$A$ carpenter has constructed a toy as shown in the adjoining figure. If the density of the material of the sphere is $12$ times that of the cone,the position of the centre of mass of the toy is given by

$A$ uniform circular disc has radius $r$. $A$ square portion of diagonal $r$ is cut from it. The centre of mass of the remaining disc from the centre of the disc is

$A$ 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 $\alpha$?

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

The coordinates of the centre of mass of a uniform $L$-shaped plate of mass $3 \ kg$ shown in the figure is:

$A$ 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,