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(D) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-\frac{bt}{2m}}$.Given that at $t = 2 \, s$,the amplitude becomes $1/3

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$3^3$

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(D) The amplitude of a damped oscillator at time $t$ is given by $A(t) = A_0 e^{-\frac{bt}{2m}}$.Given that at $t = 2 \, s$,the amplitude becomes $1/3$ of the original amplitude $A_0$:$\frac{A_0}{3} = A_0 e^{-\frac{b(2)}{2m}}$$\frac{1}{3} = e^{-\frac{b}{m}}$Now,we need to find the amplitude at $t = 6 \, s$:$A(6) = A_0 e^{-\frac{b(6)}{2m}} = A_0 (e^{-\frac{b}{m}})^3$Substituting the value $e^{-\frac{b}{m}} = 1/3$:$A(6) = A_0 \left(\frac{1}{3}\right)^3 = \frac{A_0}{27}$Given that $A(6) = \frac{A_0}{n}$,we have:$\frac{A_0}{n} = \frac{A_0}{27}$$n = 27 = 3^3$.

The amplitude of a damped oscillator becomes one third in $2 \, s$. If its amplitude after $6 \, s$ is $1/n$ times the original amplitude,then the value of $n$ is

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