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. $10\text{sec.}$
• B. $50\text{sec.}$
• C. $100\text{sec.}$
• D. $20\text{sec.}$

Answer 1

The correct option is D $20\text{sec.}$

The amplitude drops to half its value for every $10$ oscillations.

$\therefore$Time of half the amplitude $=2\text{sec}$

Now, for damped oscillations,

$A=A_0{e}^{-kt}$

$\frac{A_0}{2}=A_0{e}^{-k\times 2}\Rightarrow 2=e^{2k}$

$\therefore k=\frac{\ln 2}{2}$

Also, $\frac{A_0}{1000}=A_0{e}^{-kt}$

$1000=e^{kt}$

$\ln 1000=kt$

$3\ln 10=\left(\frac{\ln 2}{2}\right)t$

$t=6\times \frac{\ln 10}{\ln 2}$

$\therefore t\approx 20\text{sec.}$

Hence, $\left(C\right)$ is the correct answer.