Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A window washer weighing 160 pounds is attached to a rope hanging from the roof of the building whose windows he is washing. The rope weighs \(0.6 \mathrm{lb} / \mathrm{ft}\). Right now he is working 50 feet down from the rooftop. (a) How much work is required to bring him to the windows that are 25 feet from the rooftop? (b) How much work will it take to bring him from where he is to the roof?

Step-by-Step Solution

Answer

The work done to bring the window washer to the windows 25 feet from the rooftop is 4750 ft.lb. The work done to bring the window washer to the roof is 9125 ft.lb.

Step 1: Calculate the total weight

Start by calculating the total weight that needs to be lifted. In this case, it's the weight of the window washer and the weight of the part of the rope below him, which is \(0.6 lb/ft \times 50ft = 30 lb\). So, the total weight is \(160lb + 30lb = 190lb\). This gives the force of gravity that needs to be overcome to lift the window washer and the rope.

Step 2: Calculate the work done to lift the window washer to 25 feet

Now, calculate the work to lift the window washer 25 feet. Since work is force times distance, and the force is constant (190 lb), the work done is \(190 lb \times 25 ft = 4750 ft.lb\).

Step 3: Calculate the work done to lift the window washer to the roof

Now, let's calculate the work to lift the window washer from 50 feet to the roof. Again, the work done is force times distance, but this time, we need to subtract the weight of the rope that is above the window washer, because it's not being lifted. Thus, in the first 25 feet, the work done is \(190 lb \times 25 ft = 4750 ft.lb\). In the next 25 feet, we subtract the weight of 25 feet of rope from the total weight, which gives us a reduced force of \(190lb - (0.6 lb/ft \times 25ft) = 175 lb\). Thus, the work done in this segment is \(175 lb \times 25 ft = 4375 ft.lb\). So, the total work done to lift the window washer to the roof is \(4750 ft.lb + 4375 ft.lb = 9125 ft.lb\).

Key Concepts

Physics problems

In physics, problems often involve real-life situations where forces interact to produce motion or change. These scenarios allow us to apply basic principles, such as calculating work. By understanding how to approach these problems systematically, we can find accurate solutions.

When tackling a problem like the one involving a window washer, it is essential to break the problem down into parts. First, identify the forces at play and where these forces need to be applied.

In this exercise, the forces include the weight of both the window washer and the rope he uses. They represent constant forces pulling downward due to gravity, which must be overcome by performing work to move the washer.

The complexity arises from the varying weight of the rope as different sections are pulled up, requiring you to calculate correctly at each stage. It involves understanding how gravity acts over distances, and how to apply formulas such as work = force x distance.

Force and distance

In physics, force is an essential concept and is simply any interaction that, when unopposed, changes the motion of an object. It's often expressed in Newtons (N) but in this problem, we're using pounds.

Distance refers to how far an object will move or has to be moved to achieve its intended position. It's a crucial aspect when calculating work, as it determines how much force needs to be exerted over how much space.

• Understanding the relationship between force and distance helps in calculating work more accurately.
• In this exercise, the distance is the vertical height the window washer needs to be moved.
• The force, in this case, is the total weight of the window washer and the segment of the rope being moved up.

Work is calculated using the formula: \( \text{Work} = \text{Force} \times \text{Distance} \). By knowing both the force that needs overcoming and how far it must act, one can correctly calculate the work needed in any given problem.

Work-energy principle

The work-energy principle is a core concept in physics that states that the work done by all forces acting on a body will result in a change in its kinetic energy. In simple terms, it emphasizes how work leads to motion or change in motion.

To understand this in the context of the window washer problem, consider how lifting the washer changes his position, overcoming gravitational potential energy, which constitutes "work". As energy is transferred via work, the system's energy state transitions.

• This principle helps explain why moving the window washer requires effort directly related to the energy it takes to change his position.
• In practical terms, work results in raising him higher against gravity, indicating a gain in gravitational potential energy.

Incorporating the rope's weight at different distances adds a layer of complexity, showing how work needed changes with different force magnitudes. This insight highlights the interconnection between work, energy, and force—bounded within real-world physical limitations.