Find the centre of mass of `a` unifrom disc of radius a from which a circulr section of radius `b` has been removed. The centre of hole is at a distance `c` from the centre of the disc.

Explanation

Suppose the circular disc of radius a with centre `O` is made up of
(i) circular section of radius b with centre `O_(1)` and (ii) remaining portion of disc with c.m at `O_(2)`.
Taking `O` as origin, and `O_(1),O_(2)` on X-axis,` (y = 0, z = 0)`, Fig. the position of c.m of disc is given by
`x_(cm) = (m_(1)x_(1) + m_(2)x_(2))/(m_(1) + m_(2))` ..(i)
If `sigma` is surface density of material of the disc,
`m_(1) = pi b^(2) sigma, x_(1) = c`
`m_(2) = pi(a^(2) - b^(2)) sigma`
`x_(2) = ?`
`m_(1) + m_(2) = pia^(2) sigma`
From (i), `0 = (pib^(2) sigma xx c+ pi (a^(2) - b^(2)) sigma xx x_( 2))/(pia^(2) sigma) :. x_(2) = (-cb^(2))/((a^(2) - b^(2)))`
Hence c.m of rest of the portion of the disc lies on the left of `O`, at a distance `x_(2)` as shown in the Fig.