$A$ disc of mass $M$ and radius $R$ is rolling on a horizontal surface as shown in the figure. What is the magnitude of the angular momentum of the disc about the origin $O$?

Two bodies rotate with kinetic energies $E_1$ and $E_2$. Their moments of inertia about their axes of rotation are $I_1$ and $I_2$. If $I_1 = \frac{I_2}{3}$ and $E_1 = 27 E_2$,then the ratio of their angular momenta $L_1$ to $L_2$ is:

$A$ cylinder of mass $M$ and radius $R$ is rotating about its axis with angular velocity $\omega$. $A$ particle of mass $m$ moving with velocity $v$ hits the cylinder and sticks to its rim. Calculate the angular velocity of the cylinder after the impact.

$A$ mass $m$ is moving with a constant velocity $v$ along a line parallel to the $X$-axis. Its angular momentum with respect to the origin or the $Z$-axis is:

$A$ small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x-y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system,calculated about $O$ and $P$ are denoted by $\vec{L}_O$ and $\vec{L}_P$ respectively,then

Two wheels are connected by a belt. The radius of the larger wheel is three times that of the smaller one. What is the ratio of the rotational inertia of the larger wheel to the smaller wheel,when both wheels have the same angular momentum?