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 the entire chain with a 500 -pound load attached to it.

Short answer

The work done by the winch in lifting the entire chain with a load attached to it is 8175 lb*ft.

Step-by-step solution

Calculate the total weight of the chain and the load

The chain is 15-foot long and weighs 3 pounds per foot, so the total weight of the chain (\(Wc\)) is \(3 lbs/ft * 15 ft = 45 lbs\). A 500-pound load is attached to it, so the total weight (\(Wt\)) to be lifted is \(Wc + load = 45 lbs + 500 lbs = 545 lbs\).

Convert the weight to force

In the context of physics, the weight is a force and is measured in pounds in the British measurement system. Therefore, the force (\(F\)) needed to lift the chain and the load is \(545 lbs\).

Calculate the work done by the winch

The winch winds up the entire chain which is 15 feet. So, by using the work formula \(W = Fd\), the work (\(W\)) done by the winch in winding up the chain and the load is \(W = F * d = 545 lbs * 15 ft = 8175 lb*ft\).

Key concepts

Chain Lifting Work

When we discuss chain lifting work in calculus, we're delving into an interesting real-world application of the principles of integration in physics. Imagine you have a chain suspended from a winch, like the one mentioned in the exercise - a 15-foot chain that weighs 3 pounds per foot. Lifting the chain doesn't require the same force throughout because as you lift, the length of the hanging chain decreases, and thus the force needed changes with it.

To calculate the total work done in lifting the chain, we first determine the force required to lift a small element of the chain and then integrate this force over the path it's lifted. It's important to note that the force needed to lift the last segment of the chain is less than the force needed at the beginning because there's less chain to lift.

In our example, the work is not only for lifting the chain but also includes a significant load attached to it. This changes the dynamic as you no longer can consider the weight to be distributed. The entire 500-pound load must be considered in every step of the lifting process, contributing to a constant force that compounds the difficulty of the problem.

Force Calculation

The concept of force calculation is central to solving any work-related problem in physics and calculus. Force is intuitively understood as a push or pull on an object. In the context of the chain lifting problem, force represents the total weight that needs to be overcome to lift the chain and the attached load off the ground.

Mathematically, calculating the force (\(F\)) is a matter of multiplying the weight (in pounds, lbs) of the object by the acceleration due to gravity, which is a constant (\(9.8 m/s^2\) or approximately 32 feet per second squared in the Imperial system of measurements). However, since the weight given already accounts for gravitational acceleration, the force in our problem is equal to the total weight being lifted. In our exercise, after adding the weight of the load to the total weight of the chain, we have a single value for force, 545 pounds, which simplifies our calculations.

Work-Energy Principle

The work-energy principle is a fundamental concept that connects the dots between work, force, and movement. It simply states that work done by a force on an object results in a change in the object’s kinetic energy. In the context of lifting a stationary object like the chain in our problem, the principle tells us that the work done by the winch is stored as potential energy in the lifted chain and load.

To quantify the work done, we use the formula \(W = Fd\), where \(W\) is work, \(F\) is the constant lifting force, and \(d\) is the distance over which the force is applied. In our case, the work \(W\) against gravity is calculated by multiplying the total force required to lift the chain and the load (545 pounds) by the height the chain is lifted (15 feet), resulting in \(W = 545 lbs * 15 ft = 8175 lb*ft\).

This calculation tells us the amount of energy transferred from the winch to the chain and load's potential energy, and it is essential in understanding the physical requirements of the lifting process and designing a system capable of completing the task.