Spring 2002 Lecture #17


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1 Sprng 22 Lecture #17 r. Jaehn Yu 1. Cndtns fr Equlbrum 2. Center f Gravty 3. Elastc Prpertes f Slds Yung s dulus Shear dulus ulk dulus Tday s Hmewrk Assgnment s the Hmewrk #8!!! 2 nd term eam n Wednesday, Apr. 1. Wll cver chapters 113.
2 Cndtns fr Equlbrum What d yu thnk des the term An bject s at ts equlbrum mean? The bject s ether at rest (Statc Equlbrum) r ts center f mass s mvng wth a cnstant velcty (ynamc Equlbrum). When d yu thnk an bject s at ts equlbrum? Translatnal Equlbrum: Equlbrum n lnear mtn Is ths t? d d C Apr. 1, Sprng 22 The abve cndtn s suffcent fr a pntlke partcle t be at ts statc equlbrum. Hwever fr bject wth sze ths s nt suffcent. One mre cndtn s needed. What s t?  Let s cnsder tw frces equal magntude but ppste drectn actng n a rgd bject as shwn n the fgure. What d yu thnk wll happen? The bject wll rtate abut the C. The net trque actng n the bject abut any as must be. τ r an bject t be at ts statc equlbrum, the bject shuld nt have lnear r angular speed. v ω C 2
3 re n Cndtns fr Equlbrum T smplfy the prblems, we wll nly deal wth frces actng n y plane, gvng trque nly alng zas. What d yu thnk the cndtns fr equlbrum be n ths case? The s pssble equatns frm the tw vectr equatns turns t three equatns. τ τ z y What happens f there are many frces eertng n the bject? If an bject s at ts translatnal statc equlbrum, and f the net trque actng n the bject s abut ne as, the net trque must be abut any arbtrary as. r Or 5 O Net rce eertng n the bject Net trque abut O Pstn f frce abut O Apr. 1, 22 O ' Sprng 22 τ O r1 1 + r2 2 + r3 3 + r r ' r r' τ r r' τ O 3 ' r r' Net trque abut O ( ) ( ) τ O' r 1 + r' r1 r' 1 + r2 r'
4 Center f Gravty Revsted When s the center f gravty f a rgd bdy the same as the center f mass? Under the unfrm gravtatnal feld thrughut the bdy f the bject. m g m 3 g 3 C CG m 1 g 1 m 2 g 2 Let s cnsder an arbtrary shaped bject The center f mass f ths bject s Apr. 1, Sprng 22 y C C m m m y m m Let s nw eamne the case wth gravtatnal acceleratn n each pnt s g Snce the CG s the pnt as f all the gravtatnal frce s eerted n, the trque due t ths frce becmes ( m g m g + ) m g + m g CG If g s unfrm thrughut the bdy ( m1 + m2 + ) gcg ( m11 + m22 + ) CG m m C g m y Generalzed epressn fr dfferent g thrughut the bdy 4
5 Eample 12.1 A unfrm 4. N bard supprts a father and daughter weghng 8 N and 35 N, respectvely. If the supprt (r fulcrum) s under the center f gravty f the bard and the father s 1. m frm CG, what s the magntude f nrmal frce n eerted n the bard by the supprt? g 1m n g Apr. 1, Sprng 22 g Therefre the magntude f the nrmal frce etermne where the chld shuld st t balance the system. The net trque abut the fulcrum by the three frces are Therefre t balance the system the daughter must st τ Snce there s n lnear mtn, ths system s n ts translatnal equlbrum y g + g + g n n N g + g 1. g g 8 1.m 1.m 2. 29m g 35 5
6 τ Snce the nrmal frce s Eample 12.1 Cntnued Apr. 1, Sprng 22 etermne the pstn f the chld t balance the system fr dfferent pstn f as f rtatn. The net trque abut the as f rtatn by all the frces are ( 1. + / 2) n / 2 g / 2 g / 2 + g The net trque can be rewrtten Therefre g 1m Rtatnal as n /2 g τ g 8 1.m 1.m 2. 29m g 35 n g + g + g / 2 + g ( 1. + / 2) ( g + g + g ) / 2 g g 1. g g g / 2 What d we learn? N matter where the rtatn as s, net effect f the trque s dentcal. 6
7 Eample 12.2 A persn hlds a 5.N sphere n hs hand. The frearm s hrzntal. The bceps muscle s attached 3. cm frm the jnt, and the sphere s 35.cm frm the jnt. nd the upward frce eerted by the bceps n the frearm and the dwnward frce eerted by the upper arm n the frearm and actng at the jnt. Neglect the weght f frearm. O d U l mg rm the rtatnal equlbrum cndtn Thus, the frce eerted by the bceps muscle s rce eerted by the upper arm s Apr. 1, Sprng 22 Snce the system s n equlbrum, frm the translatnal equlbrum cndtn d y τ mg l U U mg + d mg l mg l N d 3. U mg N 7
8 Eample 12.3 A unfrm hrzntal beam wth a length f 8.m and a weght f 2N s attached t a wall by a pn cnnectn. Its far end s supprted by a cable that makes an angle f 53. wth the hrzntal. If 6N persn stands 2.m frm the wall, fnd the tensn n the cable, as well as the magntude and drectn f the frce eerted by the wall n the beam. 2m 53. 8m rm the rtatnal equlbrum Usng the translatnal equlbrum Rcsθ Rsnθ θ tan R 6Ν Rcsθ θ Rsnθ T cs53. T sn T 2Ν Tsn53 Tcs53 τ T sn53. T 313N 8313 sn cs N + 2N 71.7 rst the translatnal equlbrum, usng cmpnents y Rcsθ T cs53. Rsnθ + T sn53. 6N 2N 8.6N 2.2N 4.m And the magntude f R s T cs cs53. R 582N csθ cs71.1 Apr. 1, Sprng 22 8
9 Eample 12.4 A unfrm ladder f length l and weght mg5 N rests aganst a smth, vertcal wall. If the ceffcent f statc frctn between the ladder and the grund s µ s.4, fnd the mnmum angle θ mn at whch the ladder des nt slp. l θ Thus, the nrmal frce s Apr. 1, Sprng 22 n O The mamum statc frctn frce just befre slppng s, therefre, rm the rtatnal equlbrum f P mg θ rst the translatnal equlbrum, usng cmpnents n mg 5N y f P mg ma f s µ sn.4 5N 2N l τo mg csθ mn + Plsnθ 2 1 mg 1 5N tan tan 2P 4N mn + mn n 51 P 9
10 Elastc Prpertes f Slds We have been assumng that the bjects d nt change ther shapes when eternal frces are eertng n t. It ths realstc? N. In realty, the bjects get defrmed as eternal frces act n t, thugh the nternal frces resst the defrmatn as t takes place. efrmatn f slds can be understd n terms f Stress and Stran Stress: A quantty prprtnal t the frce causng defrmatn. Stran: easure f degree f defrmatn It s emprcally knwn that fr small stresses, stran s prprtnal t stress The cnstants f prprtnalty are called Elastc dulus Elastc dulus stress stran Three types f Elastc dulus 1. Yung s mdulus: easure f the resstance n length 2. Shear mdulus: easure f the resstance n plane 3. ulk mdulus: easure f the resstance n vlume Apr. 1, Sprng 22 1
11 Yung s dulus Let s cnsder a lng bar wth crss sectnal area A and ntal length L. L Tensle stress A:crss sectnal area TensleStress Yung s dulus s defned as L f L + L e After the stretch e Y e A Tensle stran Tensle Tensle Stress Stran e n e L n TensleStran A L L L Used t characterze a rd r wre stressed under tensn r cmpressn What s the unt f Yung s dulus? Epermental Observatns 1. r fed eternal frce, the change n length s prprtnal t the rgnal length 2. The necessary frce t prduce a gven stran s prprtnal t the crss sectnal area Apr. 1, Sprng 22 rce per unt area Elastc lmt: amum stress that can be appled t the substance befre t becmes permanently defrmed 11
12 Shear dulus Anther type f defrmatn ccurs when an bject s under a frce tangental t ne f ts surfaces whle the ppste face s held fed by anther frce. f s A h ed face After the stress f s Shear stress Shear Stress Tangental rce Surface Area the frce apples A Shear stran Shear Stran h Shear dulus s defned as S Shear Shear Stress Stran A h Apr. 1, Sprng 22 12
13 ulk dulus ulk dulus characterzes the respnse f a substance t unfrm squeezng r reductn f pressure. After the pressure change V V ulk dulus s defned as ecause the change f vlume s reverse t change f pressure. Apr. 1, Sprng 22 Nrmal rce Surface Area the frce apples Vlume stress Pressure pressure If the pressure n an bject changes by P /A, the bject wll underg a vlume change V. Vlume Vlume Stress Stran V A V A P V V Cmpressblty s the recprcal f ulk dulus 13
14 Eample 12.7 A sld brass sphere s ntally under nrmal atmspherc pressure f N/m 2. The sphere s lwered nt the cean t a depth at whch the pressures s N/m 2. The vlume f the sphere n ar s.5m 3. y hw much ts vlume change nce the sphere s submerged? Snce bulk mdulus s The pressure change P s The amunt f vlume change s P V V V PV rm table 12.1, bulk mdulus f brass s N/m 2 P P f P Therefre the resultng vlume change V s V Apr. 1, Sprng 22 V f V The vlume has decreased m 3
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