$A$ uniform disc of mass $M$ and radius $R$ is rotating about a horizontal axis passing through its centre with angular velocity $\omega$. $A$ piece of mass $m$ breaks from the disc and flies off vertically upwards. The angular speed of the disc will be

$A$ boy stands with his hands folded at the center of a platform rotating about its central axis. The kinetic energy of the system is $K$. The boy now stretches his hands out, which doubles the moment of inertia of the system. The kinetic energy of the system will become

$A$ stone of mass $m$,tied to the end of a string,is whirled around in a circle on a horizontal frictionless table. The length of the string is reduced gradually,keeping the angular momentum of the stone about the centre of the circle constant. Then,the tension in the string is given by $T = Ar^n$,where $A$ is a constant and $r$ is the instantaneous radius of the circle. The value of $n$ is equal to

In the above problem,the angular velocity of the system after the particle sticks to it will be ....... $rad/s$. (in $.3$)

$A$ boy is standing at the center of a turntable holding two blocks in his hands. The rotational kinetic energy of the system about the axis passing through the center is $K$. When the boy extends his arms,the moment of inertia of the system about the axis of rotation doubles. What will be the new rotational kinetic energy of the system?

The angular speed of a body changes from $\omega_1$ to $\omega_2$ without applying a torque but due to changes in moment of inertia. The ratio of radii of gyration in two cases is