A mass $‘m’$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

A rod $P$ of length $1\ m$ is hinged at one end $A$ and there is a ring attached to the other end by a light inextensible thread . Another long rod $Q$ is hinged at $B$ and it passes through the ring. The rod $P$ is rotated about an axis which is perpendicular to plane in which both the rods are present and the variation between the angles $\theta $ and $\phi $ are plotted as shown. The distance between the hinges $A$ and $B$ is ........ $m.$

The centre of mass of two particles lies

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

A $T$ shaped object with dimensions shown in the figure, is lying a smooth floor. A force $\vec {'F'}$ is applied at the point $P$ parallel to $AB$, such that the object has only the translational motion without rotation. Find the location of $P$ with respect to $C$

A particle of mass $= 5\ units$ is moving with a uniform speed $V = 3 \sqrt 2\ units$ in the $XOY$ Plane along the line $Y = X+4 $ . The magnitude of the angular momentum of the particle about the origin is ...... $unit.$