Locate `c.m.` of thin , uniform semicircular plate.

Explanation

`Mass//area = (M)/(pi R^(2) //2) = ( 2M)/(pi R^(2))`
Here , the elements is semicircular ring of radius and thickness for
`dA = pi r dr`
`dm = ( 2M)/( pi R^(2)) xx pi r dr = ( 2M)/(R^(2)) rdr`
The coordinates of the `c.m.` of element , `y = ( 2r)/(pi)`
`y_(c.m.) = (int y dm)/(int dm) = (int_(0)^(R) (2r)/(pi) ( 2M rdr)/(R^(2)))/(M)`
`= ( 4)/(pi R^(2)) int_(0)^(R) r^(2) dr = (4)/(pi R^(2)) {:r^(3))/(3:}_(0)^(R)`
`= (4)/(pi R^(2)) ((R^(3))/(3)) = ( 4 R)/(3 pi)`