A particle of mass $m$ is moving with constant velocity $v$ parallel to the $x$-axis as shown in the figure. Its angular momentum about origin $O$ is ……….

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 thin uniform rod, pivoted at $O$, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time, $t =0$, a small insect starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at $t = T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $(|\vec{\tau}|)$ on the system about $O$, as a function of time is best represented by which plot?

A uniform rod $A B$ of mass $2 \mathrm{~kg}$ and Length $30 \mathrm{~cm}$ at rest on a smooth horizontal surface. An impulse of force $0.2\  \mathrm{Ns}$ is applied to end $B.$ The time taken by the rod to turn through at right angles will be $\frac{\pi}{\mathrm{x}}\  \mathrm{s}$, where  X=____

$A$ hollow sphere of radius $R$ and mass $m$ is fully filled with water of mass $m$. It is rolled down a horizontal plane such that its centre of mass moves with a velocity $v$. If it purely rolls