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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 1/1000 of the original amplitude is close to :

20 s

100 s

10 s

50 s

Solution:

Correct option is B. 20 s
A = A_0 e^-γ t
A = A_0/2 after 10 oscillations
∵ After 2 seconds
A_0/2 = A_0 e^-γ (2)
2 = e^2 γ
ln 2 = 2 γ
γ = ln 2/2
∵ A = A_0 e^-γ t
ln A_0/A = γ t
ln 1000 = ln 2/2 t
2 (3 ln 10/ln2) = t
6 ln 10/ln 2 = t
t = 19.931 sec ≈ 20 sec