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The lamina in the figure that follows is bounded above by the lines with equations y=x+2aand y=-x+2aand below by thex-axis on the interval -axa.The density of the lamina is constant.

Short Answer

Expert verified

The Center of mass of lumina is atx,y=0,7a9.

Step by step solution

01

Step 1. Given information.   

The given lamina is the following.

The density of the given lamina is constant.

02

Step 2. x coordinate Center of mass of the left lamina  

substituting ρ(x,y)=kin the formula of the center of mass xfor left lamina.

role="math" localid="1650352640964" x¯=Ωxρ(x,y)dAΩρ(x,y)dAx¯1=a00x+2axkdydxa00x+2akdydxx¯1=ka0[y]0x+2axdxka0[y]0x+2adxx1¯=ka0[x2+2ax]dxka0[x+2a]dxx¯1=[x33+ax2]a0[x22+2ax]a0x1¯=[2a33][3a22]=49a

03

Step 3. y coordinate the Center of mass of the left lamina  

substituting ρ(x,y)=kin the formula of the center of mass yfor left lamina.

role="math" localid="1650352548682" y¯=Ωyρ(x,y)dAΩρ(x,y)dAy1=a00x+2aykdydxa00x+2akdydxy1=ka0[y22]0x+2adxka0[y]0x+2adxy1=a0[(x2+4ax+4a2)]2]dxa0[x+2a]dxy1=12x33+2ax2+4a2xa0x22+2ax-aay1=12(a332a3+4a3)(a222a2)y1=79a
So the center of mass of the left lamina is atx1,y1=-4a9,7a9.

04

Step 4. x coordinate Center of mass of the right lamina 

substituting ρ(x,y)=kin the formula of the center of mass xfor the right lamina.
role="math" localid="1650352880244" x¯=Ωxρ(x,y)dAΩρ(x,y)dAx2=0a0x+2axkdydx0a0x+2akdydxx2=k0a[y]0x+2axdxk0a[y]0x+2adxx2=0a[x2+2ax]dx0a[x+2a]dxx1=[x33+ax2]0a[x22+2ax]0ax2=[a33+a3][a22+2a2]x2=49a

05

Step 5. y coordinate the Center of mass of the right lamina 

substituting ρ(x,y)=kin the formula of the center of mass yfor the right lamina.

role="math" localid="1650353155945" y¯=Ωyρ(x,y)dAΩρ(x,y)dAy2=0a0x+2aykdydx0a0x+2akdydxy2=0a[y22]0x+2ak0a[y]0x+2adxy2=0a[(x24ax+4a2)2]dx0a[x+2a]dxy2=12[x332ax2+4a2x]0a[x22+2ax]0ay2=12(a332a3+4a3)(a22+2a2)y2=79a
So the center of mass of the right lamina is atx2,y2=4a9,7a9.

06

Step 6. Center of mass of composition of the lamina.    

The Center of mass of composition of the left and right lamina is following.

x¯=m1x¯1+m2x¯2m1+m2x¯=m(49a)+m(49a)m+mx¯=0y¯=m1y¯1+m2y¯2m1+m2y¯=m(79a)+m(79a)m+m)y¯=79a

So the center of mass of the lamina is atx,y=0,7a9.

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