P is a fixed smooth cylinder of radius R and Q is a disc of mass M and radius R. A light thread is tightly wound on Q and its end is connected to a rope ABC. The rope has a mass m and length `(piR)/(2)` and is initially placed on the cylinder with its end A at the top. The system is released from rest. The rope slides down the cylinder as the disc rolls without slipping. The initial separation between the disc and the cylinder was `L = (piR)/(2)` (see fig). Find the speed with which the disc will hit the cylinder. Assume that the rope either remains on the cylinder or remains vertical, it does not fly off the cylinder.

Answer

`V = sqrt(((8 + 3 pi^(2))/(pi))((mgR)/(3M + 8m)))`