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 $\frac{1}{1000}$ of the original amplitude is close to :

• A. 100 s
• B. 20 s
• C. 10 s
• D. 50 s

Answer 1

Answer: (B)

$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\left(2\right)} $
$ 2 = e^{2\gamma} $
$ \ell n^{2} = 2\gamma $
$ \gamma = \frac{\ell n^{2}}{2} $
$\because A = A_{0}e^{-\gamma t} $
$\ell n \frac{A_{0}}{A} = \gamma t $
$\ell n1000 = \frac{\ell n^{2}}{2} t $
$ 2\left(\frac{3\ell n10}{\ell n2}\right) = t $
$ \frac{6\ell n10}{\ell n2} = t $
t = 19.931 sec
t $\approx$ 20 sec