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