Question

A crane lifts a \(2,000 \mathrm{lb}\) load vertically \(30 \mathrm{ft}\) with a \(1^{\prime \prime}\) cable weighing \(1.68 \mathrm{lb} / \mathrm{ft}\) (a) How much work is done lifting the cable alone? (b) How much work is done lifting the load alone? (c) Could one conclude that the work done lifting the cable is negligible compared to the work done lifting the load?

Short answer

(a) 756 ft-lb; (b) 60,000 ft-lb; (c) Yes, it's negligible.

Step-by-step solution

Calculate the weight of the entire cable

The length of the cable is the same as the distance the load is lifted, which is 30 feet. Given that each foot of the cable weighs 1.68 lb, the total weight of the cable is calculated as: \( \text{Total weight} = 1.68 \text{ lb/ft} \times 30 \text{ ft} = 50.4 \text{ lb} \).

Compute work done lifting the cable

To lift a cable, work is continuously done against gravity as each added piece of cable must be lifted from its initial position to the final position. If the cable is raised uniformly, on average, the center of mass of the cable is lifted halfway (15 ft) over its entire length. Thus, the work done lifting the cable is: \[ \text{Work} = \text{Weight of Cable} \times \text{Average Lift Height} = 50.4 \text{ lb} \times 15 \text{ ft} = 756 \text{ ft-lb}. \]

Calculate work done lifting the load

The load itself is lifted directly to the height of 30 feet. The work done in lifting the load to this height is given by: \[ \text{Work} = \text{Weight of Load} \times \text{Lift Height} = 2000 \text{ lb} \times 30 \text{ ft} = 60,000 \text{ ft-lb}. \]

Compare work done on the cable and the load

The work done to lift the cable is 756 ft-lb, while the work done to lift the load is 60,000 ft-lb. To evaluate whether the work on the cable is negligible, compare these values. The work on the cable is significantly smaller than the work on the load. Specifically, the work on the cable is only about 1.26% of the work done on the load: \[ \frac{756}{60,000} \times 100 \approx 1.26\% \].

Key concepts

Lifting Cable Calculations

When you need to calculate the work done to lift a cable, you start by determining the total weight of the cable. The weight is important because work is defined as the force applied over a distance.
For this particular problem, each foot of the cable weighs 1.68 lb and the cable is 30 feet long. Multiply the weight per foot by the length of the cable to find the total weight of the cable:

• Total weight of the cable = 1.68 lb/ft × 30 ft = 50.4 lb

Next, you calculate the average lift height. Since the cable is lifted from a horizontal position to a vertical one, its center of mass is raised through a distance equal to half of its length. This is because, on average, each piece of the cable has been lifted across its full length.
Finally, the work done on lifting the cable can be calculated using:

• Work = Weight of Cable × Average Lift Height = 50.4 lb × 15 ft = 756 ft-lb

Lifting Load Work

Lifting a load vertically requires us to calculate the work based on the load's weight and the distance it's raised. Work in this scenario is calculated by multiplying the weight of the load by the distance it moves.
In this example, the load weighs 2,000 lb, and it is lifted directly 30 feet up.
Therefore, the calculation for work done to lift just the load is:

• Work = Weight of Load × Lift Height = 2,000 lb × 30 ft = 60,000 ft-lb

This tells us how much energy is required to raise the load to its specified height. It's a straightforward application of the work formula, highlighting the linear relationship between weight and lift distance.

Work Against Gravity

When lifting objects, you're performing work against the force of gravity. This means you are exerting a force upward to counteract the gravitational force pulling the object downwards.
In our scenario, both the cable and the load require work to lift because they both have weight which is consistently being pulled by gravity. The heavier the object or the greater the distance you lift it, the more work you need to do.

• Work against gravity is given by: Work = Weight × Height

However, two different components (the cable and the load) show that while gravity affects both, the impact is very different. Heavier objects like our load require significantly more work to lift.

Physics Problems Solutions

Solving physics problems often involves applying fundamental principles like the work-energy principle to practical scenarios.
To tackle these types of problems efficiently:

• Break down the problem into smaller parts
• Calculate each part using the relevant formulas
• Use consistent units throughout
• Pay attention to what each component contributes to the overall system

Here, evaluating the work done both on the cable and the load separately helps in understanding how different parts of one system demand different energy levels.
By calculating and then comparing the work on the cable (756 ft-lb) and on the load (60,000 ft-lb), we help visualize why it's easy to conclude that the work needed to lift the cable is negligible in comparison to lifting the full load.