If the positions of two like parallel forces on a light rod are interchanged, their resultant shifts by one-fourth of the distance between them, then the ratio of their magnitude 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 :$(\mathrm{m} = 3 \mathrm{kg}, \mathrm{g} = 10 \mathrm{m} {\mathrm{s}}^{-2}, \mathrm{and} \mathrm{r} =0.1 \mathrm{m})$

The free end of the string wound on the surface of a solid cylinder of mass $M=1 \mathrm{kg}$ and radius $R=\frac{2}{3} \mathrm{m}$ is pulled up by a force $F$ as shown. If there is sufficient friction between cylinder and floor then the upper limit to the angular acceleration $\left(\mathrm{in} \mathrm{rad} {\mathrm{s}}^{-2}\right)$ of the cylinder for which it rolls without slipping is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$ 

A uniform solid cylinder of mass $m=1 \mathrm{kg}$ rests on two horizontal planks. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically down with a constant force $F$. 

$\left(\mathrm{a}\right)$ Find the maximum magnitude of the force $F \left(\mathrm{in} \mathrm{N}\right)$ which still does not bring about any sliding of the cylinder, if the coefficient of friction between the cylinder and the planks is equal to $\mu =\frac{1}{3}$.

$\left(\mathrm{b}\right)$ The acceleration $a_{\max }$ of the axis of the cylinder rolling on the planks is $\frac{n}{3} \mathrm{m} {\mathrm{s}}^{-2}$, where $n$ is

A uniform disc of mass $\mathrm{M} = 1\mathrm{kg},$ radius $\mathrm{R} = 1 \mathrm{m}$ is moving towards right on smooth horizontal surface with velocity ${\mathrm{v}}_{0 }= 20 \mathrm{m} {\mathrm{s}}^{-1}$ & having angular velocity ${\mathrm{\omega }}_0 = 4 \mathrm{rad} {\mathrm{s}}^{-1}$ about the perpendicular axis outward the plane of disc passing through centre of disc. Suddenly top point of the disc gets hinged about a fixed smooth axis. The angular velocity (in $\mathrm{rad} {\mathrm{s}}^{-1}$) of disk about new rotation axis is:

A regular hexagonal uniform block of mass $m=4\sqrt{3} \mathrm{kg}$ rests on a rough horizontal surface with a coefficient of friction $\mu$ as shown in the figure. A constant horizontal force is applied on the block as shown. If the coefficient of friction is sufficient to prevent slipping before toppling, then the minimum force (in $\mathrm{N}$) required to topple the block about its corner $A$ is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$