Question

From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

Answer 1

Let mass per unit area of the original disc=$\sigma$

Thus mass of original disc=$M=\sigma \pi R^2$

Radius of smaller disc=$R/2$.

Thus mass of the smaller disc=$\sigma \pi (R/2{)}^2=M/4$

After the smaller disc has been cut from the original, the remaining portion is considered to be a system of two masses. The two masses are:

M(concentrated at O), and -M(=M/4) concentrated at O'

(The negative sign indicates that this portion has been removed from the original disc.)
Let x be the distance through which the centre of mass of the remaining portion shifts from point O.

The relation between the centres of masses of two masses is given as:

$x=(m_1{r}_1+m_2{r}_2)/(m_1+m_2)$

$=(M\times 0-(M/4)\times (R/2\left)\right)/(M-M/4)=-R/6$

(The negative sign indicates that the centre of mass gets shifted toward the left of point O)