# Forces and potential energy

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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
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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?  If one moves
downwards on this map?

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

```

```
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?  At which of these places
is there a stable equilibrium?

```

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 component notation. Don't forget the units!