Question

In damped oscillations, if the amplitude becomes half of its maximum value in $1\text{s}$, then after $2\text{s}$ amplitude will be
($A_o$ is initial amplitude)

• A. $\frac{A_o}{4}$
• B. $\frac{A_o}{2}$
• C. $A_o$
• D. $\frac{\sqrt{3}{A}_o}{2}$

Answer 1

The correct option is A $\frac{A_o}{4}$

We know that amplitude of damped oscillation decreases exponentially and given as,
$A=A_o{e}^{-rt}....\left(1\right)$
where $r=\frac{b}{2m}$
$A=\frac{A_o}{2}$ at $t=1$ minute
Substituting in Eq.$\left(1\right)$,
$\frac{A_o}{2}=A_o{e}^{-r}$
$e^{-r}=\frac{1}{2}....\left(2\right)$
Putting $t=2\text{minute}$ in Eq.$\left(1\right)$,
$A=A_o{e}^{-2r}$
Using Eq.$\left(2\right)$ we get
$A=A_o\times (e^{-r}{)}^2$
$\Rightarrow A=A_o\times \frac{1}{4}=\frac{A_o}{4}$