$A$ solid sphere of mass $m$,radius $R$,having moment of inertia about an axis passing through its center of mass as $I$ is recast into a disc of thickness $t$ whose moment of inertia about an axis passing through the rim (edge) and perpendicular to its plane remains $I$. Then the radius of the disc is:

In the given figure,a ring of mass $m$ is kept on a horizontal surface,and a body of equal mass $m$ is attached through a string wound on the ring. When the system is released,the ring rolls without slipping. Consider the following statements and choose the correct option.
$(i)$ Acceleration of the centre of mass of the ring is $\frac{g}{3}$.
$(ii)$ Acceleration of the hanging particle is $\frac{2g}{3}$.
$(iii)$ Frictional force (on the ring) acts in the forward direction.
$(iv)$ Frictional force (on the ring) acts in the backward direction.

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to the disc and passing through the centre) and rotating with angular speeds $\omega_1$ and $\omega_2$ are brought into contact face to face with their axes of rotation coincident.
$(a)$ Does the law of conservation of angular momentum apply to the situation? Why?
$(b)$ Find the angular speed of the two-disc system.
$(c)$ Calculate the loss in kinetic energy of the system in the process.
$(d)$ Account for this loss.

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. $(a)$ What is the angular speed of the two-disc system? $(b)$ Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$.

$A$ sphere rotates with an angular velocity $\omega_0$ at a temperature of $0^{\circ}C$. What will be its new angular velocity when the temperature is increased to $100^{\circ}C$? (Given: $\alpha = 2.0 \times 10^{-5} \, ^{\circ}C^{-1}$)

$A$ uniform bar of length $12 \text{ cm}$ and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with the same speed $v$ and in the same plane as the bar. These masses strike the bar simultaneously and get stuck to it. After the collision, the entire system is rotating with an angular frequency $\omega$. The ratio of $v$ and $\omega$ is: