Question

In damped oscillations, damping force is directly proportional to the speed of the oscillator. If amplitude becomes half of its initial value in $1\text{s}$, then after $2\text{s}$, amplitude will be:
[$A_0$ is 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

The correct option is A $\frac{1}{4}{A}_0$

We know that,
Amplitude of damped oscillation $\left(A\right)=A_0{e}^{-\gamma t}$
As $A=\frac{A_0}{2}$ at $t=1\text{s}$
So, $\frac{A_0}{2}=A_0{e}^{-\gamma }$
or $e^r=2...\left(i\right)$
After two seconds,
$A=A_0{e}^{-2\gamma }$
$A=A_0(e^{\gamma }{)}^{-2}$
Using (1), we get
$A=\frac{A_0}{4}$