(a) A uniform chain is lying in form of on arc of a circle of radius R. The arc subtends an angle of `2 alpha` at the centre of the circle. Find the distance of the centre of mass of the chain from the centre of the circle.

(b) A uniform chain of length `(piR)/(2)` is lying symmetrically on the top of a fixed smooth half cylinder (see figure) of radius R. The chain is pulled slightly from one side and released. It begins to slide. Find the speed of the chain when its one end just touches the floor. What is speed of centre of mass of the chain at this instant ?

(c) In part (b) assume that the half cylinder is not fixed and can slide on the smooth floor. Find the displacement of the cylinder by the time one end of the chain touches the floor. Mass of cylinder is equal to that of the chain. For part (b) and (c) assume that the chain remains in contact with the cylinder all the while.

Answer

(a) `(R sin alpha)/(alpha)`
(b) `V=2sqrt(((sqrt(2)-1)gR)/(pi));V_(cm)=(4sqrt(2))/(pi)sqrt(((sqrt(2)-1)gR)/(pi))`
(c) `(R)/(pi)`