When an oscillator completes $100$oscillations, its amplitude reduces to $\dfrac{1}{3}$of its initial value. What will be its amplitude when it completes $200$oscillations?
A. $\dfrac{1}{8}$
B. $\dfrac{2}{3}$
C. $\dfrac{1}{6}$
D. $\dfrac{1}{9}$

Hint:In this question, we need to determine the amplitude of the oscillator when it completes 200 oscillations such that when it completes 100 oscillations, its amplitude reduces to $\dfrac{1}{3}$ of its initial value. For this, we will use the fact that for damped oscillation, the amplitude after time `t’ is given by $A = {A_0}{e^{ - bt}}$.

Complete step by step answer:
This is a case of damped oscillation. In damped oscillation, the amplitude of the particle decreases exponentially with respect to time. If the particle at the time $t = 0$has its amplitude ${A_0}$and after time t its amplitude becomes A,
Then, $A = {A_0}{e^{ - bt}}..........\left( i \right)$
Where b is damping constant from the given question,
Case (i):When an oscillator completes $100$oscillations, i.e., at the time $t = 100T$the amplitude reduces to $\dfrac{1}{3}$of its initial value i.e. $A = \dfrac{{{A_0}}}{3},$
So, from the equation $A = {A_0}\,\,\,\,{e^{ - bt}}$ , we have
\[\Rightarrow \dfrac{{{A_0}}}{3} = {A_O}\,\,\,{e^{ - b\left( {100T} \right)}}\]
$\Rightarrow {e^{ - 100T}} = \dfrac{1}{3}..........\left( {ii} \right)$
Case (ii): When the particle completes $200$oscillation, i.e., $t = 200T$then, the amplitude$ = $?
Let the new amplitude be A then from equation (i),
\[
\Rightarrow A = {A_0}\,\,{e^{ - bt}} \\
\Rightarrow A = {A_0}\,\,{e^{ - b\left( {200T} \right)}} \\
\Rightarrow A = {A_0}{\left[ {{e^{ - b\left( {100T} \right)}}} \right]^2}...........\left( {iii} \right) \\
 \]
Now, from equation (ii) and (iii), we have
$
\Rightarrow A = {A_0}{\left[ {\dfrac{1}{3}} \right]^2} = {A_0} \times \dfrac{1}{9} \\
\Rightarrow A = \dfrac{{{A_0}}}{9} \\
$
So, the amplitude will become $\dfrac{1}{9}$of its initial value.
Hence, the correct option is (D).

Note: The force applied into the particle during damping oscillations is given by $F = - kx - bv$, and the differential equation for it is given by $\dfrac{{{d^2}x}}{{d{t^2}}} + \dfrac{6}{m}\,\,\dfrac{{dx}}{{dt}} + \dfrac{k}{m}n = 0$. The damped oscillation means an oscillation that fades away with time. For example, a swinging pendulum, weight on spring in a resistor – inductor-capacitor (RLC) circuit.