$A$ uniform disc of mass $100 \ kg$ and radius $2 \ m$ is rotating at $1 \ rad/s$ about a perpendicular axis passing through its centre. $A$ boy of mass $60 \ kg$ standing at the centre of the disc suddenly jumps to a point which is $1 \ m$ from the centre of the disc. The final angular velocity of the boy (in $rad/s$) is

In the absence of external torque for a body revolving about any axis, the quantity that remains constant is

$A$ circular platform is mounted on a frictionless vertical axle. Its radius $R = 2\, m$ and its moment of inertia about the axle is $200\, kg\, m^2$. It is initially at rest. $A$ $50\, kg$ man stands on the edge of the platform and begins to walk along the edge at the speed of $1\, m/s$ relative to the ground. Time taken by the man to complete one revolution is

$A$ thin uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega$. Another disc of the same thickness and radius but of mass $\frac{1}{8} M$ is placed gently on the first disc coaxially. The angular velocity of the system is now

$A$ circular disc with moment of inertia $I_t$ is rotating in a horizontal plane about its symmetry axis with a constant angular speed $\omega_i$. Another disc with moment of inertia $I_b$ is placed coaxially onto the rotating disc. Initially,the second disc has zero angular speed. Finally,both discs rotate with a constant angular speed $\omega_f$. What is the energy lost due to friction by the rotating disc initially?

$A$ 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 $\frac{M}{2}$ is placed gently on the first disc co-axially. The new angular velocity will be