So, I know that the integral of the force should be the work. Since I am given the density and length I think I would be able to multiply them together to get the total mass. And since force is mass times acceleration, 9.8m/s^2, I get force is 98N. From this, I then thought that I could just find then have the integral of 98x dx on the interval 0 to 5. Is this correct or is there something I am missing? I do not understand the steps I should take in order to solve work problems such as this. Thank you.
2026-04-04 07:05:21.1775286321
A 5 m chain with density 2 kg/m hangs off the roof of a building. How much work is required to pull the chain to the roof?
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You can calculate this as an integral if you wanted to, but there's a simpler way of doing this. For a 5m chain at 2km/m that makes the whole chain 10kg. Since you're given a constant density, you know that the chain has a uniform density (or at the least we can assume as much). This places the centre of mass 2.5m below the roof. You can now treat the chain as a point mass of 10kg that you have to move up 2.5m. Use GPE to calculate which should be trivial and you have an answer.
You could then check this with the integral $$ W=\int_0^519.6x\mathrm{d}x=\int_0^519.6(5-x)\mathrm{d}x $$ You can prove the integral equivalence with a simple change of variable here.