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Question

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii R1 and R2 (figure 9E5).
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Solution

We can use concept of parts cut out of solid bodies for finding COM =A1x1A2x2A1A2 ..(1),
Here we use concept of COM of semi circular plate (=4R3π)
now considering full plate of massM and area A2=πR222
Then area for part R_1 radius will be A=πR212
now using these in eq 1 we get required COM=4(R32R31)3π(R22R21)
=4(R2R1)(R21+R22+R1R2)3π(R2R1)(R2+R1)
=4(R21+R22+R1R2)3π(R1+R2) this is required center of mass above the center.

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