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
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?  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:

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 component notation. Don't forget the units!