Question

In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value $a_0$ at the end of $100$ oscillations. When the oscillator completes $200$ oscillations, its amplitude must be

• A. $\frac{a_0}{2}$
• B. $\frac{a_0}{4}$
• C. $\frac{a_0}{6}$
• D. $\frac{a_0}{9}$

Answer 1

The correct option is D $\frac{a_0}{9}$

In damped oscillation, amplitude goes on decaying exponentially.
i.e $a=a_0{e}^{-\gamma t}$ where $\gamma$ = damping coefficient.
Given, $\frac{a_0}{3}=a_0{e}^{-\gamma \times 100T}$
where $T$ = time period of one oscillation
or $\frac{1}{3}=e^{-100\gamma T}......\left(i\right)$
After $200$ oscillations,
$a=a_0{e}^{-\gamma \times 200T}$
or $a=a_0[e^{-100\gamma T}{]}^2$
or $a=a_0\times {\left[\frac{1}{3}\right]}^2$ [From Eq. $i$]
or $a=\frac{a_0}{9}$