A 3.0 m long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that x = 1.6 m of the chain remains on the top level and y = 1.4 m hangs vertically. (See the figure.) At this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 1.6 m remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 32 N/m.)
I'd look at this situation this way: The picture of chain lying on scaffold at the start has the center of mass (CM) of hanging part of chain at y/2 = 1.4/2 = 0.7 m from top of scaffold. At the finish, the entire chain is hanging in air with its CM at a distance = (1.6 + 1.4)/2 = 3/2 = 1.5 m from top of scaffold. So the change in CM position during the entire downward movement equals: 1.5 - 0.7 = 0.8 m. In first picture described above, the gravitational force on the hanging chain = (32)(1.4) = 44.8 N. In last picture entire chain hangs vertically off scaffold, gravitational force = chain weight = (32)(3) = 96 N. The average weight of chain pulling downward during the chain's movement = (44.8 + 96)/2 = 70.4 N Work done by the force of gravity on chain = (70.4)(0.8) = 56 J ANS
