The amplitude of a damped oscillator decrease | vedclass

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(C) The amplitude of a damped oscillator is given by $A(t) = A_{0} e^{-\frac{bt}{2m}}$,where $A_{0}$ is the initial amplitude.According to the problem

Vedclass · Accepted Answer

$0.729$

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(C) The amplitude of a damped oscillator is given by $A(t) = A_{0} e^{-\frac{bt}{2m}}$,where $A_{0}$ is the initial amplitude.According to the problem,after $t = 5\,s$,the amplitude becomes $0.9 A_{0}$:$0.9 A_{0} = A_{0} e^{-\frac{b(5)}{2m}}$$e^{-\frac{5b}{2m}} = 0.9$ ............... $(i)$We need to find the amplitude after another $10\,s$,which means at a total time $t = 5\,s + 10\,s = 15\,s$:$A(15) = A_{0} e^{-\frac{b(15)}{2m}}$$A(15) = A_{0} (e^{-\frac{5b}{2m}})^3$ ............... $(ii)$Substituting equation $(i)$ into equation $(ii)$:$A(15) = A_{0} (0.9)^3$$A(15) = A_{0} (0.729)$Thus,the amplitude decreases to $0.729$ times its original magnitude.Therefore,$\alpha = 0.729$.

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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