Damped Harmonic Oscillator
For mx'' + cx' + kx = 0 the discriminant c^2 - 4mk splits behavior into overdamped (two real roots, slow non-oscillatory decay), critically damped (repeated root, fastest return to equilibrium without oscillation), and underdamped (complex roots, decaying oscillation at quasi-frequency sqrt(k/m - c^2/4m^2)). Critical damping c = 2*sqrt(mk) is the design target for shock absorbers and meter needles.
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For mx'' + cx' + kx = 0 the discriminant c^2 - 4mk splits behavior into overdamped (two real roots, slow non-oscillatory decay), critically damped (repeated root, fastest return to equilibrium without oscillation), and underdamped (complex roots, decaying oscillation at quasi-frequency sqrt(k/m - c^2/4m^2)). Critical damping c = 2*sqrt(mk) is the design target for shock absorbers and meter needles.
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