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.

$A$ circular plate $A$ of radius $1.5 r$ is removed from one edge of a uniform circular plate $B$ of radius $2 r$. The distance of the centre of mass of the remaining portion from the centre of the plate $B$ is

The position of the centre of mass of the uniform plate as shown in the figure is

From a uniform circular disc of radius $2 \,cm$ (its centre of mass is at $O$), a circular portion of radius $1 \,cm$ is removed such that the shift in centre of mass is maximum. The disc is now rotated by an angle $\theta$ about an axis perpendicular to its plane and passing through $O$. If the magnitude of displacement of the new centre of mass is $\frac{1}{\sqrt{3}} \,cm$, then the $\theta$ is (in $^{\circ}$)

$A$ 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 $\alpha$?

$A$ uniform thin metal plate of mass $10 \ kg$ with the dimensions shown in the figure is given. The ratio of the $x$ and $y$ coordinates of the center of mass of the plate is $\frac{n}{9}$. The value of $n$ is: