When an oscillator completes 100 oscillations | vedclass

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(B) In damped oscillations,the amplitude decays exponentially with time,given by $a = a_0 e^{-bt}$,where $b$ is the damping constant and $t$ is the ti

Vedclass · Accepted Answer

$\frac{1}{9}$

Vedclass · Answer

(B) In damped oscillations,the amplitude decays exponentially with time,given by $a = a_0 e^{-bt}$,where $b$ is the damping constant and $t$ is the time.Let $T$ be the time period of one oscillation. After $n$ oscillations,the time elapsed is $t = nT$.Initially,after $100$ oscillations,the amplitude is $a = \frac{a_0}{3}$.So,$\frac{a_0}{3} = a_0 e^{-b(100T)}$,which implies $e^{-100bT} = \frac{1}{3}$.After $200$ oscillations,the time elapsed is $t = 200T$.The amplitude $a'$ will be $a' = a_0 e^{-b(200T)}$.This can be written as $a' = a_0 (e^{-100bT})^2$.Substituting the value from the first step: $a' = a_0 (\frac{1}{3})^2 = \frac{a_0}{9}$.Thus,the amplitude becomes $\frac{1}{9}$ of the initial value.

When an oscillator completes $100$ oscillations,its amplitude is reduced to $\frac{1}{3}$ of its initial value. What will be its amplitude after it completes $200$ oscillations?

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