The amplitude of a damped oscillator is known to decrease to 0.9 times its original amplitude in $5…
The amplitude of a damped oscillator is known to decrease to 0.9 times its original amplitude in $5 \mathrm{~s}$. Approximately, by how many times its original amplitude will decrease after another $20 \mathrm{~s}$ ?
0.73
0.9
0.59
0.26
Solution
Initial amplitude, $A_i=A_0$
Final amplitude $A_f=0.9 A_0$
Case 1 Time $t_1=5 \mathrm{~s}$
Case 2 Time $t_2=20 \mathrm{~s}$
Let, amplitude after $20 \mathrm{~s}$ be $A^{\prime}$.
As we know that,
$
\begin{aligned}
& A=A_0 e^{-\alpha t} \\
& \therefore \quad 0.9 A_0=A_0 e^{-5 \alpha} \\
& \Rightarrow \quad e^{-5 \alpha}=0.9 \\
& A^{\prime}=A_0 e^{-20 \alpha} \\
&
\end{aligned}
$
Again,
$
A^{\prime}=A_0 e^{-20 \alpha}
$
Now on powering Eq. (i) by 4 , we get
$
\begin{aligned}
& \Rightarrow \quad\left(e^{-5 \alpha}\right)^4=(0.9)^4 \\
& \Rightarrow \quad e^{-20 \alpha}=(0.9)^4 \\
& \therefore \text { From Eq. (ii), } A^{\prime}=A_0 e^{-20 \alpha}=A_0(0.9)^4 \\
& =0.6 A_0 \simeq 0.59 A_0 \\
&
\end{aligned}
$