Mechanics: Chapter 14 – Potential Energy Diagrams and Equilibrium

Since the Work done,

  • W = \int F \cdot dr

And, for a conservative force,

  • W = - \Delta U

One can deduce that, for a conservative force,

  • \Delta U = - \int F \cdot dr

In other words, the change in the potential energy of an object under the influence of a conservative force as it moves from position r_{1} to r_{2} is

  • \Delta U = - \int_{r_{1}}^{r_{2}} F \cdot dr = \int_{r_{2}}^{r_{1}} F \cdot dr

Similarly, because of the Fundamental Theorem of Calculus, we can deduce that,

  • F = -\frac{dU}{dr}

potential energy diagram is a diagram with the potential energy on one axis and the position on the other. The local maxima on a potential energy diagram are said to be points of unstable equilibrium, while the local minima are points of stable equilibrium.

Here is a graph of what is called the Lennard-Jones potential for the interaction of a pair of atoms:

Lennard-Jones Plot

Now the local minimum on this graph (which also happens to be the absolute minimum) is the point where the system reaches stable equilibrium. If given an equation for the potential energy U, we can use elementary calculus to find the minimum point which would be the point at which stable equilibrium is reached.

diagram

In the above potential energy diagram there are two points of unstable equilibrium (two local maxima) and three points of stable equilibrium (three local minima).

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