The Electrical-Mechanical Analogy — Question 1

Question 1

The oscillator analogy pairs capacitance with compliance

A mass on a spring obeys $m\ddot x+kx=0$. An ideal LC circuit obeys $L\ddot q+q/C=0$.

1. Identify the analogues of position, velocity, mass, and spring stiffness by matching equations and energies.

2. Use the correspondence to derive the electrical angular frequency.

Question 1 - Solution

The oscillator analogy pairs capacitance with compliance

(a) Method and calculation

$$x\leftrightarrow q,\quad\dot x\leftrightarrow i,\quad m\leftrightarrow L,\quad k\leftrightarrow1/C.$$$$\boxed{\tfrac12m\dot x^2+\tfrac12kx^2\ \leftrightarrow\ \tfrac12Li^2+\frac{q^2}{2C}}.$$

(b) Interpretation and further calculation

$$\omega_m^2=k/m\Rightarrow\omega_e^2=(1/C)/L.$$$$\boxed{\omega_e=1/\sqrt{LC}}.$$

Physical check

Larger capacitance corresponds to a softer spring, not a stiffer one.