$A$ $5\, m$ long pole of $3\, kg$ mass is placed against a smooth vertical wall as shown in the figure. Under equilibrium condition,if the pole makes an angle of $37^{\circ}$ with the horizontal,the frictional force between the pole and the horizontal surface is

$A$ rod of length $l$ and mass $m$ lies on a fixed horizontal smooth table. $A$ cord is led through a pulley,and its horizontal part is attached to one end of the rod,while its vertical part is attached to a block of mass $m_1$. Assume the pulley and the cord are ideal. The maximum possible acceleration of the rod's centre of mass $C$ (for all possible values of masses $m$ and $m_1$) at the moment of releasing the block $m_1$ is $\frac{g}{n}$. Find the value of $n$.

$A$ uniformly thick wheel with moment of inertia $I$ and radius $R$ is free to rotate about its centre of mass (see fig). $A$ massless string is wrapped over its rim and two blocks of masses $m_{1}$ and $m_{2}$ $(m_{1} > m_{2})$ are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when $m_{1}$ descends by a distance $h$ is

$A$ bob of mass $m$ is tied by a massless string whose other end is wound on a flywheel (disc) of radius $R$ and mass $m$. When released from rest,the bob starts falling vertically downwards. If the bob has covered a vertical distance $h$,then the angular speed of the wheel will be (There is no slipping between the string and the wheel,$g$ is the acceleration due to gravity).

$A$ body rotates about a stationary axis. If the angular deceleration is proportional to the square root of the angular speed, then the mean angular speed of the body, given $\omega_0$ as the initial angular speed, is

$A$ mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming the pulley to be a perfect uniform circular disc,the acceleration of the mass $m$,if the string does not slip on the pulley,is