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} (A)/(2)=A\cdot e^(-k\cdot1) \\ (1)/(2)=e^(-k) \\ \ln ((1)/(2))=-k \\ k=-\ln ((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} (A)/(5)=A\cdot e^(-0.693\cdot t) \\ (1)/(5)=e^(-0.693\cdot t) \\ \ln ((1)/(5))=-0.693t \\ t=(\ln ((1)/(5)))/(-0.693) \\ t\approx2.32\sec \end{gathered}`

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