Problem 26 - University Physics with Modern Physics

Question

A 75-kg roofer climbs a vertical 7.0-m ladder to the flat roof of a house. He then walks 12 m on the roof, climbs down another vertical 7.0-m ladder, and finally walks on the ground back to his starting point. How much work is done on him by gravity (a) as he climbs up; (b) as he climbs down; (c) as he walks on the roof and on the ground? (d) What is the total work done on him by gravity during this round trip? (e) On the basis of your answer to part (d), would you say that gravity is a conservative or nonconservative force? Explain.

Step-by-Step Solution

Answer

(a) -5145 J, (b) 5145 J, (c) 0 J, (d) 0 J, (e) Gravity is a conservative force.

Step 1: Understanding Work Done by Gravity

Work done by gravity can be calculated using the formula: \( W = mgh \), where \( m \) is mass, \( g \) is acceleration due to gravity, and \( h \) is the height change. Gravity does positive work when an object moves downward and negative work when an object moves upward.

Step 2: Calculate Work Done As He Climbs Up

When the roofer climbs up the ladder, gravity does negative work since he moves against gravity. The height \( h \) is 7.0 m. Using \( W = mgh \): \( W = 75 \text{ kg} \times 9.8 \text{ m/s}^2 \times 7.0 \text{ m} = 5145 \text{ J} \). This is negative: \( -5145 \text{ J} \).

Step 3: Calculate Work Done As He Climbs Down

When descending, gravity does positive work. The height \( h \) is again 7.0 m. Using \( W = mgh \): \( W = 75 \text{ kg} \times 9.8 \text{ m/s}^2 \times 7.0 \text{ m} = 5145 \text{ J} \). This time the work is positive: \( 5145 \text{ J} \).

Step 4: Work Done As He Walks on the Roof and Ground

While walking on flat surfaces, such as the roof and ground, there is no vertical displacement; hence, gravity does no work, i.e., \( W = 0 \text{ J} \).

Step 5: Calculate Total Work Done by Gravity During the Trip

Sum the work done during each stage: Ascent: \(-5145 \text{ J}\), Descent: \(5145 \text{ J}\), and walks: \(0 \text{ J}\). Total work done: \(-5145 \text{ J} + 5145 \text{ J} + 0 \text{ J} + 0 \text{ J} = 0 \text{ J} \).

Step 6: Determine if Gravity is Conservative or Nonconservative

Since the total work done by gravity in a closed path is zero, gravity is a conservative force.

Key Concepts

Conservative Force

The concept of a conservative force is central to understanding how nature's forces work, especially in physics. A conservative force, like gravity, is one where the total work done in moving an object between two points does not depend on the path taken. This means that if you take an object from point A to point B and back to point A, the total work done by a conservative force is zero. Hence, the energy associated with conservative forces is path-independent.

An essential feature of conservative forces is that they conserve mechanical energy. This means that any work done against them, for example, by lifting an object, is stored as potential energy, which can be fully recovered. In our exercise, when the roofer climbs up and down the ladder, the work done by gravity cancels out because gravity is a conservative force. So, the net work in the entire round-trip remains zero.

Ladder Height

The height of the ladder is critical in determining the work done by gravity. The higher the ladder, the more potential energy is involved when climbing. In our scenario, the roofer climbs a ladder that is 7.0 meters high. This height is directly proportional to the work done according to the formula: \[ W = mgh \]where:

• \( m \) is the mass of the object, here 75 kg,
• \( g \) is the acceleration due to gravity, approximately 9.8 m/s^2,
• \( h \) is the height climbed, here 7.0 m.

Understanding ladder height helps in calculating how much energy is being stored as gravitational potential energy and how it changes.

Potential Energy Change

When the roofer climbs up or down, there is a change in gravitational potential energy. Potential energy is the energy that is stored by virtue of the roofer's position relative to the Earth. When he climbs up the ladder to the rooftop at a height of 7.0 m, his potential energy increases. It's calculated using:\[ ext{Potential Energy} = mgh \]This increase in potential energy represents the work done against gravity, which is a conservative force. Conversely, as the roofer descends the ladder, potential energy is converted back into kinetic energy. This means the potential energy decreases by the same amount it increased while climbing up. In these scenarios, potential energy changes are reversible and consistent with the conservative nature of gravity.

Closed Path

In physics, a closed path or loop is where the object returns to its original position after moving through an entire sequence of movements. In our scenario, the roofer starts and ends at the same point after climbing the ladder, walking on the roof and ground.

This closed path is significant when considering forces like gravity. For a conservative force, the work done over a closed path is zero. In this exercise, the work done against gravity as the roofer moves up is exactly countered by the work done by gravity as he moves down. The walks on flat ground involve no vertical displacement, resulting in no work done by gravity. Therefore, the concept of a closed path helps illustrate why the total work done by gravity is zero, reinforcing the conservative nature of gravitational force.