# If a 200 pound cable is 100 feet long and hangs vertically from the top of a tall building, how much work is required to lift the cable to the top of the building?

Oct 29, 2014

Since the density of the cable is

$\frac{200}{100} = 2$ lb/ft,

the small piece of the cable of length $\mathrm{dx}$ pulls down with the force

$2 \mathrm{dx}$ lb.

Let $x$ be the distance of the piece of the cable from the top of the building. To lift this piece up to the top, it requires the force of

$\mathrm{dW} = 2 \mathrm{dx} \cdot x = 2 x \mathrm{dx}$ ft$\cdot$lb .

Since the cable is $100$ ft long,

$W = {\int}_{0}^{100} 2 x \mathrm{dx} = {\left[{x}^{2}\right]}_{0}^{100} = 10000$ ft$\cdot$lb

I hope that this was helpful.