$A$ circular plate of radius $r$ is removed from a uniform circular plate $P$ of radius $4 r$ to form a hole. If the distance between the centre of the hole formed and the centre of the plate $P$ is $2 r$,then the distance of the centre of mass of the remaining portion from the centre of the plate $P$ is

$A$ uniform rectangular thin sheet $ABCD$ of mass $M$ has length $a$ and breadth $b$,as shown in the figure. If the shaded portion $HBGO$ is cut off,the coordinates of the centre of mass of the remaining portion will be

From a homogeneous square plate,a triangle is cut out as shown in the figure. The side of the square is $a$,and the apex of the triangle is at the center of the square. The distance from the center of the square to the center of mass of the remainder of the plate is:

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.

$A$ solid cone is placed on a horizontal surface has height $h$,radius $R$ and apex angle $\theta$ as shown. If the gravitational potential energy of the cone does not change as the position of the cone is changed from figure $(A)$ to figure $(B)$,then,

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 $^{\circ}$)