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 F is conservative, or path-independent, iff any of the following conditions apply (and any of the conditions implies the others):

  • The line integral over a closed path C of F dot ds=0 along any closed path C (the force does no work over any closed path)
  • curl F=0
  • The force has an associated potential function V such that F=-grad V

For any conservative force, the connection between the force and its potential energy function is F=-grad U 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 F=-(dU/dx)(i-hat). This is actually fairly obvious when you think about it: multiply both sides by dot i-hat dx to get F dot dx=-dU, then integrate: Int F dot dx=-Int dU, or W=-U. Since we’ve defined W=Delta E, that makes perfect sense.

Potential Energy Diagram

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 y=U(x) 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 B and E 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 C 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 A and D 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
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 K_i, what points can the object get to? The object can get to any point with U is less than or equal to K_i; the graphical interpretation of this is that the object can get to any point below the line U=K_i. A common follow-up takes the form “at a point with U=U_1, what is the kinetic energy of the particle at a particular point?” Its kinetic energy is K_1=K_i-U_1; given the mass, of course, you can then find the particle’s speed.

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