$A$ train is moving slowly at $2 \, m/s$ next to a railway platform. $A$ man,$1.5 \, m$ tall,alights from the train such that his feet are fixed on the ground. Taking him to be a rigid body,the instantaneous angular velocity (in $rad/s$) is

$A$ disc of mass $M$ and radius $R$ is moving with an angular velocity $\omega$ as shown in the figure. Find the angular momentum of the disc about the reference point $O$.

$A$ circular platform rotates in a horizontal plane about a vertical axis passing through its center. $A$ tortoise sits on the edge of the platform. The platform is given an initial angular velocity $\omega_0$. The tortoise now moves along a chord of the platform with a constant velocity (relative to the platform). How does the angular velocity $\omega(t)$ of the platform change with time $t$?

$A$ thin uniform circular disc of mass $M$ and radius $R$ is rotating with angular velocity $\omega$ in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of the same radius but of mass $\frac{M}{3}$ is placed gently on the first disc co-axially. The new angular velocity will be:

$A$ disc of mass $1 \, kg$ and radius $0.1 \, m$ is rotating with angular velocity $20 \, rad/s$. What is the angular velocity (in $rad/s$) if a mass of $0.5 \, kg$ is placed on the circumference of the disc?

$A$ 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