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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 ฯ(x,y)=kyin the formula of the center of mass x.

role="math" localid="1650347825085" xยฏ=โˆฌฮฉxฯ(x,y)dAโˆฌฮฉฯ(x,y)dAxยฏ1=โˆซโˆ’33โˆซ46xkydydxโˆซโˆ’33โˆซ46kydydxxยฏ1=kโˆซโˆ’33y2246xdxkโˆซโˆ’33y22โˆ’36dxxยฏ1=2kโˆซโˆ’33xdx2kโˆซโˆ’33dxxยฏ1=2kx22โˆ’332k[x]โˆ’33xยฏ1=0

03

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

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

role="math" localid="1650348806680" yยฏ=โˆฌฮฉyฯ(x,y)dAโˆฌฮฉฯ(x,y)dAy1=โˆซโˆ’33โˆซ46ykydydxkโˆซโˆ’33โˆซ46kydydxy1=kโˆซโˆ’33y33โˆ’46kโˆซโˆ’33[y2]-46dxy1=76kโˆซโˆ’33dx15kโˆซโˆ’33dxy1=76k[x]โˆ’3315k[x]โˆ’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 ฯ(x,y)=kyin the formula of the center of mass x.

role="math" localid="1650348765861" xยฏ=โˆฌฮฉxฯ(x,y)dAโˆฌฮฉฯ(x,y)dAx2=โˆซโˆ’11โˆซ04xkydydxkโˆซโˆ’11โˆซ04kydydxx2=kโˆซโˆ’11y2204xdxkโˆซโˆ’11y2204dxx2=8kโˆซโˆ’11xdx8kโˆซโˆ’11dxx2=8kx22โˆ’118k[x]โˆ’11x2=0

05

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

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

y2=โˆฌฮฉyฯ(x,y)dAโˆฌฮฉฯ(x,y)dAy2=โˆซโˆ’11โˆซ04ฮฉykydydxโˆซโˆ’11โˆซ04kydydxy2=kโˆซโˆ’11y3304dxkโˆซโˆ’11y2204dxy2=643โˆซโˆ’11dx8โˆซโˆ’11dxy2=8[x]โˆ’113[x]โˆ’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.

xยฏ=m1xยฏ1+m2xยฏ2m1+m2xยฏ=m(0)+m(0)m+mxยฏ=0yยฏ=m1yยฏ1+m2yยฏ2m1+m2yยฏ=m(7615)+m(83)m+myยฏ=(11615)2yยฏ=5815

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

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