If the radius of the earth is reduced to three-fourth of its present value without any change in its mass,then the duration of the day on earth will be . . . . . . hours $30$ minutes.

$A$ 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:

$A$ uniform rod of length $6L$ and mass $8m$ is pivoted at its centre $C$. Two masses $m$ and $2m$ with speeds $2v$ and $v$ respectively,as shown in the figure,strike the rod and stick to it. Initially,the rod is at rest. If it rotates with angular velocity $\omega$ due to the impact,then $\omega$ will be:

Suppose all the people in the world line up at the equator and all start running at speed $v_{rel}$ relative to the surface of the Earth along the equatorial circle. The initial angular velocity of the Earth is $\omega_0$. Let the moment of inertia of the Earth be $I_E$,the moment of inertia of all people be $I_P$,and the radius of the Earth be $R$. Which of the following statements is correct?

$A$ 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