A point mass $m$ collides with a disc of mass $m$ and radius $R$ resting on a rough horizontal surface as shown in figure. Its collision is perfectly inelastic and immediately system starts pure rolling. Find angular velocity of the system after pure rolling starts 

In figure the uniform gate weighs $300 \mathrm{N}$ and is $3 \mathrm{m}$ wide and $2 \mathrm{m}$ high. It is supported by a hinge at the bottom left corner and a horizontal cable at the top left corner, as shown, the tension in the cable and the force that the hinge exerts on the gate (magnitude & direction.

A uniform rod AB of mass $m$ and length $\ell$ at rest on a smooth horizontal surface. An impulse $P$ is applied to the end $B$. The time taken by the rod to turn through a right angle is

Determine the acceleration $a$ of the supporting surface required to keep the centre $G$ of the circular pipe in a fixed position w.r.t. ground during the motion. No slipping takes place between the pipe and its support.

A bar of mass $M$ and length $L$ is in pure translatory motion with its centre of mass velocity $V$. It collides with and sticks to a second identical bar which is initially at rest. (Assume that it becomes one composite bar of length $2L$). The angular velocity of the composite bar will be: