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Question

A rectangular lamina in the xy plane is shown below:


Density of this lamina is constant along x axis and varies along y axis. If the variation in density is given as ρ(y)=16ρ0y2, then

A
xCOM=l2
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B
yCOM=b2
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C
xCOM=l4
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D
yCOM=3b4
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Solution

The correct options are
A xCOM=l2
D yCOM=3b4

Taking an element parallel to x-axis at a distance y from the x axis.
Area of element =dA=ldy
and Mass of element =dm=ρdA=16ρ0y2ldy

Therefore, ycom=b0ydmb0dm
=b0y×16ρ0y2ldyb016ρ0y2ldy
=16ρ0lb0y3dy16ρ0lb0y2dy
=[y44]b0[y33]b0=34b

Density is constant along x - axis
xcom=l2
Thus, coordinates of COM =[l2,3b4]

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