Question

A rope of length \(\ell\) ft hangs over the edge of tall cliff. (Assume the cliff is taller than the length of the rope.) The rope has a weight density of \(d \mathrm{lb} / \mathrm{ft}\). (a) How much work is done pulling the entire rope to the top of the cliff? (b) What percentage of the total work is done pulling in the first half of the rope? (c) How much rope is pulled in when half of the total work is done?

Short answer

(a) Work = \( \frac{d \ell^2}{2} \); (b) First half = 25\% of total work; (c) Intervals up to \( \frac{\ell}{\sqrt{2}} \) involve half the work, about 70.7% of the rope.

Step-by-step solution

Understand the Concept of Work

The work done in pulling a segment of rope is given by the formula \( W = F \cdot d \), where \( F \) is the force exerted and \( d \) is the distance over which the force is exerted. In this case, the force is the weight of the rope, which varies with the length of rope that still needs to be lifted.

Set Up the Integral for Total Work

For a small segment \( dx \) of the rope at height \( x \) from the top, the force is \( d \cdot x \), since \( x \) feet of rope below it is being lifted. The work done to lift this segment to the top is \( d \cdot x \cdot dx \). Therefore, the total work \( W \) is the integral: \[ W = \int_0^\ell d \cdot x \cdot dx = d \int_0^\ell x \cdot dx. \]

Calculate Total Work for Lifting the Rope

Compute the integral calculated in the previous step: \[ W = d \left[ \frac{x^2}{2} \right]_0^\ell = d \cdot \frac{\ell^2}{2}. \] Thus, the total work done is \( \frac{d \ell^2}{2} \).

Calculate Work for the First Half of the Rope

When pulling the first half of the rope, we determine the work done for \( x \) ranging from \( 0 \) to \( \frac{\ell}{2} \):\[ W_1 = d \int_0^{\frac{\ell}{2}} x \cdot dx = d \cdot \left[ \frac{x^2}{2} \right]_0^{\frac{\ell}{2}} = d \cdot \frac{\left(\frac{\ell}{2}\right)^2}{2} = \frac{d \ell^2}{8}. \]

Calculate Percentage of Total Work in First Half

The percentage of work done pulling in the first half of the rope is \[ \frac{W_1}{W} \times 100\% = \frac{\frac{d \ell^2}{8}}{\frac{d \ell^2}{2}} \times 100\% = 25\%. \]

Determine How Much Rope is Pulled in with Half the Total Work

We need to find the distance \( x \) where half the total work \( \frac{d \ell^2}{4} \) is done:\[ d \int_0^x x \cdot dx = \frac{d \ell^2}{4}, \] \[ \frac{d x^2}{2} = \frac{d \ell^2}{4}, \] \[ x^2 = \frac{\ell^2}{2}, \] \[ x = \frac{\ell}{\sqrt{2}}. \] Thus, approximately \( 0.707 \ell \) ft of the rope is pulled in when half of the work is completed.

Key concepts

Integration

Integration is a fundamental concept in calculus that helps us calculate the area under curves. In the context of work done in physics, it allows us to find the total work by adding up tiny amounts of work done over very small distances.
Calculus-based work problems, like pulling a rope, require us to integrate because the force changes with the length of the rope that remains to be lifted. With integration, we can compute the work done over continuously varying distances.
In our rope example, integration helps us find the total work needed to lift each small segment of the rope. We express this small work segment as a product of force, which depends on the length of the rope left, and a tiny distance increment, denoted as \( dx \).

• Integration formula used: \( W = \int_0^\ell d \cdot x \cdot dx \)
• This formula calculates continuous summing of small work segments from the bottom to the top of the cliff.

This approach ensures that each piece of rope is factored into our calculation as we lift it.

Work-Energy Principle

The work-energy principle is a cornerstone of mechanics that describes how work can change the energy of a system. In simpler terms, work is done when a force causes an object to move, transferring energy to that object.
When lifting a rope over a cliff, work is done on the rope as a force is applied against gravity. Because the weight of the rope exerting this force varies as parts of the rope are lifted, the work done increases as more of the rope is lifted.
The work-energy principle shows us the relationship between work done and energy change in this scenario.

• The work-energy equation here is \( W = F \times d \), demonstrating that work depends on both the force applied and the distance moved.
• In our case, the force is the weight of a segment of the rope, and the distance is the height the rope has moved against gravity.

As we calculate, notice how the effort, or input of energy, results in the lifting of the rope heightening its position.

Applied Calculus

Applied calculus refers to using calculus in practical real-world problems. When it comes to problems like pulling a rope, applied calculus helps us model and solve how much work is needed, considering varying forces.
This application takes theoretical calculus and applies it to everyday physics problems to derive precise solutions. Features of applied calculus in this context include:

• Translating physical scenarios into mathematical formulas or equations.
• Using integrals to handle continuously changing variables, such as rope length.
• Breaking down complex problems into smaller, solvable elements, much like handling different segments of the rope in the integral.

In the exercise, applied calculus allows us to compute how much work is done for different portions of the rope or under different conditions. It translates a physical action into a computation that gives us insights into the work performed.

Physical Applications

Physics often employs calculus to solve complex real-world problems, especially those involving motion or forces. Our rope problem is a great representation of physical applications in calculus.
Physical applications involve taking a physical scenario and using calculus to determine unknowns like work or energy. This type of problem usually involves some assumptions such as uniform weight density or constant forces.

• In the rope problem, we assumed the cliff is tall enough and the rope has a constant weight per unit.
• The integration provides a method to calculate continuous changes.
• Understanding physical applications further helps in solving mechanical work problems which are common in engineering and physics.

Through these physical applications, calculus provides a robust toolset for solving wide-ranging problems from engineering structures to rope-climbing challenges. The methods used are widely applicable to numerous physical science challenges beyond just pulling a rope.