Objectives. Learning Outcome. 7.1 Centre of Gravity (C.G.) 7. Statics. Determine the C.G of a lamina (Experimental method)

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1 Ojectves 7 Statcs 7. Cete of Gavty 7. Equlum of patcles 7.3 Equlum of g oes y Lew Sau oh Leag Outcome (a) efe cete of gavty () state the coto whch the cete of mass s the cete of gavty (c) state the coto fo the equlum of a patcle () solve polems volvg foces equlum at a pot (e) efe toque as = 3 (f) state the cotos fo the equlum of a g oy (g) sketch a lael the foces whch act o a patcle a a g oy (h) use the tagle of foces to epeset foces equlum () solve polems volvg foces equlum Statcs Statcs equlum s a state whee alace foces actg o a g oy o a patcle so that t emas at est. 7. Cete of Gavty (C.G.) he Cete of Gavty (C.G.) of a oy s the pot though whch the whole weght of the oy appeas to act o a egula oject, the C.G s always at the cete Cete of Gavty (C.G.) Deteme the C.G of a lama (Epemetal metho) Le of acto of ts weght must pass though the cete of gavty. hug feely at hug feely at at est, aw vetcal le 7 at est, aw vetcal le G s cete of gavty 8

2 he Cete of Gavty of a Lama (athematcal metho) ssumpto: Lama s mae up of patcles cooate of CG ahatta e ss (H.K.) Lt. cooate of each patcle he Cete of Gavty of a Lama If g = costat ( a ufom gavtatoal fel), g = g = g 3 =... = g = g, the ( m ) Smlaly, y takg momets aout the -as, ( m y ) y Cete of ass (C..) vs. Cete of Gavty (C.G.) Cete of mass of a oy s efe as the pot whee the ete mass of the oy acts. Cete of gavty of a oy s efe as the pot whee the weght of the oy acts. he Cete of Gavty of a Lama akg momets aout the y-as, X avg = ( ) = (m g ) + (m g ) + (m 3 g 3 ) (m g ) ( m g ) ( m g ) he Cete of Gavty of a Lama Cooates of cete of gavty: ( m ) ( m y ) Note: hese ae fact the cooates of the cete of mass. Eample of C.G.: y If a oy s ufom a egula shape, the the c.g. s at the geometcal cete. c.g. c.g. c.g. c.g. Cete of cyle 3 7. Equlum of patcles oy s alace f pvote at a pot whch passes though c.g.. c.g. 7. Equlum of patcles he coto fo a patcle statc equlum state. (a) Resultat foce s zeo. he ufom ule aove s alace. If = - he the esultat foce, et = + = 6

3 7. Equlum of patcles hus, + = he vecto sum of the foces must e zeo. Eample Let thee foces,, a 3 actg o a patcle O, whch s equlum, as show: (y) (y) () 3() O () 7 he, (compoets ) () + () + 3() =, a (compoets y) (y) + (y) + 3(y) = 3 3(y) 8 lteatve metho Close polygo Vecto sum of the foces must e zeo = 3 C 3 O 9 Cose thee foces actg o a patcles s equlum. he foces ca e joe to fom a polygo. he esultat foce = 3 Close polygo If the foces actg o a patcle s ot equlum, a esultat foce s estg. he legth of the ses epesetg the magtue of the foces. 3 he ecto of foce s epesete y ts aow. 4 Resultat, 3 3 Resultat Resultat oce, = Equlum of a Rg oy he followg shows a o alace, whch s so calle statc equlum. he weght of the o s suppote y the eacto, R R 7.3 Equlum of a Rg oy wo cotos ae ecessay: No esultat foce (R = weght) No esultat toque (omet = ) R Suppot weght weght 3 4

4 oque If a foce, s apple o the oe e of the o, t wll stat to otate aout G. toque s pouce y the foce, whee toque, = G oque oque ca e pouce y two paallel foces of the magtue, ut opposte ecto of a stace fom the otatg as. he pa of foces = a couple = m = m weght 5 6 oque he a couple (toque), f ests, wll cause a momet (toque) aout the otatg as, whch s gve y = = () Hece the oy s ot equlum. (eteal foce) = m = m 7 oe aout oque oque s the teecy of a foce to otate a oject aout a as. oque,, s a vecto quatty. Cose a oject pvotg aout the pot y the foce eg eete at a stace. omet am Le of cto 8 oe aout oque he le that etes out of the tal of the foce vecto s calle the le of acto. he pepecula stace fom the pvotg pot to the le of acto s calle omet am. omet am oe aout oque Le of cto he thee ae moe tha oe foce eg eete o ceta pots of the oject, oe ca sum up the toque geeate y each foce vectoally. he coveto fo sg of the toque s postve f otato s coute-clockwse a egatve f clockwse. s 9 3 oe aout oque agtue of toque s efe as the pouct of the foce eete o the oject to otate t a the momet am. s S omet am Le of cto 7.3 Equlum of a Rg oy If a mass of kg s put o oe e, the o wll otate clockwse ecto. hle the foce, = N he toque, = = ()() = Nm Suppot = m = N 3 3

5 7.3 Equlum of a Rg oy If aothe mass of kg s apple o the othe e, the system wll e statc equlum aga, whee R = +, o R + + =. R = N m m = N Suppot =N Eample he fgue shows a o of legth m a weght kg estg o suppots at e a. the eacto at a. (g = ms - ) Equlum of a Rg oy I ths case, the sum of momet/toque = = ()() ()() =. m m = N Suppot =N Decto of Rotato Soluto Let R = eacto at pot, R = eacto at pot, = foce ue to weght of the o. 34 R R = N Soluto Sce the system s equlum, Resultat oce = Soluto Sum of momet aout, = R (.5 + ) () = R = /.5 =6.67 N om (), R = 6.67 =3.33 N R R R R m = N 37 = N 38 Soluto OR, Sum of momet aout, = R (.5 + ) (.5) = R = 5/.5 = 3.33 N R = 3.33 = 6.67N R R m = N 39 Eample ufom hozotal eam wth a legth of l = 8. m a a weght of = N s attache to a wall y a p coecto a suppote y a cale as show. peso of weght p = 6 N stas a stace =. m fom the wall. 4

6 Eample Soluto the teso the cale as well as the magtue a ecto of the foce eete y the wall o the eam. 4 alyze Daw a fee oy agam Use the pvot the polem (at the wall) as the pvot hs wll geeally e easest Note thee ae thee ukows (, R, q) 4 Soluto he foces ca e esolve to compoets the fee oy agam. Soluto pply the two cotos of equlum to ota thee equatos Solve fo the ukows akg momet aout R, 43 z ( s )( l ) l p ( ) l s p l ( ) (6N )(m) (8m)s 53 (N)(4m) 33N 44 Soluto Eample 3 Coseg sum of a sum of y, y R s ta R cos ta Rcos cos Rs s p s s s cos (33N) cos53 R cos cos 7.7 p s p N safe whose mass s = 43 kg s hagg y a ope fom a oom wth mesos a =.9 m a =.5 m as show the fgue. he ufom eam has a mass m of 85 kg; the mass of the cale a ope ae eglgle. a Eample 3 Soluto a) hat s the teso c the cale;. e., what s the magtue of the foce c o the eam a fom the hozotal cale? ) hat s the foce at the hge? Cose the sum of a sum of y, ; et, et, h c h c et, y et, y v ; c v 47 h 48

7 Soluto Soluto akg momet aout the hge et, z et, z ca ca c a a h v ( m ) g a c v h c 693N h v c 547N h 693N v 79N v h c 49 5 Summay Statcs Equlum of atcles Equlum of Rg oes Cete of Gavty Cete of ass Resultat oce = agle of foces oque, = Resultat foce = Resultat toque = 5