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Use the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.

Short Answer

Expert verified

The Center of mass of the lamina is atX,Y=0,5815.

Step by step solution

01

Step 1. Given information.     

Given lamina is a composition of rectangles.

Density is proportional to the distance from the x-axis.

02

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

substituting ρ(x,y)=kyin the formula of the center of mass x.

role="math" localid="1650347825085" x¯=Ωxρ(x,y)dAΩρ(x,y)dAx¯1=3346xkydydx3346kydydxx¯1=k33y2246xdxk33y2236dxx¯1=2k33xdx2k33dxx¯1=2kx22332k[x]33x¯1=0

03

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

substituting ρ(x,y)=kyin the formula of the center of mass y.

role="math" localid="1650348806680" y¯=Ωyρ(x,y)dAΩρ(x,y)dAy1=3346ykydydxk3346kydydxy1=k33y3346k33[y2]-46dxy1=76k33dx15k33dxy1=76k[x]3315k[x]33y1=7615

So the center of mass of horizontal lamina isx1,y1=0,7615.

04

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

substituting ρ(x,y)=kyin the formula of the center of mass x.

role="math" localid="1650348765861" x¯=Ωxρ(x,y)dAΩρ(x,y)dAx2=1104xkydydxk1104kydydxx2=k11y2204xdxk11y2204dxx2=8k11xdx8k11dxx2=8kx22118k[x]11x2=0

05

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

substituting ρ(x,y)=kyin the formula of the center of mass y.

y2=Ωyρ(x,y)dAΩρ(x,y)dAy2=1104Ωykydydx1104kydydxy2=k11y3304dxk11y2204dxy2=64311dx811dxy2=8[x]113[x]11y2=83

So the center of mass of vertical lamina isx2,y2=0,83.

06

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

Considering the mass of each lamina is m then the Center of mass of composition of the lamina is following.

x¯=m1x¯1+m2x¯2m1+m2x¯=m(0)+m(0)m+mx¯=0y¯=m1y¯1+m2y¯2m1+m2y¯=m(7615)+m(83)m+my¯=(11615)2y¯=5815

So the center of mass of the lamina is atX,Y=0,5815.

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