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:

$A$ 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

When does the magnitude of angular momentum become zero?

$A$ thin rod of mass $M$ and length $L$ is struck at one end by a ball of clay of mass $m$,moving with speed $v$ as shown in the figure. The ball sticks to the rod. After the collision,the angular momentum of the clay-rod system about $A$,the midpoint of the rod,is:

$A$ solid cylinder of mass $20 \ kg$ rotates about its axis with an angular speed of $100 \ rad \cdot s^{-1}$. The radius of the cylinder is $0.25 \ m$. The magnitude of its angular momentum about the axis of the cylinder is ........ $J \cdot s$.

$A$ body of mass $m$ is thrown with velocity $u$ from the origin of a coordinate system at an angle $\theta$ with the horizontal. The magnitude of the angular momentum of the particle about the origin at the time $t$ when it is at the maximum height of the trajectory is proportional to