Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_{1}$ and $R_{2}$ (figure $9 - E 5$ ).

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Solution

We can use concept of parts cut out of solid bodies for finding COM $= \frac{A_{1} x_{1} - A_{2} x_{2}}{A_{1} - A_{2}}$ ..(1),
Here we use concept of COM of semi circular plate ( $= \frac{4 R}{3 π}$ )
now considering full plate of mass $M$ and area $A_{2} = \frac{π R_{2}^{2}}{2}$
Then area for part R_1 radius will be $A = \frac{π R_{1}^{2}}{2}$
now using these in eq 1 we get required COM $= \frac{4 (R_{2}^{3} - R_{1}^{3})}{3 π (R_{2}^{2} - R_{1}^{2})}$
$= \frac{4 (R_{2} - R_{1}) (R_{1}^{2} + R_{2}^{2} + R_{1} R_{2})}{3 π (R_{2} - R_{1}) (R_{2} + R_{1})}$
$= \frac{4 (R_{1}^{2} + R_{2}^{2} + R_{1} R_{2})}{3 π (R_{1} + R_{2})}$ this is required center of mass above the center.

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