Two discs $A$ and $B$ are mounted coaxially on a vertical axle. The discs have moment of inertia $I$ and $2 I$ respectively about the common axis. Disc $A$ is imparted an initial angular velocity $2 \omega$ using the entire potential energy of a spring compressed by a distance $x_1$. Disc $B$ is imparted an angular velocity $\omega$ by a spring having the same spring constant and compressed by a distance $x_2$. Both the discs rotate in the clockwise direction.

The loss of kinetic energy during the above process is :

A solid sphere of radius $R$ has moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$. If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to $I$, then the value of $r$ is equal to

A wooden log of mass M and length L is hinged by a frictionless nail at O; a bullet of mass m strikes with velocity $$v$$ and sticks to it. Find angular velocity of the system immediately after the collision about O.

A cylinder of mass m and radius R rolls down on an inclined plane of inclination $$\theta$$. Calculate the linear acceleration of axis of cylinder.