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 .... $s$
JEE MAIN 2019, Diffcult
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$A=A_{0} e^{-\gamma t}$
$A=\frac{A_{0}}{2}$ 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}$
$\because A=A_{0} e^{-\gamma t}$
$\ell n \frac{\mathrm{A}_{0}}{\mathrm{A}}=\gamma \mathrm{t} ; \quad \ell \mathrm{n} 1000 \frac{\ell n 2}{2} \mathrm{t}$
$2\left(\frac{3 \ell n 10}{\ell n 2}\right)=t ; \quad \frac{6 \ell n 10}{\ell n 2}=t$
$t=19.931 \mathrm{sec}$
$t \approx 20 \mathrm{sec}$
$A=\frac{A_{0}}{2}$ 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}$
$\because A=A_{0} e^{-\gamma t}$
$\ell n \frac{\mathrm{A}_{0}}{\mathrm{A}}=\gamma \mathrm{t} ; \quad \ell \mathrm{n} 1000 \frac{\ell n 2}{2} \mathrm{t}$
$2\left(\frac{3 \ell n 10}{\ell n 2}\right)=t ; \quad \frac{6 \ell n 10}{\ell n 2}=t$
$t=19.931 \mathrm{sec}$
$t \approx 20 \mathrm{sec}$
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