The amplitude of a damped oscillator is known | vedclass

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(C) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-\alpha t}$.Given that at $t_1 = 5 \,s$, $A(5) = 0.9 A_0$.Substituting

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$0.59$

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(C) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-\alpha t}$.Given that at $t_1 = 5 \,s$, $A(5) = 0.9 A_0$.Substituting this into the equation: $0.9 A_0 = A_0 e^{-5\alpha}$, which implies $e^{-5\alpha} = 0.9$.We need to find the amplitude after another $20 \,s$, which means at total time $t_2 = 5 \,s + 20 \,s = 25 \,s$.Let $A'$ be the amplitude at $t_2 = 25 \,s$.$A' = A_0 e^{-25\alpha} = A_0 (e^{-5\alpha})^5$.Substituting $e^{-5\alpha} = 0.9$:$A' = A_0 (0.9)^5 = A_0 	imes 0.59049$.Thus, the amplitude decreases to approximately $0.59$ times its original amplitude.

The amplitude of a damped oscillator is known to decrease to $0.9$ times its original amplitude in $5 \,s$. Approximately, by how many times its original amplitude will it decrease after another $20 \,s$?

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