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:

An object of mass $m$ is projected from the origin in a vertical $xy$ plane at an angle $45^{\circ}$ with the $x$-axis with an initial velocity $v_0$. The magnitude and direction of the angular momentum of the object with respect to the origin,when it reaches the maximum height,will be [$g$ is the acceleration due to gravity].

$A$ constant torque of $200 \, N-m$ turns a flywheel, which is at rest, about an axis through its centre and perpendicular to its plane. If its moment of inertia is $50 \, kg-m^{2}$, then in $4 \, s$, what will be the change in its angular momentum?

Three equal masses $m$ are kept at vertices $(A, B, C)$ of an equilateral triangle of side $a$ in free space. At $t = 0$,they are given an initial velocity $\vec{V}_A = V_0 \hat{u}_{AC}, \vec{V}_B = V_0 \hat{u}_{BA}$ and $\vec{V}_C = V_0 \hat{u}_{CB}$. Here,$\hat{u}_{AC}, \hat{u}_{CB}$ and $\hat{u}_{BA}$ are unit vectors along the edges of the triangle. If the three masses interact gravitationally,then the magnitude of the net angular momentum of the system about the centroid of the triangle is:

$A$ particle is moving along a straight line parallel to $x-$ axis with constant velocity. Its angular momentum about the origin

$A$ particle of mass $2\, kg$ is moving such that at time $t$,its position,in meters,is given by $\vec{r}(t) = 5\hat{i} - 2t^2\hat{j}$. The angular momentum of the particle at $t = 2\, s$ about the origin in $kg\, m^2\, s^{-1}$ is: