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…

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

It’s not so much that they behave as opposites as it is that you’re comparing to the reciprocal of the analogous quantity, a quirk of how electrical quantities are most commonly described. It works better if you compare spring stiffness to conductivity instead of resistance, force to current through the resistor, and extension/compression of the spring to voltage across the resistor. In this analogy, stiffness describes how effectively the spring transmits force, conductivity describes how effectively the resistor transmits voltage. Resistance would correspond to spring compliance (or “mushiness”): it takes less force to produce a given displacement of a highly compliant spring, and less current to produce a given voltage across a high resistance resistor.

You could also use inductors or capacitors, which are a bit more like springs in that they actually store energy. There is a non-standard unit for reciprocal capacitance (“elastance”, measured with the non-standard unit “daraf”). I’m not aware of a reciprocal quantity for inductance ever being used.

force/stiffness = displacement

current/conductivity = voltage

charge/darafs = voltage