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  4. The displacement of a damped harmonic oscillator is given by x(t ) = e-01.1t cos (10 π t + φ ) . Here t is in seconds. The time taken for its amplitude of vibration 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 $x(t ) = e^{-01.1t} \cos (10 \pi t + \phi )$ . Here $t$ is in seconds.
The time taken for its amplitude of vibration to drop to half of its initial value is close to :

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Solution:

$A = A_{0} e^{-0.1t} = \frac{A_{0}}{2} $
$ ln 2 = 0.1 t$
$ t = 10 ln2 = 6.93 \approx 7 \sec$