State the equivalent physical quantities in linear motion corresponding to the moment of inertia and torque in rotational motion.

One twirls a circular ring (of mass $M$ and radius $R$) near the tip of one's finger as shown in Figure $1$. In the process,the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone,shown by the dotted line. The radius of the path traced out by the point where the ring and the finger are in contact is $r$. The finger rotates with an angular velocity $\omega_0$. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger are in contact (Figure $2$). The coefficient of friction between the ring and the finger is $\mu$ and the acceleration due to gravity is $g$.
$(1)$ The total kinetic energy of the ring is
$[A]$ $M \omega_0^2 R^2$ $[B]$ $\frac{1}{2} M \omega_0^2(R-r)^2$ $[C]$ $M \omega_0^2(R-r)^2$ $[D]$ $\frac{3}{2} M \omega_0^2(R-r)^2$
$(2)$ The minimum value of $\omega_0$ below which the ring will drop down is
$[A]$ $\sqrt{\frac{g}{\mu(R-r)}}$ $[B]$ $\sqrt{\frac{2 g}{\mu(R-r)}}$ $[C]$ $\sqrt{\frac{3 g}{2 \mu(R-r)}}$ $[D]$ $\sqrt{\frac{g}{2 \mu(R-r)}}$
Given the answers to questions $(1)$ and $(2)$:

$A$ uniform disc is rolling on a horizontal surface. At a certain instant,$B$ is the point of contact and $A$ is the topmost point of the disc,where $R$ is the radius of the disc.

The graphs below show angular velocity $\omega$ as a function of time $t$. In which one is the magnitude of the angular acceleration $\alpha$ constantly decreasing?

Two identical rings $A$ and $B$ of same mass $M$ and radius $R$ are revolving. Ring $A$ revolves around its own diameter, and ring $B$ revolves about a tangential axis in its own plane. Both rings $A$ and $B$ have the same rotational kinetic energy. The ratio of the angular velocity of ring $B$ $(\omega_B)$ to that of ring $A$ $(\omega_A)$ is:

$A$ yo-yo is resting on a rough horizontal table. Forces $F_1, F_2$ and $F_3$ are applied separately as shown. The correct statement is