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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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Most popular questions from this chapter

\(\mathrm{A} 2.50 \mathrm{~kg}\) mass is pushed against a horizontal spring of force constant \(25.0 \mathrm{~N} / \mathrm{cm}\) on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store \(11.5 \mathrm{~J}\) of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur?

A 90.0 kg mail bag hangs by a vertical rope 3.5 m long. A postal worker then displaces the bag to a position 2.0 m sideways from its original position, always keeping the rope taut. (a) What horizontal force is necessary to hold the bag in the new position? (b) As the bag is moved to this position, how much work is done (i) by the rope and (ii) by the worker?

CALC The potential energy of a pair of hydrogen atoms separated by a large distance \(x\) is given by \(U(x)=-C_{6} / x^{6},\) where \(C_{6}\) is a positive constant. What is the force that one atom exerts on the other? Is this force attractive or repulsive?

You are designing a delivery ramp for crates containing exercise equipment. The \(1470 \mathrm{~N}\) crates will move at \(1.8 \mathrm{~m} / \mathrm{s}\) at the top of a ramp that slopes downward at \(22.0^{\circ} .\) The ramp exerts a \(515 \mathrm{~N}\) kinctic friction force on cach crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of \(5.0 \mathrm{~m}\) along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the largest force constant of the spring that will be needed to meet the design criteria.

The food calorie, equal to \(4186 \mathrm{~J},\) is a measure of how much energy is released when the body metabolizes food. A certain fruit-and-cereal bar contains 140 food calories. (a) If a 65 kg hiker eats one bar, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes into increasing gravitational potential energy? (b) If, as is typical, only \(20 \%\) of the food calories go into mechanical energy, what would be the answer to part (a)? (Note: In this and all other problems, we are assuming that \(100 \%\) of the food calories that are eaten are absorbed and used by the body. This is not true. A person's "metabolic efficiency" is the percentage of calories eaten that are actually used; the body eliminates the rest. Metabolic efficiency varies considerably from person to person.
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