Problem 53 - University Physics with Modern Physics

Question

A uniform \(3.00-\mathrm{kg}\) rope 24.0 \(\mathrm{m}\) lies on the ground at the mtop of a vertical cliff. A mountain climber at the top lets down half of it to help his partner climb up the cliff. What was the change in potential energy of the rope during this maneuver?

Step-by-Step Solution

Answer

The change in potential energy of the rope is approximately -89 J.

Step 1: Understand the Problem

We are asked to find the change in potential energy when a mountain climber lets down half of a uniform rope. The rope has a mass of 3.00 kg and its full length is 24.0 m. Thus, the climber lets down 12.0 m of rope.

Step 2: Determine the Mass of the Lowered Rope

Since the entire rope is 3.00 kg and the climber lets down half of it, the mass of the lowered rope is half of the total mass, which is 1.50 kg.

Step 3: Calculate the Initial Potential Energy

Initially, all parts of the rope are at the top of the cliff. The potential energy of the entire rope is 0 since the reference point (the ground at the base of the cliff) is below it. We simplify this by saying the initial potential energy of the lowered section, where its center-of-mass is 24 m above the ground, is given by:\[ PE_{ ext{initial}} = mgh = 1.50 imes 9.81 imes 24 \]

Step 4: Calculate the Final Potential Energy of the Lowered Section

After the rope is lowered, the center of the mass of the lowered section is placed halfway through its half-length from the top of the cliff. This means the center of mass has descended 24.0 m - 12.0/2 m = 18.0 m from the ground. Therefore, the final potential energy for the lowered mass is:\[ PE_{ ext{final}} = 1.50 imes 9.81 imes 18 \]

Step 5: Calculate the Change in Potential Energy

The change in potential energy is found by calculating the difference between the final and initial potential energies. Thus,\[\Delta PE = PE_{ ext{final}} - PE_{ ext{initial}} \]Substituting the calculated values from Step 3 and Step 4 gives the energy change:\[\Delta PE = (1.50 \times 9.81 \times 18) - (1.50 \times 9.81 \times 24)\]

Step 6: Compute the Change in Potential Energy

Solving the previous equation, we get:\[\Delta PE = (1.50 \times 9.81 \times 18) - (1.50 \times 9.81 \times 24)\]\[\Delta PE = 264.735 - 353.7 = -88.965 \, \text{Joules}\]Thus, the change in potential energy is approximately -89 J.

Key Concepts

Gravitational Potential Energy

Potential energy is the energy stored within an object due to its position relative to a reference point, usually the ground. In particular, gravitational potential energy is the energy an object possesses because of its height above the ground. The key formula for gravitational potential energy is given by:

• \(PE = mgh\)

where:

• \(m\) is the mass of the object,
• \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\) on Earth),
• \(h\) is the height of the object above the ground.

In the given exercise, when the climber lets down the rope, he changes its position relative to the ground. Initially, all sections of the rope are at the top, each with its potential energy dependent on its height. Lowering half of the rope changes its gravitational potential energy since the mass distribution is different now.
This energy change helps us understand how energy transfers as the position (or height) of an object changes.

Work and Energy

Work and energy are closely related concepts in physics. When you lower an object, such as the rope in our exercise, gravity does work on it. Work done by a force can be calculated as the force multiplied by the distance over which it acts. For gravitational forces, this means:

• \(W = F \times d\)

Since gravity acts vertically downward, the work done corresponds to the change in potential energy. Thus, the negative sign in the calculated change in potential energy (-89 J) indicates that energy has been transferred out of the system as work done by gravity. When an object's gravitational potential energy decreases, such as when it is lowered, this energy is transferred from the object. This concept highlights the conservation of energy, where energy is neither created nor destroyed but merely changes form or position from one system to another.

Mechanical Energy

Mechanical energy is the sum of potential energy and kinetic energy in a system. It plays a vital role in analyzing simple mechanical processes like the one in our rope exercise.

• \(E_{mech} = PE + KE\)

Our exercise mainly deals with the potential energy part since the rope isn't moving sideways, and we focus on its vertical motion due to gravity.
Mechanical energy helps us understand how energy conversion occurs in systems. Even though our primary concern was potential energy, should the climber's partner start climbing using the rope, potential energy would convert into kinetic energy as they ascend. By grasping the concept of mechanical energy, students can appreciate the bigger picture of energy transformations occurring in everyday scenarios, further linking it to the work-energy principle.