Question

Amplitude of damped oscillations reduces twice during one second. Find the time of its five-fold decrease.

Answer 1

Use the exponential decrease formula.

$A=A_0e^{-kt}$

First, find the constant k. The final amplitude would be half, and t = 1.

$\begin{gathered} \frac{A}{2}=A\cdot e^{-k\cdot1} \\ \frac{1}{2}=e^{-k} \\ \ln (\frac{1}{2})=-k \\ k=-\ln (\frac{1}{2})\approx0.693 \end{gathered}$

Then, we find the time of its five-fold decrease, which means the final amplitude is a fifth part of the initial amplitude.

$\begin{gathered} \frac{A}{5}=A\cdot e^{-0.693\cdot t} \\ \frac{1}{5}=e^{-0.693\cdot t} \\ \ln (\frac{1}{5})=-0.693t \\ t=\frac{\ln (\frac{1}{5})}{-0.693} \\ t\approx2.32\sec \end{gathered}$

Therefore, the time of its five-fold decrease is 2.32 seconds.