Question

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

Answer 1

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

\[\text{dm} = \frac{M\pi rdr}{\frac{\pi\left( R_2^2 - R_1^2 \right)}{2}} = \frac{M}{\frac{\left( R_2^2 - R_1^2 \right)}{2}}\text{rdr}\]
The centre of mass is given as:
\[y_{cm} = \frac{\int y \text{dm}}{M}\]
\[y_{cm} = \int_{R_1}^{R_2} \left( \frac{2r}{\pi} \right) . \frac{M}{\frac{\left( R_2^2 - R_1^2 \right)}{2}} \times \frac{\text{rdr}}{M}\]
\[ = \frac{2}{\pi\frac{\left( R_2^2 - R_1^2 \right)}{2}} \int_{R_1}^{R_2} r^2 dr\]
\[ = \frac{2}{\pi\frac{\left( R_2^2 - R_1^2 \right)}{2}}\left[ \frac{\left( R_2^3 - R_1^3 \right)}{3} \right]\]
\[ = \frac{4\left( R_1^2 + R_1 R_2 + R_2^2 \right)}{3\pi\left( R_1 + R_2 \right)}\]