Problem 20 - Essential Calculus

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.

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Physics

In the context of lifting a chain, understanding physics is crucial. The concept of work is fundamental here.
Work is defined as the product of force and the distance over which this force is applied. If you are lifting a chain, the force can be understood as the weight of the chain segment being lifted.
The weight is the gravitational force acting on the chain. It is dependent on the mass of the chain and the acceleration due to gravity. However, here it’s simplified to weight since mass is given in pounds.

• Weight of the chain is affected by the height from which it is being lifted.
• Total weight of the chain is calculated as length times weight per unit length.

Calculating work means figuring out how much force is applied and over what distance. For this chain, one-third is lifted from the ground, moving through a vertical distance of 5 feet. The work done is measured in foot-pounds, indicating the weight force acting over the distance.

Calculus

Calculus plays an essential role when dealing with continuously changing quantities like lifting a chain. It helps in calculating the work done with changes in force as the chain is being lifted.
When a chain is lifted, its weight felt by the winch changes.
At any moment, the amount of chain, hence the weight, varies as more chain is gathered.

• This is where integration comes in since it sums up infinitesimally small elements over a distance.
• Integration assists in finding the total work done by accumulating individual bits of work as tiny chain increments are lifted.

The average weight is found by taking the sum of initial and final weights at two specific points: when the whole part is on the ground and when the segment is completely lifted. This produces an average value that simplifies the work calculation into a straightforward multiplication rather than using full integration here.

Lifting Force

The concept of lifting force is directly tied to how we understand and calculate work. When lifting a chain, the force applied corresponds to the weight of that portion of the chain.
This challenge in lifting stems from needing to counteract the gravitational force pulling the chain down.
The lifting force must exceed the weight of the chain segment to achieve movement.

• The weight of this chain section can fluctuate from its full weight at ground level to nothing when lifted entirely.
• The variation in force shows that the force applied by the winch decreases as more chain is wound up.

To make calculations simple in this exercise, the average force is used. The winch applies a force equivalent to lifting the average weight of 7.5 pounds over the 5-foot distance. This simplification helps to efficiently compute the work with the understanding that lifting force is not constant over the entire operation.