Question

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

Answer 1

Answer:

The time for five-fold decrease = 2.32 seconds

Explanation:

The final amplitude of a damped oscillation is given as:

$A=A_0e^{-kt}$

The amplitude reduces two-folds during one second

That is:

t = 1 second

A = 0.5A₀

$\begin{gathered} 0.5A_0=A_0e^{-kt} \\ \frac{0.5A_0}{A_0}=e^{-kt} \\ 0.5=e^{-k(1)} \\ 0.5=e^{-k} \\ \ln 0.5=-k \\ k=-\ln 0.5 \\ k=0.693 \end{gathered}$

For a five-fold decrease

$\begin{gathered} \frac{A_0}{5}=A_0e^{-kt} \\ 0.2A_0=A_0e^{-kt} \\ \frac{0.2A_0}{A_0}=e^{-kt} \\ 0.2=e^{-kt} \\ \ln 0.2=-kt \\ \ln 0.2=-0.693t \\ -1.609=-0.693t \\ t=\frac{-1.609}{-0.693} \\ t=2.32 \end{gathered}$

The time for five-fold decrease = 2.32 seconds