Question

When an oscillator completes $100$ oscillations its amplitude reduces to $\frac{1}{3}$ of its initial value. What will be its amplitude when it completes $200$ oscillations?

• A. $\frac{1}{8}$
• B. $\frac{2}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{9}$

Answer 1

Answer: (D)

$\frac{1}{9}$

Its is a damped oscillation, where amplitude of oscillation at time $t$ is given by $A=a_0{e}^{-\gamma t}$
where $a_0=$ initial amplitude of oscillation
$\gamma =$ damping constant
As per question, $\frac{a_0}{3}=a_0{e}^{-\gamma 100/v}...\left(i\right)$
(where $v$ is the frequency of oscillation)
and $A=a_0{e}^{-\gamma 200/v}...\left(ii\right)$
From $\left(i\right)$; $\frac{a_0}{3}=a_0{e}^{-\gamma \times 100/v}...\left(iii\right)$
Dividing equation $\left(ii\right)$ by $\left(iii\right)$, we have
$\frac{A}{a_0\left(1/3\right)}=\frac{e^{-\gamma \times 200/v}}{e^{-\gamma \times 100/v}}=e^{-\gamma \times 100/v}=\frac{1}{3}$
or $A=a_0\times \frac{1}{3}\times \frac{1}{3}=\frac{1}{9}{a}_0$