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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 $\frac{1}{1000}$ of the original amplitude is close to _______ $(l n 10 = 2.3)$

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50 s

100 s

10 s

20 s

answer is D.

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Detailed Solution

$A = A_{0} e^{- r t} \frac{A_{0}}{2} = A_{0} e^{- r (10 T)} 10 r T = l n 2 \dots \dots \dots . . (1) \frac{A_{0}}{1000} = A_{0} e^{- r t}, r t = l n 1000 \frac{r t}{10 r T} = \frac{2.3 \times 3}{0.693} \approx 10 t = 100 T = 100 \times 0.2 = 20 s$

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