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

Answer 1

The correct option is B $20s$

$A=A_0{e}^{-\gamma t}$
$A=\frac{A_0}{2}$ after 10 oscillations
$\because$ After 2 seconds
$\frac{A_0}{2}=A_0{e}^{-\gamma \left(2\right)}$
$2=e^{2\gamma }$
$\ell n2=2\gamma$
$\gamma =\frac{\ell n2}{2}$
$\because A=A_0{e}^{-\gamma t}$
$\ell n\frac{A_0}{A}=\gamma t$
$\ell n1000=\frac{\ell n2}{2}t$
$2\left(\frac{3\ell n10}{\ell n2}\right)=t$
$\frac{6\ell n10}{\ell n2}=1$
$t=19.931sec$
$t\approx 20sec$