Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii R₁ and R₂.
Figure

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Solution

Let the mass of the plate be M.
Consider a small semicircular portion of mass dm and radius r, as shown in fig.

$d m = \frac{M π r d r}{\frac{π (R_{2}^{2} - R_{1}^{2})}{2}} = \frac{M}{\frac{(R_{2}^{2} - R_{1}^{2})}{2}} r d r$
The centre of mass is given as:
$y_{c m} = \frac{\int y d m}{M}$

$y_{c m} = \int_{R_{1}}^{R_{2}} (\frac{2 r}{π}) . \frac{M}{\frac{(R_{2}^{2} - R_{1}^{2})}{2}} \times \frac{r d r}{M} = \frac{2}{π \frac{(R_{2}^{2} - R_{1}^{2})}{2}} \int_{R_{1}}^{R_{2}} r^{2} d r = \frac{2}{π \frac{(R_{2}^{2} - R_{1}^{2})}{2}} [\frac{(R_{2}^{3} - R_{1}^{3})}{3}] = \frac{4 (R_{1}^{2} + R_{1} R_{2} + R_{2}^{2})}{3 π (R_{1} + R_{2})}$

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