Equilibrium and Potential Energy Diagrams

Potential energy is associated only with conservative forces. Recall the definition of a conservative force from the discussion of friction (not a conservative force): A force is conservative, or path-independent, iff any of the following conditions apply (and any of the conditions implies the others):

• along any closed path (the force does no work over any closed path)
• 
• The force has an associated potential function such that

For any conservative force, the connection between the force and its potential energy function is in three dimensions. Because the gradient and partial derivative are unfamiliar to Calculus BC students, in Physics C we will work only with one-dimensional motion, so we simply have . This is actually fairly obvious when you think about it: multiply both sides by to get , then integrate: , or . Since we’ve defined , that makes perfect sense.

One problem that seldom comes up on the AP free-response section but puts in frequent appearances on the multiple-choice section has to do with potential energy diagrams. These are plots of and often look something like the graph pictured to the right.

There are two basic things to know about potential energy diagrams: equilibrium points and accessibility. A local maximum is said to be a point of unstable equilibrium, because an object placed at such a point will not return to its equilibrium position after being displaced slightly. Points and are examples of unstable equilibrium points. A local minimum is a point of stable equilibrium, since an object placed at a local minimum will return to its equilibrium position after a slight displacement. Point is an example of such a point. A point where the second derivative is zero is a point of neutral equilibrium—a saddle point. An object in neutral equilibrium, if displaced slightly, will stay in its displaced position. Points and are examples.

Although it’s not a rigorous treatment, when presented with a potential energy diagram, you can think about points of equilibrium in terms of what would happen to a small marble placed at a given point. If it’s placed at a local maximum and displaced slightly, it will roll down—unstable equilibrium. Placed at a local minimum, upon displacement it will return to the local minimum—stable equilibrium. Finally, a marble on a saddle point will stay wherever you displace it to—neutral equilibrium.

The cones below provide another example of the three types of equilibrium.

A cone in stable equilibrium	A cone in unstable equilibrium	A cone in neutral equilibrium

The other issue with potential energy diagrams is that of accessibility. If an object has a particular initial kinetic energy , what points can the object get to? The object can get to any point with ; the graphical interpretation of this is that the object can get to any point below the line . A common follow-up takes the form “at a point with , what is the kinetic energy of the particle at a particular point?” Its kinetic energy is ; given the mass, of course, you can then find the particle’s speed.