The amplitude of a damped oscillator decrease | vedclass

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

Vedclass · Accepted Answer

$0.729$

Vedclass · Answer

(C) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-kt}$.Given that at $t = 5 \ s$,$A(5) = 0.9 A_0$.Substituting this into the equation: $0.9 A_0 = A_0 e^{-k(5)} \Rightarrow e^{-5k} = 0.9$.We need to find the amplitude after another $10 \ s$,which means at total time $t = 5 + 10 = 15 \ s$.$A(15) = A_0 e^{-k(15)} = A_0 (e^{-5k})^3$.Substituting the value of $e^{-5k} = 0.9$:$A(15) = A_0 (0.9)^3 = A_0 (0.729)$.Thus,$\alpha = 0.729$.

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5 \ s$. In another $10 \ s$,it will decrease to $\alpha$ times its original magnitude,where $\alpha$ equals:

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