1. Tardigrade
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  4. The displacement of a damped harmonic oscillator is given by x(t)=e- 0 . 1 tcos(10 π t + varphi) , where t is the time in seconds. The time taken for the amplitude of the oscillator to drop to half of its initial value is close to

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Q. The displacement of a damped harmonic oscillator is given by $x\left(t\right)=e^{- 0 . 1 t}cos\left(10 \pi t + \varphi\right)$ , where $t$ is the time in seconds. The time taken for the amplitude of the oscillator to drop to half of its initial value is close to

NTA AbhyasNTA Abhyas 2022

Solution:

$\frac{A_{0}}{2}=A_{0}e^{- 0 . 1 t}\Rightarrow e^{0 . 1 t}=2\Rightarrow 0.1t=ln2$
Therefore,
$t=\frac{ln 2}{0 . 1}=10ln2=6.93\approx7s$