A chain 64 meters long whose mass is 20 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 3 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.
[Math] A chain 64 meters long whose mass is 20 kilograms is hanging over the edge of a tall building…
calculus
Best Answer
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 $\delta x$ denote a small length of chain. To figure out the total work, we break up the chain into small pieces of equal length $\delta x,$ 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 $x\le 3,$ or simply moves $3$ meters if $x\ge 3.$
The work required to lift a piece is $xF$ if $x\le 3,$ or $3F$ otherwise, where $F$ is the force due to gravity. Assuming the chain has constant density, the force acting on a small $\delta x$ of chain is $F=mg=(\frac{20}{64}\delta x)(9.8) = 3.0625 \delta x.$
So to move a small $\delta x$-length of chain takes $3.0625x\delta x$ Joules of work if $0\le x\le 3.$ Summing up the work over these small lengths corresponds to integrating $\int_0^3 3.0625xdx.$
The work required to move the rest of the chain is easier, since all the remaining $\delta x$-lengths move the same distance: $3$ meters. So we can compte this all at once as $3F=3mg=3(\frac{20}{64}61)(9.8)=560.4375$ Joules of work.