A damped harmonic oscillator has 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..............sec.
Approximate answer nearest to integer.

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answer is 20.

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Detailed Solution

$\begin{array}{l} T = \frac{1}{f} = \frac{1}{5} \sec t = 10 T = 2 \sec \\ a = a_{0} e^{- kt} \Rightarrow \frac{a_{0}}{2} = a_{0} e^{- k \times 2} \Rightarrow e^{2 k} = 2 \Rightarrow 2 k = \ln 2 . . . (1) \\ \frac{a_{0}}{1000} = a_{0} e^{- kt} \Rightarrow e^{kt} = 1000 \\ \Rightarrow kt = \ln 1000 \\ From (1) \Rightarrow t = \frac{\ln (1000)}{\frac{\ln (2)}{2}} ≃ 20 s \end{array}$