RESONANCE
Push a damped spring: displacement peaks BELOW ω₀, velocity peaks AT ω₀ — the exact curve acres draws for RLC current.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A mass on a damped spring, pushed rhythmically by an outside force. Push at the right beat and it swings enormously; push wrong and it barely stirs. The question everyone thinks they can answer: what IS the right beat? It turns out to be not quite ω₀.
Lag
In steady state the mass settles into a sinusoid at the driving frequency, but lagging the push by a phase φ. That lag runs from zero (slow pushing, the mass keeps up) through exactly 90° at ω₀, to 180° (fast pushing, the mass fights the drive).
Two peaks
And there are TWO resonances hiding here. The displacement swings widest at ω_r = √(ω₀²−2γ²), a little BELOW ω₀. But the velocity — and the power the drive delivers — peak exactly AT ω₀. Most demonstrations conflate them; they are genuinely different frequencies.
Sweep
Sweep the drive and the operating point climbs the amber curve to its peak and back down. The sharpness Q sets how narrow that peak is: the half-power band has width ω₀/Q. High Q gives a razor peak; low Q a broad, gentle hump.
Audit
Audited: displacement peaks at √(ω₀²−2γ²), velocity at ω₀, both exact; bandwidth ω₀/Q. The whole curve is the SAME one the AC-resonance module draws for current — a mass-and-spring and an RLC circuit obey one equation, confirmed executably to twelve digits, and cross-checked by integrating the equation of motion outright.