$A$ 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

In the cube of side $a$ shown in the figure,the vector from the central point $G$ of the face $ABOD$ to the central point $H$ of the face $BEFO$ is:

$A$ circular disc of radius $R$ has a uniform thickness. $A$ circular hole of diameter equal to the radius of the disc has been cut out as shown. The distance of the centre of mass of the remaining disc from point $O$ is

$A$ 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 hole of radius $R/2$ is cut from a circular disc of mass $M$ and radius $R$ such that the circumference of the hole passes through the center of the disc. What is the moment of inertia of the remaining part about an axis perpendicular to the plane of the disc and passing through its center?

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