# Forces and potential energy

Not all forces have associated potential energies. For example,

• gravity DOES have a potential energy
• a spring DOES have a potential energy
• but
• friction DOES NOT have a potential energy

We call forces which do have associated potential energies conservative forces. One way you can recognize a conservative force is to take an object on a round trip and see what happens to it.

• If the net work done by the force during the round trip is ZERO, then the force is conservative
• If the net work done by the force during the round trip is NOT zero, then the force is non-conservative

Conservative        Non-conservative
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Examples                  gravity            friction
spring force       air resistance
electricity        water resistance

Work done by
force during
a round trip              zero               negative
(closed path)

Does it have
a potential energy?        yes                 no

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Here's a somewhat more sophisticated map showing potential energy -- electric potential energy in this case -- as a function of position on a piece of paper.

Q:  What is the change in potential energy
per meter at location A, if one
moves to the right on this map?

Q:  What is the change in potential energy
per meter at location A, if one moves
upward on this map?

Q:  In which direction is the change in potential
energy largest at point B?

Q:  What is the size of the electric force at point B?

Q:  What is the size, and direction, of the
force on a particle at point A?
Express this force in unit-vector notation.

Q:  At what locations is the total force on
a particle zero?

We call locations at which the net force on a particle equilibrium points. In theory, if you place a particle at one of these locations, since it feels no force, it should not accelerate -- or move.

But not all equilibria are created equal. Suppose you move the particle just a TEENY LITTLE BIT away from the point of equilibrium. What happens next?

1. If the force pushes the particle back toward the point of equilibrium -- a "restoring" force -- then the particle will go back; it might oscillate back and forth, or eventually settle down at the equilibrium point. We call such a location a stable equilibrium.
2. If the force pushes the particle away from the point of equilibrium, then the particle will move farther away -- and feel a stronger force -- and move farther away -- and never come back. We call such a location an unstable equilibrium.

Q:  Are there any stable equilibrium points
on the map of electric field?

Q:  Are there any unstable equilibrium points
on the map of electric field?

If you are interested in orbits and rockets and space travel, you might read about the important case of Lagrangian points in the Earth-Moon, or Sun-Earth, or Sun-Jupiter, systems. Things get a little more complicated --- but useful! -- when one adds rotation to the mix....

Image courtesy of The American Physics Society

Suppose that in some region of space, the potential energy U (measured in Joules) is a function of position (x, y) (measured in meters) like this:

1. What are the units of q?
2. What is the force on a particle in the x-direction?
3. What is the force on a particle in the y-direction?
4. Suppose the numerical value of the coefficient is q = 0.25 . Express the force on a particle at (3 m, 6 m) as a vector, in unit-vector notation. Don't forget the units!