Question

When an oscillator completes $100$ oscillations its amplitude reduced to $ \frac{1}{3} $ of 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)

Its is a damped oscillation, where amplitude of oscillation at time $t$ is given by $A = a _{0} e ^{-\gamma r }$
where $a _{0}=$ initial amplitude of oscillation
$\gamma=$ damping constant
As per question, $\frac{a_{0}}{3}=a_{0} e^{-\gamma 100 / v} $...(i)
(where $v$ is the frequency of oscillation)
and $A = a _{0} e ^{-\gamma 200 / v }$....(ii)
From (i); $ \frac{a_{0}}{3}=a_{0} e ^{-\gamma \times 100 / v}$
Dividing equation (ii) by (iii), we have
$\frac{ A }{ a _{0}(1 / 3)}=\frac{ e ^{-\gamma \times 200 / v }}{ e ^{-\gamma \times 100 / v }}= e ^{-\gamma \times 100 / v }=\frac{1}{3}$
or $A=a_{0} \times \frac{1}{3} \times \frac{1}{3}=\frac{1}{9} a_{0}$