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Calculate the work done by gravity on weighted chain

by RoRi

As in the previous exercise we have a chain weighing 2 pounds per foot with a 50 pound weight attached to one end. However, now we assume the chain is 60 feet long and is dropped to the ground from a building of height 100 ft from an initial position with the weighted end of the chain hanging 10 feet over the ledge of the building. Calculate the amount of work done by gravity for the chain to fall all the way to the ground.

Now we have a piecewise defined force function: one piece while part of the chain is still above the ledge, and the other piece once the entire chain is over the ledge. This function is

    \[ f(x) = \begin{cases} 170 + 2x & \text{for } 0 \leq x \leq 50 \\ 270 & \text{for } 50 < x \leq 90. \end{cases} \]

So, the work done is

    \begin{align*} W &= \int_0^{50} (170 + 2x) \, dx + \int_{50}^{90} 270 \, dx \\ &= 170 (50) + (50)^2 + 270 (90 - 50) \\ &= 21800 \text{ ft-lbs}. \end{align*}

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