Harmonic mean has come up in a couple posts this week (with numbers and functions). This post will show how harmonic means come up in physical problems involving springs and resistors.

Suppose we have two springs in series with stiffness k_{1} and k_{2}:

Then the combined stiffness k of the two springs satisfies 1/k = 1/k_{1} + 1/k_{2}. Think about what this says in the extremes. If one of the springs were infinitely stiff, say k_{2} = ∞. Then k = k_{1}. It would be as if the second spring were not there. Being infinitely stiff, we could think of it as an extension of the block it is attached to. Now think of one of the springs having no stiffness at all, say k_{1} = 0. Then k = 0. One mushy spring makes the combination mushy.

Next think of two springs in parallel:

Now the combined stiffness of the two springs is k = k_{1} + k_{2}. Again think of the two extremes. If one spring is infinitely stiff, say k_{1} = ∞, then k = ∞ and the combination is infinitely stiff. And if one spring has no stiffness, say k_{2} = 0, then k = k_{1}. We could imagine the spring with no stiffness isn’t there.

The stiffness of springs in series adds harmonically. The stiffness of the combination is half the **harmonic mean** of the two individual stiffnesses.

Electrical resistors combine in a way that is the opposite of mechanical springs. Resistors in **parallel** combine like springs in **series**, and *vice versa*.

If two resistors have resistance r_{1} and r_{2}, the combined resistance r of the two resistors in parallel satisfies 1/r = 1/r_{1} + 1/r_{2}. If one of the resistors has infinite resistance, say r_{2} = ∞, then r = r_{1}. It would be as if the second resistor were not there. All electrons would flow through the first resistor.

If the two resistors were in series, then r = r_{1} + r_{2}. If one resistor has infinite resistance, so does the combination. Electrons cannot flow through the combination if they cannot flow through one of the resistors. And if one resistor has zero resistance, say r_{2} = 0, then r = r_{1}. Since the second resistor offers no resistance to the flow of electrons, it may as well not be there.

These physical problems illustrate why zeros as handled specially in the definition of means.

Image credit: Wikipedia

Wonderful explanation, thanks! I always wondered about the practical example of harmonic means…