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 .... \(s\)

\(A=A_{0} e^{-\gamma t}\)
\(A=\frac{A_{0}}{2}\) after \(10\) oscillations
\(\because\) After 2 seconds
\(\frac{A_{0}}{2}=A_{0} e^{-\gamma(2)} \quad ; \quad 2=e^{2 \gamma}\)
\( \ell n 2=2 \gamma \quad ; \quad \gamma=\frac{\ell n 2}{2}\)
\(\because A=A_{0} e^{-\gamma t}\)
\(\ell n \frac{\mathrm{A}_{0}}{\mathrm{A}}=\gamma \mathrm{t} ; \quad \ell \mathrm{n} 1000 \frac{\ell n 2}{2} \mathrm{t}\)
\(2\left(\frac{3 \ell n 10}{\ell n 2}\right)=t ; \quad \frac{6 \ell n 10}{\ell n 2}=t\)
\(t=19.931 \mathrm{sec}\)
\(t \approx 20 \mathrm{sec}\)