A rod can rotate about a fixed vertical axis, passing through one of its end. The mass is non-uniformly distributed along the length of the rod. A horizontal force of constant magnitude and always perpendicular to the rod is applied at the end. Which of the following quantity (after one rotation) will not depend on the information that through which end the axis passes? (Assuming initial angular velocity to be zero)

A particle of mass $m$ is moving horizontally at speed $v$ perpendicular to a uniform rod of length $d$ and mass $M=6m$. The rod is hinged at centre $O$ and can freely rotate in the horizontal plane about a fixed vertical axis passing through its centre $O$. The hinge is frictionless. The particle strikes and sticks to the end of the rod. The angular speed of the system just after the collision

A small block of mass $m$ is rigidly attached at $P$ to a ring of mass $3m$ and radius $r$. The system is released from rest at $\mathrm{\theta }=90°$ and rolls without sliding. The angular acceleration of the ring just after release is

A box of dimensions $l$ and $b$ is kept on a truck moving with acceleration $a$. If the box does not slide, the maximum acceleration for it to remain in equilibrium (with respect to the truck) is-

Two steel balls of equal diameter are connected by a rigid bar of negligible weight as shown and are dropped in the horizontal position from height $h$ above the heavy steel and brass base plates. If the coefficient of restitution between the ball and steel base is $0.6$ and that between the other ball and the brass base is $0.4$. The angular velocity of the bar immediately after the rebound is $n\times {10}^{-2} \mathrm{rad} {\mathrm{s}}^{-1}$ where $n$ is: (Assume the two impacts are simultaneous.) $\left(g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$

Each of the double pulleys shown has a centroidal mass moment of inertia of ${\mathrm{mr}}^2,$ inner radius r and an outer radius $2\mathrm{r}.$ Assuming that the bearing friction of hinge at A and at B is equivalent to torque of magnitude $\frac{\mathrm{mgr}}{4}$ then the tension (in N) in the string connecting the pulleys is :$(\mathrm{m} = 3 \mathrm{kg}, \mathrm{g} = 10 \mathrm{m} {\mathrm{s}}^{-2}, \mathrm{and} \mathrm{r} =0.1 \mathrm{m})$