Question

A uniform cylinder of radius R and mass M can rotate freely about a stationary horizontal axis 0 (Fig. 1.59). A thin cord of length 1 and mass m is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as a function of the length x of the hanging part of the cord. The wound part of the cord is supposed to have its centre of gravity on the cylinder axis.

Answer 1

Let us use the equation dM_z/dt =N_z  relative to the axis through O .....(1) 

For this purpose, let us find the angular momentum of the system M_z about the given rotation axis and the corresponding torque N_z. The angular momentum is

The downward pull of gravity on the overhanging part is the only external force, which exerts a torque about the z-axis, passing through O and is given by, 

Note: We may solve this problem using conservation of mechanical energy of the system (cylinder + thread) in the uniform field of gravity.