$A$ small disc of radius $2\, cm$ is cut from a disc of radius $6\, cm$. If the distance between their centres is $3.2\, cm$,what is the shift in the centre of mass of the disc in $cm$?

The equilateral triangle $ABC$ is cut from a thin solid sheet of wood. $D, E$ and $F$ are the midpoints of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_0$. If the smaller triangle $DEF$ is removed from $ABC$,the moment of inertia of the remaining figure about the same axis is $I$. Then

$A$ uniform square plate has a side of length $2R$. $A$ circular piece of maximum possible area is cut and removed from one of the quadrants of the plate as shown in the figure. Calculate the shift in the centre of mass of the 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,

$A$ sphere of diameter $r$ is cut from a sphere of radius $r$ such that the centre of mass of the remaining mass is at the maximum distance from the original centre. What is this distance?

$A$ circular portion of radius $R_2$ has been removed from one edge of a circular disc of radius $R_1$. The correct expression for the centre of mass for the remaining portion of the disc is