We neglect that moment of inertia of the axle because we are told that its radius is small compared to the length of the chain. The work required to lift up the chain is equal to the difference in gravitational potential energy between the initial and final configuration. It is easy to show that the gravitational potential energy of an object is equal to the height of the center of mass (above the reference point) multiplied by the weight of the object. Let be the up direction. Then
Therefore, if we take the initial location of the center of mass to be the reference level for the gravitational potential energy, then the initial potential energy is . We assume that, after the chain has been completely wound, it can be approximated by a point mass at the location of the axle. The final potential energy is then where is the mass of the chain, and is the length of the chain. , so is the difference in gravitational potential energy between the initial and the final configurations. That is also the amount of work needed to wind the chain. Therefore, answer (C) is correct.