$A$ disc of mass $M$ and radius $R$ is rolling with angular speed $\omega$ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is

$A$ thin ring of mass $M$ and radius $R$ is rotating with an angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. If $4$ point masses each of mass $m$ are placed gently at the opposite ends of two mutually perpendicular diameters,what will be the new angular velocity of the system?

$A$ circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. $A$ tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity $\omega_0$. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform $\omega(t)$ will vary with time $t$ as:

Two coaxial discs,having moments of inertia $I_1$ and $\frac{I_1}{2}$,are rotating with angular velocities $\omega_1$ and $\frac{\omega_1}{2}$ respectively,about their common axis. They are brought into contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies,then $(E_f - E_i)$ is

$A$ thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect,the angular speed of the disc

$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity: