A 17-foot chain that weighs 4 pounds per cubic foot hangs from a winch 17 feet above ground level. Find the work done by the winch in winding up the entire chain.
A)346.80 ft-lb
B)1,040.40 ft-lb
C)867.00 ft-lb
D)462.40 ft-lb
E)578.00 ft-lb

Question

Quizplus · Accepted Answer

The work done by the winch in winding up the chain can be calculated using the formula for work, $W = \int_{a}^{b} F(x) \, dx$, where $F(x)$ is the force function and $x$ is the distance over which the force is applied. Here, the force varies as the chain is wound up because less chain is hanging from the winch as it winds up, reducing the weight being lifted at any moment.Given that the chain weighs 4 pounds per cubic foot and is 17 feet long, we first find the weight per foot of the chain. Since the weight is given in pounds per cubic foot but we're not given the volume of the chain, we proceed by considering the weight per foot directly given by the problem's context, which implies a linear density rather than a volumetric one. This was a mistake in interpreting the given information as implying a volumetric density; the correct approach is to consider the chain's total weight and how it changes as the chain is lifted, without needing to calculate a volume or use the given density in a volumetric sense.The total weight of the chain is the force that needs to be overcome to lift it. As the chain is lifted, the force required decreases linearly because less of the chain's length is being supported. The average force over the distance can be used to simplify the calculation, which is the average weight of the chain as it is being lifted.The total weight of the chain is $17 \text{ feet} \times 4 \text{ pounds/foot} = 68 \text{ pounds}$. The work done to lift this weight over 17 feet can be calculated by considering the average position of the chain, which is at half its length when considering the work done over the entire lifting process. Thus, the average height at which the chain is lifted is $\frac{17}{2} = 8.5 \text{ feet}$.The work done, $W$, is then the weight of the chain multiplied by this average height: $W = 68 \text{ pounds} \times 8.5 \text{ feet} = 578 \text{ ft-lb}$. This calculation simplifies the integral by considering the average force over the distance lifted, which is appropriate given the linear variation in force as the chain is wound up.

A 17-Foot Chain That Weighs 4 Pounds Per Cubic Foot