The amplitude of a damped oscillator becomes one third in $2\, sec$. If its amplitude after $6\, sec$ is $1/n$ times the original amplitude then the value of $n$ is
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since $A_{t}=A_{0} e^{\frac{-b t}{2 m}}$
When $t=2$ sec, $\quad \frac{A}{3}=A e^{\frac{-2 b}{2 m}}$
$\frac{1}{3}=e^{-b / m}$
When $t=6$ sec
$\frac{A_{0}}{n}=A_{0} e^{\frac{-6 b}{2 m}}$
$\frac{1}{n}=\left(e^{\frac{-b}{m}}\right)^{3}$
$\frac{1}{n}=\left(\frac{1}{3}\right)^{3}$
$n=3^{3}$
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