$A$ particle of mass $m = 5$ is moving with a uniform speed $v = 3\sqrt{2}$ in the $XOY$ plane along the line $Y = X + 4$. The magnitude of the angular momentum of the particle about the origin is .......

$A$ particle of mass $2 \ kg$ located at the position $(\hat{i} + \hat{j}) \ m$ has a velocity $2(\hat{i} - \hat{j} + \hat{k}) \ m/s$. Its angular momentum about the $z$-axis in $kg \cdot m^2/s$ is:

$A$ particle is moving in a circular path of radius $a$ with a constant speed $v$ as shown in the figure. The centre of the circle is marked by $C$. The angular momentum about the origin $O$ can be written as:

$A$ particle is rotating in a circle with uniform speed as shown. The angular momentum of the particle with respect to the origin is .........

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:

$A$ uniform rod $AB$ of mass $m$ and length $l$ is at rest on a smooth horizontal surface. An impulse $P$ is applied to the end $B$. The time taken by the rod to turn through a right angle is