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 oscillations, amplitude goes on decaying exponentially as,
$a=a_0{e}^{-bt}....\left(i\right)$

where $b=$ damping coefficient, and $a$ is the amplitude at any time $t$

The amplitude reduces to $\frac{1}{3}\text{rd}$ after $100$ oscillations i.e after $100$ complete cycles.

$\Rightarrow t=100\times T$

where $T=$time to complete one cycle or oscillation i.e. time period.

Substituting in Eq.$\left(i\right)$

$\frac{a_0}{3}=a_0{e}^{-b\times 100T}$

$\frac{1}{3}=e^{-100bT}...\left(ii\right)$

when the oscillator completes $200$ oscillations,

$a=a_0{e}^{-b\times 200T}$
$a=a_0{\left[e^{-100bT}\right]}^2$ $a=a_0\times {\left[\frac{1}{3}\right]}^2$

[substituting from eq. $\left(ii\right)$]
$\Rightarrow a=\frac{a_0}{9}$

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