Damped oscillator relaxation time numerical

Convention used: for a damped oscillator with damping force −bv, amplitude a = a0 e^(−bt/2m) and energy E = E0 e^(−bt/m). The relaxation time is τ = m/b, the time in which the energy falls to 1/e of its initial value. Hence a = a0 e^(−t/2τ) and E = E0 e^(−t/τ).

Given τ = 50 s.

(i) Amplitude falls to 1/e³

a/a0 = e^(−t/2τ) = e^(−3)
t/2τ = 3
t = 6τ = 6 × 50 = 300 s

(ii) Energy falls to 1/e⁴

E/E0 = e^(−t/τ) = e^(−4)
t = 4τ = 4 × 50 = 200 s

Answer: the amplitude falls to 1/e³ in 300 s (5 min); the energy falls to 1/e⁴ in 200 s.

Check: at t = 300 s the energy is E0 e^(−6), consistent with E ∝ a²: (e^(−3))² = e^(−6). At t = 200 s the amplitude is a0 e^(−2).