Question

(1 point) A chain 66 meters long whose mass is 26 kilograms is hanging over the edge of a tall building and does not touch the ground. How much work is required to lift the top 8 meters of the chain to the top of the building? Use that the acceleration due to gravity is 9.8 meters per second squared. Your answer must include the correct units.

Answer 1

Let x denote the position along the chain in meters, with x=0 corresponding to the top of the chain at the top of the building. Let δx denote a small length of chain. To figure out the total work, we break up the chain into small pieces of equal length δx, compute the work to move each piece, and add it all together.

The top piece does not need to be lifted further. A piece at approximately x meters down the chain needs to move x meters if x≤8, or simply moves 8 meters if x≥8

The work required to lift a piece is xF if x≤8, or 8F otherwise, where F is the force due to gravity. Assuming the chain has constant density, the force acting on a small δx of chain is F=mg=(26/66)(δx)(9.8)=3.8606δx.

So to move a small δx-length of chain takes 3.8606xδx Joules of work if 0≤x≤8. Summing up the work over these small lengths corresponds to integrating 0 to 8 (3.8606)xdx.

The work required to move the rest of the chain is easier, since all the remaining δx-lengths move the same distance: 8 meters. So we can compte this all at once as 8F=8mg=8(26/66)(58)(9.8)=1791.3212 Joules of work.