Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_{1}$ and $R_{2}$ .

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Solution

The centre of mass of the blate will be on the symmetrical axis.

${¯ Y}_{c m}$ = $\frac{(\frac{π R_{2}^{2}}{2}) (\frac{4 R_{2}}{3 π}) - (\frac{π R_{1}^{2}}{2}) (\frac{4 R_{1}}{3 π})}{\frac{π R_{2}^{2}}{2} - \frac{π R_{1}^{2}}{2}}$

= $\frac{(\frac{2}{3}) R_{2}^{3} - (\frac{2}{3}) R_{1}}{\frac{π}{2} (R_{2}^{2} - R_{2}^{2})}$

= $\frac{4}{3 π} \frac{(R_{2} - R_{1}) (R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{(R_{2} - R_{1}) (R_{2} + R_{1})}$

= $\frac{4}{3 π} \frac{(R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{R_{1} + R_{2}}$ above the centre.

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