the RLC circuit
/ R-L-C /
An RLC circuit is a loop wired from three parts: a resistor R (which wastes energy as heat), an inductor L (a coil that resists changes in current, storing energy in a magnetic field), and a capacitor C (two plates that store energy in an electric field). Connect them in a ring, give the capacitor an initial charge or attach a battery, and the current sloshes back and forth around the loop — the electrical twin of a weight bouncing on a spring.
Kirchhoff's voltage law says the voltage drops around the loop sum to the applied voltage. Writing each part's voltage in terms of the charge q(t) on the capacitor gives L q'' + R q' + (1/C) q = E(t). The inductor's voltage is L q'' (it opposes acceleration of charge), the resistor's is R q' (proportional to current q'), the capacitor's is q/C (it pushes back the more charge piles up), and E(t) is the source. This is the exact same second-order equation as the spring-mass-damper, term for term.
Because the mathematics is identical, every result about mechanical vibration transfers instantly to circuits and back. The circuit has a natural ringing frequency, it can be underdamped (current oscillates as it dies), and it shows resonance — the principle behind tuning a radio to one station. Studying one teaches you both, which is the deep payoff of the analogy.
With L = 1 H, R = 2 ohm, C = 0.25 F and no source, the charge obeys q'' + 2 q' + 4 q = 0; the current rings and decays, exactly like a damped mass on a spring.
Inductor (q''), resistor (2 q'), and capacitor (4 q) play the roles of mass, damper, and spring.
Many circuits are written in terms of current i = q' instead of charge; differentiating once gives L i'' + R i' + (1/C) i = E'(t), the same form with the derivative of the source as forcing. Both descriptions are correct — just pick one and stay consistent.