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Question

Surface mass density of a semicircular disk varies with position as σ=σ0rR where σ0 is constant, R is the radius of the disc and r is measured from the centre of the disc. Find the COM of the disc.

A
(0,R2)
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B
(0,2Rπ)
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C
(0,3R2π)
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D
(0,4R3π)
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Solution

The correct option is C (0,3R2π)
Cut an elementary ring of radius r and thickness dr.


Mass of elementary ring
dm=σ(πrdr)
=σ0rRπrdr
=σ0πRr2dr
Centre of mass of elementary ring =(0,2rπ)
So, x coordinate of COM of elementary ring, ¯x=0
& ¯y=ydmdm=R02rπσ0πRr2drR0σ0πRr2dr
=2σ0RR0r3drσ0πRR0r2dr=2πR44R33=3R2π
So, coordinates of COM of semicircular disc
(xcom,ycom)=(0,3R2π)

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