A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to \frac{1}{1000} of the original amplitude is close to :

Engineering

Physics

Damped Oscillation

Question

A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to :

20 s

100 s

50 s

10 s

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A = A₀e^–γt

$A = \frac{A_{0}}{2} after 10 oscillations$

$∵$ After 2 seconds]

$\frac{A_{0}}{2} = A_{0} e^{- γ (2)}; 2 = e^{2 γ}$

ln2 = 2γ ; $γ = \frac{ℓn 2}{2}$

$∵$ A = A₀e^–γt

$ℓn \frac{A_{0}}{A} = γt; ℓn 1000 \frac{ℓn}{2} t$

$2 (\frac{3 ℓn 10}{ℓn 2}) = t; \frac{6 ℓn 10}{ℓn 2} = t$

t = 19.931 sec

t ≈ 20 sec