A uniform chain of length L and mass M is lying on a smooth table and one-third of its length is hanging vertically down over the edge of the table. If g is accelelation due to gravity, the work done to pull the hanging part on to the table is?

unit length of rope is M/L

given L/3 part of rope is hanging

we have to solve using integration methods, lets say 'x' part of rope is consided with an element 'dx' which is pulled up with limits from 0 to L/3

Potential energy = mgh

here dE= integral of ( Mx/L*g*dx)

        E = integral of ( M/L*g*x.dx) limits from 0 to L/3

    P.E = MgL/18

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regards

Ramesh