# Potential Energy and Energy Conservation (Graph)

Please see attached file for the problem

Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle.

Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc.

Consider the potential energy diagram shown. The curve represents the value of potential energy as a function of the particle's coordinate . The horizontal line above the curve represents the constant value of the total energy of the particle . The total energy is the sum of kinetic ( ) and potential ( ) energies of the particle.

The key idea in interpreting the graph can be expressed in the equation

where is the x component of the net force as function of the particle's coordinate . Note the negative sign: It means that the x component of the net force is negative when the derivative is positive and vice versa. For instance, if the particle is moving to the right, and its potential energy is increasing, the net force would be pulling the particle to the left.

If you are still having trouble visualizing this, consider the following: If a massive particle is increasing its gravitational potential energy (that is, moving upward), the force of gravity is pulling in the opposite direction (that is, downward).

If the x component of the net force is zero, the particle is said to be in equilibrium. There are two kinds of equilibrium:

• Stable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle back toward the equilibrium point (think of a ball rolling between two hills).

• Unstable equilibrium means that small deviations from the equilibrium point create a net force that accelerates the particle further away from the equilibrium point (think of a ball on top of a hill).

In answering the following questions, we will assume that there is a single varying force acting on the particle along the x axis. Therefore, we will use the term force instead of the cumbersome x component of the net force.

1. The force acting on the particle at point A is __________.

directed to the right

directed to the left

equal to zero

2. The force acting on the particle at point C is __________.

directed to the right

directed to the left

equal to zero

3. The force acting on the particle at point B is __________.

directed to the right

directed to the left

equal to zero

4. If the particle is located slightly to the left of point B, its acceleration is __________.

directed to the right

directed to the left

equal to zero

5. Name all labeled points on the graph corresponding to unstable equilibrium.

List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE.

6. Name all labeled points such that when a particle is released from rest there, it would accelerate to the left.

List your choices alphabetically, with no commas or spaces; for instance, if you choose points B, D, and E, type your answer as BDE.

7. Consider points A, E, and G. Of these three points, which one corresponds to the greatest magnitude of acceleration of the particle?

A

E

G

8. If the total energy of the particle is known, one can also use the graph of to draw conclusions about the kinetic energy of the particle since

.

As a reminder, on this graph, the total energy is shown by the horizontal line.

What point on the graph corresponds to the maximum kinetic energy of the moving particle?

9. At what point on the graph does the particle have the lowest speed?

#### Solution Preview

Please see the attachment for the solution.

Learning Goal: To be able to interpret potential energy diagrams and predict the corresponding motion of a particle.

Potential energy diagrams for a particle are useful in predicting the motion of that particle. These diagrams allow one to determine the direction of the force acting on the particle at any point, the points of stable and unstable equilibrium, the particle's kinetic energy, etc.

Consider the potential energy diagram shown. The curve represents the value of potential energy as a function of the particle's coordinate . The horizontal line above the curve represents the constant value of the total energy of the particle . The total energy is the sum of kinetic ( ) and potential ( ) energies of the particle.

The key idea in interpreting the graph can be expressed in the equation

where is the x component of the net force as function of the particle's coordinate . Note the negative sign: It means that the x component of the net force is negative when the derivative is positive ...

#### Solution Summary

This solution solves a potential energy and energy conservation problem.