When a oscillator completes 100 oscillations its amplitude reduces to $\frac{1}{3}$ of the initial value.…
When a oscillator completes 100 oscillations its amplitude reduces to $\frac{1}{3}$ of the initial value. What will be its amplitude when it completes 200 oscillations?
$\frac{1}{8}$
$\frac{2}{3}$
$\frac{1}{6}$
$\frac{1}{9}$
Solution
Let $a_0$ be the initial amplitude and $b$ be the damping constant
The used formula is $a=a_0 e^{-b t}$
In Ist case : $t=100 T$ and $a=\frac{a_0}{3}$
$\therefore \quad \frac{a_0}{3}=a_0 e^{-b(100 T)}$
$\begin{aligned}
& \Rightarrow \quad e^{-b \times 100 T}=\frac{1}{3} \\
& \text { In 2nd case : } t=200 T \\
& a=a_0 e^{-200 b T} \\
& \Rightarrow \quad a=a_0\left(e^{-100 b T}\right)^2 \\
& \Rightarrow \quad a=a_0\left(\frac{1}{3}\right)^2 \\
& \Rightarrow \quad a=\frac{a_0}{9} \\
&
\end{aligned}$