I was solving a practice Physics GRE and there was a question about springs connected in series and parallel. I was too lazy to derive the way the spring constants add in each case. But I knew how capacitances and resistances add when they are connected in series/parallel. So I reasoned that spring constants should behave as capcitances because both springs and capacitors store energy.
This line reasoning did give me the correct answer for how spring constants add, but I was just curious if this analogy makes sense, and if it does, how far one can take it. That is, knowing just that two things store energy, what all can you say will be similar for the two things.
Best Answer
Electrical analogies of mechanical elements such as springs, masses, and dash pots provide the answer. The "deep" connection is simply that the differential equations have the same form.
In electric circuit theory, the across variable is voltage while the through variable is current.
The analogous quantities in mechanics are force and velocity. Note that in both cases, the product of the across and through variables has the unit of power.
(An aside, sometimes it is convenient to use force and velocity as the across and through variables respectively while other times, it is more convenient to switch those roles.)
Now, assuming velocity is the through variable, velocity and electric current are analogous. Thus, displacement and electric charge are analogous.
For a spring, we have $f = kd \rightarrow d = \frac{1}{k}f$ while for a capacitor we have $Q = CV$.
For a mass, we have $f = ma = m\dot v $ while for an inductor we have $V = L \dot I$
Finally, for a dashpot, we have $f = Bv$ while for a resistor we have $V = RI$.
So, we have
$\frac{1}{k} \rightarrow C$
$m \rightarrow L$
$B \rightarrow R$
For a nice summary with examples, see this.
UPDATE: In another answer, RubenV questions the answer given above. His reasoning requires an update.
In fact, it is relevant and it does tell you everything about components in series or in parallel. Let's review:
When two circuit elements are in parallel, the voltage across each is identical.
When two circuit elements are in series, the current through each is identical.
This is fundamental and must be kept in mind when moving to the mechanical analogy.
In the mechanical analogy where a spring is the mechanical analog of a capacitor:
force is the analog of voltage
velocity is the analog of current.
Keeping this in mind, consider two springs connected in mechanical parallel and note that the velocity (rate of change of displacement) for each spring is identical.
But recall, in this analogy, velocity is the analog of current. Thus, the equivalent electrical analogy is two capacitors in series (identical current).
In series, capacitance combines as so:
$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$
With the spring analogy, $C \rightarrow \frac{1}{k}$ , this becomes:
$k_{eq} = k_1 + k_2$
The key point to take away from this is that mechanical parallel is, in this analogy, circuit series since, in mechanical parallel, the velocity (current) is the same, not the force (voltage).
For example, consider dash pots (resistors). Two dash pots in "parallel" combine like two resistors in series, i.e., the resistance to motion of two dash pots in "parallel" is greater then each individually.
Now, if the roles of the analogous variables are swapped, if force is like current and velocity is like voltage, then mechanical parallel is like circuit parallel. However, in this analogy, mass is like capacitance.