When an oscillator completes $100$ oscillations its amplitude reduced to $\frac{1}{3}$ of initial value. What will be its amplitude, when it completes $200$ oscillations?
AIPMT 2002, Medium
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In damped oscillation, amplitude goes on decaying exponentially, $a=a_{0} e^{-b t}$ where $b =$ damping coefficient.
Initially $, \frac{a_{0}}{3}=a_{0} e^{-b \times 100 T}, \mathrm{T}=$ time of one oscillation or $\frac{1}{3}=e^{-100 b T} \ldots(i)$
Finally, $a=a_{0} e^{-b \times 200 T}$
or $a=a_{0}\left[e^{-100 b T}\right]^{2}$
or $a=a_{0} \times\left[\frac{1}{3}\right]^{2} \quad[\text { From Eq. (i) }]$
or $a=a_{0} / 9$
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