A chain is held on a frictionless table with 1 n th of its length hanging over the edge. If the chain has a length L and a mass M , how much work is required to pull the hanging part back on the table?

Step-by-step solution

If m is the mass per unit length of the chain, the mass of the chain of length y will be y × m = y m and the force acting on it due to gravity will be m y × g = m g y (assuming that y is the length of the chain hanging over the edge). So, the work done in pulling the d y length of the chain on the table d W = F - d y [as y is decreasing] i.e., d W = m g y ( - d y ) [as F = m g y ] So, the work done in pulling the hanging portion on the table : W = - ∫ L/n 0 mgydy = MgL 2 2n 2 or W = MgL/2n 2 as M = mL