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 ---------------------------------------------------------------------- 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 ----------------------------------------------------------------------

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?

- 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**. - 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 anystableequilibrium points on the map of electric field? Q: Are there anyunstableequilibrium 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:

- What are the units of
**q**? - What is the force on a particle in the x-direction?
- What is the force on a particle in the y-direction?
- 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!

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.