Damped Oscillations

Damped oscillation: amplitude decreases over time due to friction or air drag.

Equation of motion: m·ẍ + b·ẋ + kx = 0, where b is the damping constant.

Three regimes: UNDERDAMPED (oscillates with decaying amplitude), CRITICALLY DAMPED (returns to equilibrium fastest, no oscillation), OVERDAMPED (returns slowly, no oscillation).

Underdamped solution: x(t) = A₀·e^(−bt/2m)·cos(ω'·t + φ), where ω' = √(ω₀² − γ²), γ = b/2m, ω₀ = √(k/m).

Amplitude decays exponentially: A(t) = A₀·e^(−bt/2m). Half-amplitude time: t₁/₂ = (2m·ln 2)/b.

Quality factor Q = ω₀·m/b. High Q = light damping, oscillates many times. Low Q = heavy damping.

Examples: pendulum in air, RLC circuit with R, shock absorbers (critically damped by design).

Energy decays: E(t) ∝ e^(−bt/m) (twice as fast as amplitude).