Question

A rope of total mass M and total length L has a length l of the rope hanging vertically down

off the work bench. How much work must be done to place all the rope on the workbench?

Answer 1

Work done is equal to increase in potential energy of rope, in placing all the rope on the workbench.

Increase in potential energy = mgh

mass of the part of rope that is hanging = Ml/L
height h is the height, by which center of mass of the hanging part is lifted. Center of mass of hanging part is in the center of hanging mass, that l/2 below the workbench.
Hence work done to = (Ml/L)*g*(l/2)
                                  = Ml²g/2L