Moment of inertia of a uniform quarter disc of radius $R$ and mass $M$ about an axis through its centre of mass and perpendicular to its plane is:

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: