$A$ semicircular portion of radius $r$ is cut from a uniform rectangular plate as shown in the figure. The distance of the centre of mass $C$ of the remaining plate from point $O$ is:

$A$ square shaped hole of side $l = \frac{a}{2}$ is carved out at a distance $d = \frac{a}{2}$ from the centre $O$ of a uniform circular disk of radius $a$. If the distance of the centre of mass of the remaining portion from $O$ is $-\frac{a}{X}$,find the value of $X$ (to the nearest integer).

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.

$A$ circular disc of radius $\alpha R$ has a smaller circular disc of radius $R$ removed from it such that their circumferences touch each other. The center of mass of the new disc is at a distance of $\alpha R$ from the center of the original large disc. What is the value of $\alpha$?

In the figure shown,find the distance of the centre of mass of a system of a uniform circular plate of radius $3R$ from $O$,in which a hole of radius $R$ is cut whose centre is at a distance of $2R$ from the centre of the large circular plate.

$A$ square plate of side $12 \ cm$ is shown in the figure. If a square of side $2 \ cm$ is cut from one of its corners,where will the center of mass of the remaining part be with respect to the center of the original square? The plate has uniform thickness and density.