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:

A horizontal circular platform of radius $0.5 \mathrm{m}$ and mass $0.45 \mathrm{kg}$ is free to rotate about its axis. Two massless spring toy guns, each carrying a steel ball of mass $0.05 \mathrm{kg}$ are attached to the platform at a distance $0.25 \mathrm{m}$ from the centre on either side along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have the horizontal speed of $9{ \mathrm{m} \mathrm{s}}^{-1}$ with respect to the ground. The rotational speed of the platform in ${\mathrm{rad}}^{-1}$ after the balls leave the platform is

A uniform circular disc of mass $1.5 \mathrm{kg}$ and radius $0.5 \mathrm{m}$ is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5 \mathrm{N}$ are applied simultaneously along the three sides of an equilateral triangle $XYZ$ its vertices on the perimeter of the disc (see figure) One second after applying the forces, the angular speed of the disc in $\mathrm{rad} {\mathrm{s}}^{-1}$ is

A straight rod of negligible mass is mounted on a frictionless pivot and masses $2.5 \mathrm{kg}$ and $1 \mathrm{kg}$ are suspended at distances $40 \mathrm{cm}$ and $100 \mathrm{cm}$ respectively from the pivot as shown. The rod is held at an angle $\theta$ with the horizontal and released.