The Electrical-Mechanical Analogy — Question 6

Question 6

Parallel springs correspond to series capacitors in the force-voltage analogy

Two springs exert restoring force on the same mass displacement, so their stiffnesses add. Two capacitors carry the same series charge q. Ignore resistance and any capacitor offset charge.

1. Show how the capacitor voltages reproduce additive spring forces.

2. For two identical capacitors C in series with L, compare frequency with a single C and with two identical parallel capacitors.

Question 6 - Solution

Parallel springs correspond to series capacitors in the force-voltage analogy

(a) Method and calculation

$$F=-(k_1+k_2)x,\quad v=q/C_1+q/C_2.$$$$\boxed{1/C_s=1/C_1+1/C_2\ \leftrightarrow\ k_{\mathrm{eq}}=k_1+k_2}.$$

(b) Interpretation and further calculation

$$C_s=C/2,\quad C_p=2C.$$$$\boxed{\omega_s=\sqrt2\,\omega_1,\quad\omega_p=\omega_1/\sqrt2}.$$

Physical check

The analogy follows shared variables; visually parallel mechanical elements need not correspond to parallel electrical elements.