Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii `R_1 and R_2`.

Explanation

`y_(cm)=(intydm)/M`
or `dm=(Mpirdr)/(pi(R_2^2-R_1^2)/2`
`M/(R_2^2-R_1^2)/2`

`y_(CM)=int_R_1^R_2((2r)/pi).M/(R_2^2-R_1^2)/2.(rdr)/M`
`=2/pi (R_2^2-R_1^2)/2 int_R_1^R_2r^2 dr`
`=2/pi(R_2^2-R_1^2)/2[(R_2^3-R_1^3)/3]`
`=(4(R_1^2+R_1R_2+R_2^2)/(3pi(R_1+R_2)`