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In damped oscillations, damping force is directly proportional to speed of oscillator. If amplitude becomes half of its maximum value in 1 s, then after 2 s amplitude will be (A_o-initial amplitude)

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$\frac{1}{4} A_{0}$

$\frac{1}{2} A_{0}$

$\frac{\sqrt{3} A_{0}}{2}$

$A_{0}$

answer is A.

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Detailed Solution

$A = A_{0} e^{- bt}$
Amplitude becomes half hence $\frac{A_{0}}{2} A_{0} e^{- bt} [t = 1]$
$∴ e^{- b} = \frac{1}{2} ∴$ In two seconds
$A = A_{0} {(\frac{1}{2})}^{2}; A = \frac{A_{0}}{4}$

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