A chain having mass m and length l rests on the surface of a smooth sphere of radius R > l . One end of the chain is fastened to the top of the sphere. Determine the tangential acceleration dv dt of the chain at the moment it is released and starts sliding down.

Step-by-step solution

Let the linear mass density of the chain be = m l . Consider a small element of the chain of length ds = R d at an angle from the vertical. The mass of this element is dm = R d . The tangential component of the gravitational force acting on this element is dF_t = dm g = R g d . The total tangential force acting on the chain is obtained by integrating dF_t from = 0 to = l R (since the chain has length l and one end is at the top of the sphere): F_t = ₀^ l/R R g d F_t = R g [- ]₀^ l/R F_t = R g [1 - ( l R ) ] Since t