$A$ uniform disc of mass $m_0$ rotates freely about a fixed horizontal axis passing through its centre. $A$ thin cotton pad is fixed to its rim,which can absorb water. The mass of water dripping onto the pad per unit time is $\mu$. After what time will the angular velocity of the disc get reduced to half its initial value?

Initial angular velocity of a circular disc of mass $M$ is $\omega_{1}$. Then two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

$A$ solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown in the figure. The cube is free to rotate about a fixed axis $AB$. $A$ bullet of mass $m$ $(m << M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height of $4a/3$ from the surface to impart the cube an angular speed $\omega$. It strikes the face and embeds in the cube. Then,$\omega$ is close to (Note: The moment of inertia of the cube about an axis perpendicular to the face and passing through the centre of mass is $2Ma^2/3$).

If the earth suddenly shrinks to $\frac{1}{64}$th of its original volume with its mass remaining the same,the period of rotation of earth becomes $\frac{24}{x} \text{ h}$. The value of $x$ is $.......$

$A$ 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 gently placed at opposite ends of a diameter of the ring. The new angular velocity of the ring will be

Which of the following remains constant in a central force field?