Final answer:
To determine the work required to raise one end of a 14 ft long chain weighing 80 lbs to a height of 3 ft, integrate the product of the weight per foot and the varying length being lifted, then multiply by the gravitational constant.
Step-by-step explanation:
To calculate the work required to raise one end of the chain to a height of 3 feet, we assume the chain is uniform, meaning the weight is distributed evenly along its length.
With every small piece of the chain we lift, the weight we're lifting decreases linearly since there's less chain hanging down. To solve this, we can integrate the weight function from 0 to 3 feet.
First, calculate the weight per foot by dividing the total weight of the chain by its length: weight per foot = 80 lbs / 14 ft. Then, let x be the variable representing the height in feet at which the chain is being lifted. At height x, the weight being lifted is (14 - x) times the weight per foot.
The differential work dW for a small segment dx is given by dW = (weight per foot) * (14 - x) * dx. We integrate this expression from 0 to 3 to find the total work:
Work = ∫03 ((80/14) * (14 - x)) dx
After performing the integration, we multiply the result by the gravitational constant g = 32.2 ft/s2 to convert the weight to force since work is force times distance.
Upon evaluating the integral and performing the multiplication, you would obtain the total amount of work done to lift the chain.