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Date Created: 10/21/15
Spring 2005 Math 152 9 Further Applications of Integration 95E Moments and Centers of Mass Fri 04Mar 2005 Art Belmonte Executive Summary Please see 95R the regular lecture for the full treatment The point on which a thin at plate lamina balances horizontally is called the center of mass CM or center of gravity of the plate In this situation mass is distributed continuously Let p be the mass density of the plate Here p may be constant or variable ie depend on x andor y D ifDp mid1 The moment with respect to the xaxis is Mx my whereas the moment with respect to the yaxis is My mx Recall that the directed distance from a point to the xaxis is y whereas said distance to the yaxis is x mass m center ofmass 2 y You do these problems with machines folksieither your TI89 calculator or a computer with MATLAB To convince yourself of this see the hand examples in 95R the regular lecture TI89 Examples 5606 Find the centroid center of mass of a at plate of uniform density of the region bounded by the curves y l 7 x2 andy 0 Solution Here is a diagram showing the region and its center of mass SEWER SEE6 cemrum Elf a amma First compute the mass 1 17x2 mpdA pdydx4i D 71 0 3 then compute the center of mass Here p is constant iA xdldl 1 1 17x2 Plxayldydx 0 000 040 N These two steps are easy to do on a TI89 especially if you use the Muint mulitple integral menu in the TAMJCALC package You ll nd p on the Cale menu 9 ffpy0l ixAZx7ll gtm o ffpxyy0lixAZx7llm The second command is even easier than you think Just change the gt from the store operation in the entry line to for division and postmultiply p by x y 560l10 Find the centroid center of mass of a at plate of uniform density ofthe region bounded by the curves y sinx y x an x 7 Solution Here is a diagram showing the region and its center of mass Statan SEEin CEHUDM Ufa amma ente urmass ins us 1 15 First compute the mass 712 sinx mpdA pdydxp D 0 0 then compute the center of mass Here p is constant D xayld14 712 sinx 7 f f p x y dy dx p 0 0 1 w 100039 M r l m l o ffpy0sinxx07r2 gtm o f g xyy0sinxx07r2m 8326 from Section 136 in Math 253 Gale 3 Find the mass and center of mass of the lamina at plate that occupies the triangular region D in the xyplane with vertices 0 0 l l and 4 0 and has variable density p x Solution Here is a plot showing the region and its center of mass Stewart 8326 X With our general formulation the fact that the density is variable presents no dif culty whatsoever First compute the mass 1 473y 10 mpdA xdxdyi D 0 y 3 then compute the center of mass 1 7 pixydA m D 1 1 44y xixydxdy 21 3 if 2103 10 10 W o xgtp o ffpxy473yy0 1 em 9 fHMxyxy473yy01m o DelVar p Note that NewProb clears only single letter Roman identi ersinot Greek ones MATLAB Examples We ll repeat the Tl89 examples using MATLAB The syntax for the integrals involved is almost identical Just replace f with int For brevity we ll type p for the density instead of rho for p 560x06 5606 revisited Find the centroid center of mass of a at plate of uniform density of the region bounded by the curves y l 7 x2 andy 0 Solution Here is a diagram showing the region and its center of mass SEWER SEE6 cemrum Elf a lamina 3 33 Stewart 5606 3 syms p x y m mt1ntp y o prettym 17262 X 71 1 43 p 1m 1nt1ntpX y y 01ex 2x 711 CM pretty CM 0 2 5l 560x10 56010 revisited Find the centroid center of mass of a at plate of uniform density of the region bounded by the curves y sinx y 0 x 0 an x l 2 Solution Here is a diagram showing the region and its center of mass Statan SEEin CEHUDld Ufa lamina ente urmass 3 33 Stewart 56010 syms p x y 1nt1ntp y o prettym s1nx X 0 p127 P 1m 1nt1ntpX y y O s1nx x 0 CM p12 prettyCM CM evalC M CM 10000 03927 s832x06 a Stewart 56018 Find the mass and center of mass of the lamina at plate that syms p x y occupies the triangular regionD in the xyplane with vertices m mt mtlp39 3quot 039 X 39 X39 039 1 I mt1ntpy01xX127 0 0 l l and 4 0 and has vanable dens1ty p x prettym 12 p 1092 p CM Solutlon 1m mtmtpxylyoxx 01 int1ntpx y y o 1x x 1 2 CM s1mp1eCM CM Here is a plot showing the region and its center of mass 836log2 56121092l StwanEBZ prettyCM 8 5 l t t t t t t t t t t t 7 3 6 1092 6 12 1092 CM evalC M gt CM 11175 03492 echo off diary off 560x21 3 33 Stewart 8326 3 Prove that the centroid of a triangular region is located at the point syms x y of intersection of the medians of the triangluar boundary p X m 1ntmtpx y 473w y 0 1 pretty m Solution 1 03 CM 1m 1nt1ntp x y x y 473w y o 1 pretty CM Here is a diagram showing the triangular region and its center of mass You can examine all the analytical work in the code but the picture tells the story Stewart 56021 Centroid of a triangular lamina 560x18 of mass Find the centroid of the region bounded by y x y 0 y lx andx 2 Solution Here is a diagram showing the region and its center of mass There are two subregions so we must split the integrals involved Stewart Earlma A umpuste reglun Stewart EBB18b Certruid of a umpuste reglun x Enterufmass a gt gt stewart 56021 24 U U syms a b h p x y a a Vertlces l 2 l 2 V1OO7V2hal7V3Obl7 X X Slanted s es L1 c11ne2ptV1 v2 prettyL1 The center ofmass 1s a x 7 7 8 5 l 1175 03492 y x y 7 7 N h 36ln256lZln2 L2 c11ne2ptV3 v2 prettyL2 3 Medlan llnes M1 c11ne2ptV1 v2v32 prettyM1 M2 c11ne2ptV2 v1v32 prettyM2 2axrxbbh M3 c11ne2ptV3 v1v22 prettyM3 3 Common mtersectlon of medlans Il solveM1 M2 y 12 solveM1 M3 y 13 solveM2 M3 x y I 11x I1y12x 12 x x 13x I3y prettyl 13h 13a13b 13h 13a13b 13h 13a13b 3 m 1nt1ntpy Y 8Xh arbXh b X 0 h prettym 7 b 2 7 7 ah h p b h a 12 p 7 h CM 1m 1nt1ntpx y y axh 7wa11 b x o h CM leCM7 prettyCM 3 Centrold 1s at mtersectlon of med1ans 13 h 13 a 13 13 3 echo off dlary off
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