Question

The amplitude of a damped oscillation becomes half in $1\text{minute}$. If the amplitude after $4\text{minute}$ will be $\frac{1}{y}$ times of the original, find the value of $y$.

• A. $4$
• B. $8$
• C. $16$
• D. $32$

Answer 1

The correct option is C $16$

We know that amplitude of damped oscillation varies exponentially and given as,
$A=A_o{e}^{-rt}....\left(1\right)$
where $r=\frac{b}{2m}$
As $A=\frac{A_o}{2}$ at $t=1\text{minute}$
$\Rightarrow \frac{A_o}{2}=A_o{e}^{-r}$
$\Rightarrow e^{-r}=\frac{1}{2}$
Or, $e^{+r}=(e^{-r}{)}^{-1}=2....(2)$
After $4\text{minute}$ the amplitude will be $\frac{A_o}{y}$, putting $t=4\text{minute}$ in Eq.$\left(1\right)$
$\frac{A_o}{y}=A_o{e}^{-4r}$
$y=e^{4r}=(e^r{)}^4$
Substituting from Eq.$\left(2\right)$,
$\Rightarrow y=(2{)}^4=16$