moment = m! x, where x is the length of the moment arm.

Transcription

1 th 1206 Clculus Sec. 6.7: omets d Ceters of ss I. Fiite sses A. Oe Dimesiol Cses 1. Itroductio Recll the differece etwee ss d Weight.. ss is the mout of "stuff" (mtter) tht mkes up oject.. Weight is mesure of force tht results from grvity ctig o mss. c. Newto's Secod Lw: F=mg, where m is the mout of mss, g is the ccelertio of grvity. I geerl force F, is equl to the product of mss d ccelertio, i.e. F=m System of esure of esure of esuremet ss Force U.S. slug poud = (slug)(ft/sec 2 ) Itertiol kilogrm ewto = (kilogrm)(m/sec 2 ) C-G-S grm dye = (grm)(cm/sec 2 ) Coversios: 1 poud = ewtos 1 slug = kilogrms 1 ewto = pouds 1 kilogrm = slugs 1 dye = pouds 1 grm = slugs 2. Fiite sses Alog Lie.. We egi y cosiderig idelized situtio i which mss m is cocetrted t poit. If x is the directed distce etwee this poit-mss d other poit P, the the momet of m out the poit P is give y momet = m x, where x is the legth of the momet rm. Now let's imgie msses m 1, m 2, m 3 o rigid x-xis supported y fulcrum t the origi. m 1 0 m 2 m 3 " " " " " " " " " " " " " " " " " " " " " " " " # x 1 $ x 2 x 3 % fulcrum Ech mss m k exerts dowwrd force (m k g). Ech of these msses hs tedecy to tur the xis out the origi, the wy you tur seesw. This turig effect is clled torque. Torque T=(m k g)( ) where is the siged distce from the poit of pplictio to the origi. From oservtios we kow tht msses to the left of the origi exert egtive (couterclockwise) torque, while msses to the right of the origi exert positve (clockwise) torque. (Note, i geerl Torque, T=FL, where F is force d L is the distce from where the force is pplied to the ceter or origi. Thik out turig wrech to loose olt, the loger the hdle of the wrech the more torque you c exert to loose the olt (ssumig you pply the force t the ed of the hdle.)

2 The sum of ll the torques mesures the tedecy of system to rotte out the origi. This is clled the system torque. (System Torque=m 1 gx 1 +m 2 gx 2 + m 3 gx 3 + +m gx ) The system will lce o the fulcrum if d oly if the system torque=0 (we sy such system is i equilirium/lce.) Let's exmie the system torque formul little more: T=m 1 gx 1 +m 2 gx 2 +m 3 gx 3 + +m gx = g(m 1 x 1 +m 2 x 2 +m 3 x 3 + +m x sice ech m k is momet, we cll there sum the momet of the system out the origi, o. (omet, o=product of mss m of prticle y its directed distce from tht pt x ) o = m k, where is the umer of msses. Notice tht system will e i equilirium/lce if d oly if o = 0, sice g is rrely zero. We usully c't move the msses i system, ut we c move the Fulcrum i order to mke the system lce, i.e so tht o =0, (T=0). We lel tht poit x. The system's ceter of mss, x, is give y the formul: x = m k = m k o = system momet out origi system mss m 1 0 m 2 x m 3 " " " " " " " " " " " " " " " " " " " " " " " " # x 1 $ x 2 % x 3 fulcrum ceter of mss. Formuls for omet, ss Ceter of ss for Fiite sses Alog Lie. ** omet out the origi : o = ss: m k Ceter of ss : x = o m k m k m k **

3 c. Exmples 1.) Three odies of mss 6kg, 4kg 10kg re locted t x 1 =-2, x 2 =4 x 3 =9 respectively. If the distces re mesured i meters, fid the ceter of mss. 2.) Two childre wt to lce o seesw. Oe is 35ls d sits 4 ft from the ceter. Where should the other child sit if he weighs 50ls? B. Two Dimesiol Cses 1. sses Distriuted Over Ple Regio. Developmet of formuls 1.) Suppose we hve fiite collectio of msses i ple. 2.) Ech mss m k is locted t the pt (,y k ). 3.) The system mss: m k 4.) Ech mss hs momet out ech xis (the x-xis d the y-xis)..) m k s momet out the x-xis is give y m k y k..) m k s momet out the y-xis is give y m k. 5.) The systems etire momets out the two xes will e:.) omet out the x-xis: x = m k y k m k.) omet out the y-xis: y = 6.) The system s Ceter of ss is defied to e X, Y X = y ( m k ) m k 7.) Note: with X d Y defied this wy: d Y = x.) The system will lce out the lie x = X..) The system will lce out the lie y = Y. ( ) where ( m k y k ) m k

4 2. Exmples Fid the ceter of mss with uiform desity of the regio show, ot y itegrtio, ut y loctig the ceters of the rectgles d trigles d tretig them s poit msses. NOTE 1: Although the followig exmples re ot truly fiite msses i ple, we will fid the cetroid usig the dditio of the poit msses i ech regio. NOTE 2: The cetroid of trigle is locted t the poit of itersectio of the medis. A medi is the lie from vertex to the midpoit of the opposite side. edis itersect t pt 1 3 of the wy from ech side towrd the opposite vertex. 1.) Fid the cetroid of the trigle whose vertices re (0,0), (0,2), (3,0) 2.)

5 3.) II. Uiform Desity Alog Lie or Over Ple A. Cotiuous ss Distriutio Alog Lie Wires d Thi Rods with Vryig Desity 1. Recll: Desity, δ, is defied to e mteril's mss per uit volume. However, we ofte use differet uits for our ow coveiece, (i.e. mss per uit legth or mss per uit re.). 2. For the ceter of mss of rod or thi strip of metl, we c model the distriutio of mss with cotiuous fuctio, the the summtio formuls we've see thus fr c e replced with itegrls. 3. Derivtio of Formul: Tke log, thi strip of metl lyig log the x-xis from x = to x =. Prtitio the strip ito smll pieces of mss Δm k. The kth piece is Δ uits log d lies pproximtely uits from the origi. m k " # # # # # # $ Oserve: 1.) The strip's ceter of mss is erly the sme s tht of the system of poit msses we would get y ttchig ech mss Δm k to the poit : so we hve x system momet out origi system mss 2.) The momet of ech piece of the strip out the origi is pproximtely Δm k, so the system momet is pproximtely: o # "m k 3.) If the desity of the strip t is δ( ), expressed i terms of mss/uit legth, d δ is cotiuous, the: m k " # ( )

6 system momet out origi Therefore: x system mss # # "m k "m k # # $ ( )" $ ( )" 4.) Notice tht the umertor d deomitor of the lst sum re oth Riem Sums. The umertor is Riem sum for the cotiuous fuctio xδ(x) d the deomitor is Riem sum for the fuctio δ(x), oth over the closed itervl [,]. Thus s P 0 we rrive t the followig formul: x = system momet out origi system mss = o x ( x) "# $% dx ( x) "# $% dx 4. Formuls for omet, ss Ceter of ss of Thi Rod Alog the x-xis with Desity Fuctio, δ(x): ** omet out the origi : o = ss : ( x) "# $% dx Ceter of ss : X = o x ( x) "# $% dx x ( x) "# $% dx ( x) "# $% dx 5. Commets:.) The ceter of mss of stright, thi rod or strip of costt desity lies hlfwy etwee its eds..) We c tret rod of vrile thickess s rod of vrile desity. 6. Exmple A 16cm log wire hs lier desity mesured i g/cm, give y ( x)= x, 0<x<16. Fid the ceter of mss. **

7 B. Thi, Flt Ples 1. Developmet of formuls. We ssume tht the mss is cotiuously distriuted i oe of these pltes. Ofte such plte is clled plr lmi. The just like whe we moved from the poit mss system i 1-D, to the thi strips, the formuls give ove ivolvig summtios will ecome formuls ivolvig itegrls.. Imgie cuttig plr lmi i the xy-ple ito thi strips prllel to oe of the xes. The ceter of mss of smple strip is x, y ( ). c. Ech smple strip s mss Δm k is treted s if it were cocetrted t ( x, y ). d. The momet of ech smple strip out e. The system s ceter of mss ecomes X = y " ( ) x k m k " m k d # the y xis is x "m $ % the x xis is y "m Y = x " ( ) " y k m k m k f. Tkig the limits s m k " 0 we get the ceter of mss formuls: X = y x dm dm d Y = x y dm dm 2. Formuls for omet, ss Ceter of ss of Lmi with Desity Fuctio, (x). ss : dm = " x ( ) da omet out the y # xis : y = omet out the x # xis : x = x dm = y dm = x " x ( ) da y " x ( ) da Ceter of ss : X = y, Y = x Whe itegrtig wrt x: x = x, y = t x ( ) + ( x) 2 ( ) ( x) d da = "# t x $% dx, where t ( x) =top curve, ( x) =ottom curve d =desity fuctio costt. Therefore, the formuls ove c e rewritte s

8 ** ss : dm = "(x) ( t ( x) # ( x) )dx omet out the y # xis : y = ["(x)] x ( t ( x) # ( x) )dx omet out the x # xis : x = "(x) t x ( ( ) + ( x) ) 2 ( t ( x) # ( x) )dx = 1 2 "(x) ( $% t ( x) ' 2 # $% ( x ) ' 2 )dx ** Ceter of ss : X = y, Y = x Whe itegrtig wrt y: x = r y ( ) + l( y) 2 ( ) l( y), y = y d da = "# r y $% dy, where r( y) =right curve, l( y) =left curve d ( y) =desity fuctio. ss : dm = "(y) ( r( y) # l( y) )dy omet out the x # xis : x = ["(y)] y ( r( y) # l( y) )dy omet out the y # xis : y = "(y) r y ( ( ) + l( y) ) 2 ( r( y) # l( y) )dy = 1 2 "(y)( $% r( y) ' 2 # $% l ( y ) ' 2 )dy Ceter of ss : X = y, Y = x 3. Cetroid. Def : If the desity fuctio δ(x) is costt δ, the ceter of mss is clled cetroid.. Whe the desity fuctio is costt, it ccels out of the umertor d deomitor of the formuls for X Y. So some will use δ=1, i this cse. c. Whe the desity fuctio is costt, the loctio of the ceter of mss is idepedet of the mteril with which it is mde of, ut is oly depedet o the geometry of the oject. 4. Exmples

9 . Fid the ceter of mss of uiform desity of thi plte coverig the regio ouded y y=1-x 2 d y=x-1.

10 . Fid the ceter of mss of plr lmi coverig the regio etwee the x- xis, the curve y = x, 0 x 4, if the pltes desity t the poit (x,y) is (x) = x 2.

11 c. Fid the ceter of mss of thi plte coverig the regio ouded y x = 6 y 2 d x = 3 2y if the plte s desity is(y) = y.

12 d. Clculte the momets, x d y, d the ceter of mss of lmi with desity δ=3 d the give shpe.