Question

A square plate of edge 'a' and a circular disc of same diameter are placed touching each other at the midpoint of an edge of the plate as shown in figure. If mass per unit area for the two plates are same then find the distance of centre of mass of the system from the centre of the disc.

Answer 1

Distance between center of mass and $C$ is

$x$ coordinate of center of mass is

$x=\frac{\sigma \left(\frac{\pi a^2}{4}\right)\left(\frac{a}{2}\right)+\sigma \left(a^2\right)\frac{a}{2}}{\sigma (\frac{\pi a^2}{4}+a^2)}=\frac{\frac{a^3}{2}(\frac{-\pi }{4}+1)}{a^2(1+\frac{\pi }{4})}=\frac{(4-\pi )a}{2(4+\pi )}$

Distance between center of mass and center of disc

$=\frac{a}{2}+\frac{(4-\pi )a}{2(4+\pi )}=\frac{(4-\pi +4+\pi )a}{2(4+\pi )}d=\frac{4a}{4+\pi }$