$A$ cockroach of mass $\frac{M}{2}$ starts moving with velocity $V$ on the circumference of a disc of mass $M$ and radius $R$. What will be the angular velocity of the disc?

The angular momentum of a moving body remains constant if

Given below are two statements: One is labelled as Assertion $(A)$ and the other is labelled as Reason $(R)$.
Assertion $(A)$ : In a conical pendulum,angular momentum about its vertical axis remains constant.
Reason $(R)$ : Net torque about the vertical axis of a conical pendulum is not zero.
In the light of the above statements,choose the most appropriate answer from the options given below:

$A$ smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size,each of mass $m$,which can slide freely along the rod. Initially,the two beads are at the center of the rod and the system is rotating with angular velocity $\omega_0$ about its axis perpendicular to the rod and passing through its mid-point (see figure). There are no external torques. When the beads reach the ends of the rod,the angular velocity of the system is:

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speed $\omega_1$ and $\omega_2$ are brought into contact face to face with their axes of rotation coincident. What is the loss in kinetic energy of the system in the process?

$A$ dancer is standing on a rotating platform holding two spheres in her hands. If she drops the spheres to the ground,the dancer's...