strengt 5 m.m Since te average frce n te persn is less tan te strengt f bd tissue, te persn ma escape serius injur. Certain parts f te bd, suc as te legs if landing feet first, ma get mre tan te average frce, tug, and s still sustain injur. 76. e mass can be calculated frm te equatin fr te relatinsip between stress and strain. e frce causing te strain is te weigt f te mass suspended frm te wire. E. m 9. m m 9 kg E E g 9.8 m s 77. (a) rm Eample 7-6, te ttal frce f te grund n ne leg fr a stiff-legged landing is e stress in te tibia bne is te frce divided b te crsssectinal area f te bne. Stress 5.5. m 4.5 m 8 (b) Since te stress frm abve is greater tan te cmpressive strengt f bne 8.7 m, te bne will break. (c) rm Eample 7-6, te ttal frce f te grund n ne leg fr a bent-legged landing is e stress in te tibia bne is te frce divided b te crsssectinal area f te bne. Stress.45. m 4 8. m 6 Since te stress frm abve is less tan te cmpressive strengt f bne te bne will nt break. 8.7 m, 78. e number f supprts can be fund frm te cmpressive strengt f te wd. Since te wd will be riented lngitudinall, te stress will be parallel t te grain. Cmpressive Strengt ad frce n supprts Weigt f rf Safet actr rea f supprts # supprts area per supprt # supprts Weigt f rf area per supprt Safet actr Cmpressive Strengt snw 4.6 kg 9.8 m s.4 m.9 m 5 m 6 supprts 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 44

28 Giancli Psics: Principles wit pplicatins, 6 t Editin Since tere are t be supprts, tere will be 6 supprts n eac side. at means tere will be 5 gaps n eac side between te supprts, and s Spacing. m 5 gaps. m gap 79. e tensin in te string wen it breaks is fund frm te ultimate strengt f nln under tensin, frm able 9-. ensile Strengt ensile Strengt m 5 m 9.7 rm te frce diagram fr te b, we calculate te angle f te rpe relative t te rizntal frm ewtn s nd law in te vertical directin. te tat since te tensin is te same trugut te string, te angles must be te same s tat te bject des nt accelerate rizntall. sin 5 kg 9.8 m s sin sin 8.8. m 9.7 find te eigt abve te grund, cnsider te secnd diagram.. m tan. m. m tan. m. m tan8.8.4 m. m 8. e maimum cmpressive frce in a clumn will ccur at te bttm. e bttm laer supprts te entire weigt f te clumn, and s te cmpressive frce n tat laer is. r te clumn t be n te verge f buckling, te weigt divided b te area f te clumn will be te cmpressive strengt f te material. e mass f te clumn is its vlume (area eigt) times its densit. g Cmpressive Strengt Cmpressive Strengt g te tat te area f te clumn cancels ut f te epressin, and s te eigt des nt depend n te crss-sectinal area f te clumn. (a) (b) Cmpressive Strengt 5 m 6 steel g Cmpressive Strengt 7.8 kg m 9.8 m s 7 m 6 granite g.7 kg m 9.8 m s. 6.5 m 6.4 m 5 Pearsn Educatin, Inc., Upper Saddle iver, J. ll rigts reserved. is material is prtected under all cprigt laws as te currentl eist. prtin f tis material ma be reprduced, in an frm r b an means, witut permissin in writing frm te 45