To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is $$\mu $$, the torque, applied by the machine on the mop is -

A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is -

A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30^o from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad s^$$-$$1) will be (g = 10 ms^$$-$$2)

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is $$\theta $$, thrown :