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 one-third of the chain.

Short answer

The work done by the winch in winding up one-third of the chain is 37.5 foot-pounds.

Step-by-step solution

Identify Given Values

We know the length of the chain is 15 feet and it weighs 3 pounds per foot. Therefore, the total weight of the chain is 15 feet * 3 pounds/foot = 45 pounds. As we're winding up a third of the chain, the length of chain being wound up is 15 feet / 3 = 5 feet. lso, the total weight of chain being wound up is 5 feet * 3 pounds/foot = 15 pounds.

Calculate the Average Weight of the Chain Segment Being Lifted

The weight of the chain segment changes as it is lifted, from 15 pounds when it is at ground level to 0 pounds when it is entirely wound up. So, the average weight of the chain while it is being lifted is (15 pounds + 0 pounds) / 2 = 7.5 pounds.

Calculate Work Done

Work in physics is the force required to move an object a certain distance, so the work done by the winch in lifting this third of the chain is the average force (7.5 pounds) times the distance moved (5 feet). Therefore, Work = Force * Distance = 7.5 pounds * 5 feet = 37.5 foot-pounds.