To find the amount of work required to lift the rope and bucket up 12 feet, we need to break down the problem into two parts: the work required to lift the rope and the work required to lift the bucket.

Part 1: Work to Lift the Rope

The weight of the rope is 80 lb, and it is uniformly distributed along its length. Therefore, the weight per foot of the rope is: $\text{Weight per foot of rope} = \frac{80 \, \text{lb}}{20 \, \text{ft}} = 4 \, \text{lb/ft}$

When you lift the rope, the higher you lift, the less rope remains hanging, so the force changes as you lift. To calculate the work done in lifting the rope, we need to integrate over the distance being lifted, accounting for the changing length of rope.

Let $x$ be the distance from the top of the well (in feet). Initially, the full length of the rope hangs over the well, and as you lift it, the length hanging decreases.

The rope is lifted 12 feet, so the rope hanging at height $x$ changes from 0 to 12 feet.

The force due to the remaining hanging rope at height $x$ is $4(20 - x)$ pounds, and the work required to lift this segment over an infinitesimal height $dx$ is: $dW = 4(20 - x) \, dx$

Thus, the total work done to lift the rope is: $W_{\text{rope}} = \int_0^{12} 4(20 - x) \, dx$ Solving this: $W_{\text{rope}} = 4 \int_0^{12} (20 - x) \, dx$ $W_{\text{rope}} = 4 \left[ 20x - \frac{x^2}{2} \right]_0^{12}$ $W_{\text{rope}} = 4 \left[ 20(12) - \frac{12^2}{2} \right]$ $W_{\text{rope}} = 4 \left[ 240 - 72 \right] = 4 \times 168 = 672 \, \text{ft-lb}$

Part 2: Work to Lift the Bucket

The bucket weighs 20 lb, and its weight does not change as it's lifted. The work required to lift the bucket is simply the product of the weight of the bucket and the distance it is lifted: $W_{\text{bucket}} = 20 \, \text{lb} \times 12 \, \text{ft} = 240 \, \text{ft-lb}$

Total Work

The total work is the sum of the work done to lift the rope and the work done to lift the bucket: $W_{\text{total}} = W_{\text{rope}} + W_{\text{bucket}} = 672 \, \text{ft-lb} + 240 \, \text{ft-lb} = 912 \, \text{ft-lb}$

Thus, the total work required to lift the rope and bucket up 12 feet is 912 ft-lb.

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Would you like more details on any of the steps?

Here are 5 related questions:

1. How would the work change if the bucket weighed 30 lb instead?
2. What would the work be if you lifted the bucket and rope 15 feet instead of 12 feet?
3. How does the work change if the rope weighs 60 lb instead of 80 lb?
4. How would you compute the work if the rope wasn't uniformly distributed in weight?
5. How much work is required if you lift the entire rope and bucket all the way up out of the well?

Tip: Work depends not only on the force but also on how the force varies as you move an object through a distance (as in the case of the changing length of rope).