The displacement of a damped harmonic oscillator is given by $x\left( t \right) = {e^{ - 0.1\,t}}\,\cos \left( {10\pi t + \varphi } \right)$ The time taken for its amplitude of vibration to drop to half of its initial value is close to .... $s$
JEE MAIN 2019, Medium
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$A=A_{0} e^{-0.1 t}=\frac{A_{0}}{2}$
$\ln 2=0.1 \mathrm{t}$
$t=10\, \mathrm{ln} 2=6.93 \approx 7 \mathrm{sec}$
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