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Question

A semicircular portion of radius r=10 cm is cut from a uniform rectangular plate as shown in the figure. The distance of center of mass C of the remaining plate, from point O is (COM of semicircle is at distance 4r3π from the base)


A
7.75 cm
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B
5 cm
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C
17 cm
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D
4.6 cm
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Solution

The correct option is A 7.75 cm
Remaining area = Area of rectangle - Area of semicircular portion,

Area of rectangle A1=2r2
Area of semicircular portion A2=πr22

If r1 and r2 are the positions of centre of mass of the rectangle and the semicircular portion with respect to O.
r1=r2 and r2=4r3π
(left of origin is taken -ve)

Therefore, center of mass of remaining portion
=A1×(r2)A2×(4r3π)A1A2

=⎜ ⎜ ⎜ ⎜2r2×(r2)πr22×(4r3π)2r2πr22⎟ ⎟ ⎟ ⎟
=⎜ ⎜ ⎜2r324r362r2πr22⎟ ⎟ ⎟=2r3(4π)
(Putting r=10 cm (given))
=2×103(4π)=7.75 cm

So, the distance of center of mass C of remaining plate, from point O is 7.75 cm.

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