Question

The amplitude of a damped oscillator becomes half in one minute. The amplitude after $4$ minute will be $\frac{1}{Y}$ times the original, where $Y$ is _____.

Answer 1

Amplitude of damped oscillator
$ A=A_{0} \exp (-\lambda t) $
For $ t=1 \min , $
$\frac{A_{0}}{2}=A_{0} \exp (-\lambda \times 1)$
$\therefore \exp (\lambda)=2$
For $ t=4 \min , $
$ A=A_{0} \exp (-\lambda \times 4) $
$ \Rightarrow A=\frac{A_{0}}{[\exp (\lambda)]^{4}}=\frac{A_{0}}{2^{4}} $
$\therefore A =\frac{A_{0}}{16}$