$A$ $5\, m$ 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

$A$ cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. $A$ rope of negligible mass is wrapped around the cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

How can one verify that an object is undergoing rotation about a fixed axis?

One end of a rod of length $L$ is on a horizontal plane. It is inclined at an angle $\alpha$ to the horizontal plane. When released,its angular velocity after coming to the horizontal plane is:

The linear velocity of a rotating body is given by $\overrightarrow v = \overrightarrow \omega \times \overrightarrow r,$ where $\overrightarrow \omega$ is the angular velocity and $\overrightarrow r$ is the radius vector. If the angular velocity of a body is $\overrightarrow \omega = \hat i - 2\hat j + 2\hat k$ and the radius vector is $\overrightarrow r = 4\hat j - 3\hat k,$ then find the magnitude of linear velocity $|\overrightarrow v |$.

$A$ straight rod of length $L$ is released on a frictionless horizontal floor in a vertical position. As it falls and slips,the distance of a point on the rod from the lower end,which follows a quarter circular locus,is