$A$ uniform thin rod of length $2L$ and mass $m$ lies on a horizontal table. $A$ horizontal impulse $J$ is given to the rod at one end. There is no friction. The total kinetic energy of the rod just after the impulse will be

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 cylinder of radius $1 \,m$, mass $1 \,kg$ spins about its axis with an angular velocity $20 \,rad/s$. At a certain moment, the cylinder is placed into a corner as shown in the figure. The coefficient of friction between the horizontal wall and the cylinder is $\mu$, whereas the vertical wall is frictionless. If the number of rounds made by the cylinder is $5$ before it stops, then the value of $\mu$ is (acceleration due to gravity, $g=10 \,m/s^2$)

$A$ cubical block of side $a = 30\,cm$ is moving with velocity $v = 2\,m/s$ on a smooth horizontal surface. The surface has a small bump at a point $O$ as shown in the figure. The angular velocity (in $rad/s$) of the block immediately after it hits the bump is:

Two thin circular discs of mass $m$ and $4m$,having radii of $a$ and $2a$,respectively,are rigidly fixed by a massless,rigid rod of length $l=\sqrt{24}a$ through their centers. This assembly is laid on a firm and flat surface,and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point $O$ is $\vec{L}$ (see the figure). Which of the following statement$(s)$ is(are) true?
$(A)$ The center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega/5$
$(B)$ The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81ma^2\omega$
$(C)$ The magnitude of angular momentum of the assembly about its center of mass is $17ma^2\omega/2$
$(D)$ The magnitude of the $z$-component of $\vec{L}$ is $55ma^2\omega$

$A$ rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power developed by the external torque as a function of time is: