Two particles which are initially at rest, move towards each other under the action of their internal attraction. If their speeds are $v$ and $2v$ at any instant, then the speed of centre of mass of the system will be

Two loops $P$ and $Q$ are made from a uniform wire. The radii of $P$ and $Q$ are $r_1$ and $r_2$ respectively, and their moments of inertia are $I_1$ and $I_2$ respectively. If $I_2/I_1=4$ then $\frac{{{r_2}}}{{{r_1}}}$ equals

$A$ bob of mass $m$ is attached to a string whose other end is tied to a light vertical rod as shown in figure. The bob is swinging in horizontal plane with constant angular speed $\omega$. The vertical rod is supported on a block of mass $M$ which is placed on a rough surface. What is minimum friction coefficient between ground and block for which block does not slip ?

Figure shows a thin metallic triangular sheet $ABC.$ The mass of the sheet is $M.$ The moment of inertia of the sheet about side $AC$ is

The centre of mass of a body

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