A bullet, fired into a door gets embedded exactly at it's centre, causing the door to rotate about it's vertical axis, practically without friction, with an angular velocity of $0.625 \mathrm{rads}^{-1}$. The door is 1.0 m wide and weighs 12 kg . If the mass of the bullet is 10 g , find the speed with which it was fired. (Hint: The moment of inertia of the door about the vertical axis at one end is $\frac{M L^2}{3}$.

A singer, during his performance, stands on the edge of a circular turntable, and begins to walk along its edge with a speed of $1.5 \mathrm{~ms}^{-1}$ relative to the ground. The turn table is mounted on a frictionless vertical axle. Its radius R =3m and its moment of inertia about the axle is $150 \mathrm{~kg} \mathrm{~m}^2$. It is initially at rest. If the mass of the singer is 75 kg , the time taken by the man to complete one revolution is:

Four masses each 2 kg are placed at the corners A, B, C, D of a mass less square frame. 40 kg mass is at the centre O of a square frame of side 0.2 m . It is to be rotated about an axis passing through the centre O and perpendicular to the plane of the frame. Calculate the torque in $\mathrm{N}-\mathrm{m}$ required to produce an angular acceleration of $\frac{\pi}{2} \mathrm{rads}^{-2}$.

A particle starts rotating from rest. The instantaneous angular displacement is $\theta=3 t^3-t^2$, where $\theta$ is in radian and $t$ in s; The angular velocity at $t=1 \mathrm{~s}$ is