From a semicircular disc of radius `r_(1)`, another semicircular disc of radius `r_(2)` is removed. Find the `c.m.` of the remaining position.

Explanation

The diagram is symmertical about the `y`-axis.
`m_(1) prop (pi r_(1)^(2))/(2) rArr m_(1) = (k pi r_(1)^(2))/(2) , y_(1) = ( 4 r_(1))/( 3pi)`
`y_(c.m.)=(m_(1) y_(1) - m_(2) y_(2))/(m_(2) - m_(2)) = ((kpir_(1)^(2))/(2) xx ( 4 r_(1))/( 3pi) -(k pi r_(2)^(2))/(2 ) xx (4 r_(2))/(3 pi))/((kpir_(1)^(2))/(2) - (k pi r_(2)^(2))/(2))`
`= ((4)/(3 pi) (r_(1)^(3) - r_(2)^(3)))/(r_(1)^(2) - r_(2)^(2)) = ((4)/(3 pi) (r_(1) - r_(2)) (r_(1)^(2) + r_(1) r_(2) +r_(2)^(2)))/((r_(1) + r_(2)) (r_(1) - r_(2)))`
`= (4(r_(1)^(2) + r_(1)r_(2) + r_(2)^(2)))/(3 pi(r_(1) + r_(2)))`