Question

Lifting \(a\) Chain In Exercises 19-22, consider a 15-foot chain that weighs 3 pounds per foot hanging from a winch 15 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Wind up the entire chain.

Short answer

The total work done to wind up the entire chain is \( \frac{3*15^2}{2} \) foot-pounds.

Step-by-step solution

Define the problem

The problem involves calculating the total work done to wind up a 15-foot chain weighing 3 pounds per foot that is hanging from a winch 15 feet above ground level.

Set up the integral

The work done to lift the chain is the integral of the force with respect to the distance. Since the force changes as the chain is wound up, this is an integral from 0 to 15 (the length of the chain) of the weight of the chain (3lbs/ft) times the distance from the winch (15-x feet), dx.

Solve the integral

Solving this integral yields the total work done: \[ \int_{0}^{15} 3(15-x) dx = \[ 3x(15-x)|_{0}^{15} = 3(15*15-15^2/2) - 3(0) = 3*15^2/2 \] pounds-feet.

Convert to foot-pounds

In this context, work is usually measured in foot-pounds (the amount of energy required to move a one pound weight a distance of one foot). To convert our result, we multiply by the conversion factor of 1 foot-pound/pound-foot, giving a final answer of \( \frac{3*15^2}{2} \) foot-pounds.