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Chapter 7: Problem 1

In one day, a 75 kg mountain climber ascends from the 1500 m level on a vertical cliff to the top at 2400 m. The next day, she descends from the top to the base of the cliff, which is at an elevation of 1350 m. What is her change in gravitational potential energy (a) on the first day and (b) on the second day?

Short Answer

Expert verified
The change in gravitational potential energy on the first day is 661500 J, and on the second day is -770250 J.

Step by step solution

01

Calculate the change in height on the first and second day

In order to calculate the change in gravitational energy, find the difference in initial and final heights for each day. On the first day, the initial height is 1500 m and the final height is 2400 m, and therefore the change in height is \( 2400 m - 1500 m = 900 m \). On the second day, the initial height is 2400 m and the final height is 1350 m, with the change in height equaling \( 2400 m - 1350 m = 1050 m \).
02

Calculate the change in gravitational potential energy on the first day

Use the formula for gravitational potential energy \( PE = mgh \) to calculate the change in gravitational potential energy. Here, the climber's mass \( m \) is 75 kg, the gravitational acceleration \( g \) is approximately 9.8 m/s虏, and the change in height \( h \) from Step 1 for the first day is 900 m. Therefore, the change in gravitational potential energy on the first day is \( PE = 75 kg * 9.8 m/s虏 * 900 m = 661500 J \). This is the energy gained on the first day.
03

Calculate the change in gravitational potential energy on the second day

Again, apply the formula \( PE = mgh \). The mass \( m \) is 75 kg, the gravitational acceleration \( g \) is around 9.8 m/s虏, and the change in height \( h \) from Step 1 for the second day is 1050 m. Thus, the change in gravitational potential energy on the second day is \( PE = 75 kg * 9.8 m/s虏 * 1050 m = 770250 J \). The energy is negative this time, because she is descending, therefore the energy is lost.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mechanical Energy
Mechanical energy is a term used in physics to describe the sum of potential energy and kinetic energy for an object. Potential energy can come from an object's position, as is the case with gravitational potential energy.

In the context of our mountain climber, while she scales the cliff, her mechanical energy changes primarily due to a shift in gravitational potential energy.
  • Potential energy increases as she ascends, adding energy to her mechanical system.
  • As she descends, potential energy decreases, meaning less mechanical energy overall.
The concept of mechanical energy is important because it helps us understand how the two forms of energy work together and influence motion and forces acting on an object.
Physics Problem Solving
Solving physics problems requires a systematic and logical approach. For the mountain climber question, it's crucial to break down the problem into manageable steps.

Here's a simple process to keep in mind:
  • Identify what is known (e.g., mass, initial and final heights).
  • Specify what needs to be found (e.g., change in gravitational potential energy).
  • Use relevant equations and physical laws to calculate unknowns.
For this problem, the equation for gravitational potential energy, \( PE = mgh \), plays a crucial role. By systematically using known values, such as climber's mass and height changes, we compute the energy change efficiently.
Gravitational Force
Gravitational force is the attraction between two objects with mass. Earth's gravitational pull provides a constant acceleration, denoted as \( g = 9.8 \, \text{m/s}^2 \).

This force is a crucial component of the climber's potential energy changes:
  • The gravitational pull affects both the ascent and descent.
  • With the climber gaining height, she works against this force, storing potential energy.
  • Descending, she releases this stored energy back into the system.
Understanding gravitational force helps us appreciate how potential energy is practically stored and altered, and how this process can be calculated using simple physics.
Energy Conservation
Energy conservation is a fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. This concept is underlined in the climber's journey:

  • As she climbs, chemical energy in her muscles converts into potential energy, evident through her increased height.
  • When descending, potential energy converts to kinetic energy and potentially some thermal energy due to friction.
Nothing is wasted; instead, energy is just changing its form. For students, grasping energy conservation clarifies why her potential energy reduces upon descent, reflecting just a transformation rather than a disappearance of energy.

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