Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_1$ and $R_2$ (figure $9-E5$).

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 ($=\frac{4R}{3\pi }$)

now considering full plate of mass$M$ and area $A_2=\frac{\pi R_2^2}{2}$

Then area for part R_1 radius will be $A=\frac{\pi R_1^2}{2}$

now using these in eq 1 we get required COM$=\frac{4(R_2^3-R_1^3)}{3\pi \left(R_2^2-R_1^2\right)}$

$=\frac{4(R_2-R_1)(R_1^2+R_2^2+R_1{R}_2)}{3\pi (R_2-R_1)(R_2+R_1)}$

$=\frac{4(R_1^2+R_2^2+R_1{R}_2)}{3\pi (R_1+R_2)}$ this is required center of mass above the center.