Question

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 $\left(A_{0}-\right.$ initial amplitude)

• A. $\frac{1}{4} A_{0}$
• B. $\frac{1}{2} A_{0}$
• C. $A_{0}$
• D. $\frac{\sqrt{3} A_{0}}{2}$

Answer 1

Answer: (A)

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