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Sagot :
Answer:
The time for five-fold decrease = 2.32 seconds
Explanation:The final amplitude of a damped oscillation is given as:
[tex]A=A_0e^{-kt}[/tex]The amplitude reduces two-folds during one second
That is:
t = 1 second
A = 0.5A₀
[tex]\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}[/tex]For a five-fold decrease
[tex]\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}[/tex]The time for five-fold decrease = 2.32 seconds
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