Three masses of $2\,kg$, $4\, kg$ and $4\, kg$ are placed at the three points $(1, 0, 0)$ $(1, 1, 0)$ and $(0, 1, 0)$ respectively. The position vector of its center of mass is

A thin wire of length $\ell$ and mass $m$ is bent in the form of a semicircle as shown. Its moment of inertia about an axis joining its free ends will be ...........

If a solid sphere is rolling, the ratio of its rotational energy to the total kinetic energy is given by

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat i\,\, + \,\,2\hat j\,\, + \,\,\hat k$ and  $-\,3\hat i\,\, - \,\,2\hat j\,\, + \,\,\hat k$, respectively. The centre of mass of this system has a position vector

Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the speed of $A$ is $v$ and the speed of $B$ is $2v$, the speed of centre of mass of the system is

A cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. A rope of negligible mass is wrapped around the solid cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be