Two bodies have moments of inertia $I_1 = I$ and $I_2 = 2I$ about their axes of rotation. If their rotational kinetic energies are equal,what is the ratio of their angular momenta?

$A$ bar of length $L$ carrying a small mass $m$ at one of its ends rotates with a uniform angular speed $\omega$ in a vertical plane about the midpoint of the bar. During the rotation, at some instant of time when the bar is horizontal, the mass is detached from the bar but the bar continues to rotate with the same $\omega$. The mass moves vertically up, comes back, and reaches the bar at the same point. At that place, the acceleration due to gravity is $g$.

$A$ ring of mass $m = 1 \ kg$ and radius $R = 1.25 \ m$ is kept on a rough horizontal ground. $A$ small body of same mass $m = 1 \ kg$ is stuck to the top of the ring. When it is given a slight push forward,the ring starts rolling purely on the ground. What is the maximum speed of the centre of the ring (in $m/s$)?

$A$ solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown below. The cube is free to rotate about the fixed axis $AB$. $A$ bullet of mass $m (< M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed $\omega_{c}$,so that the cube just topples over. Then,$\omega_{c}$ is (Note: the moment of inertia of the cube about an axis passing through the centre of mass and parallel to the edge is $2Ma^{2}/3$)

Two discs of moment of inertia $I_1 = 4 \ kg \ m^2$ and $I_2 = 2 \ kg \ m^2$ about their central axes and normal to their planes,rotating with angular speeds $10 \ rad/s$ and $4 \ rad/s$ respectively,are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is . . . . . . $J$.

$A$ sphere of mass $M$ and radius $R$ is attached by a light rod of length $l$ to a point $P$. The sphere rolls without slipping on a circular track as shown. It is released from the horizontal position. The angular momentum of the system about $P$ when the rod becomes vertical is: