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A damped harmonic oscillator has a frequency of 5 oscillations per second.The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to:

Option 1)
50 s

Option 2)
100 s

Option 3)
20 s

Option 4)
10 s

Answers (1)

best_answer

Resultant amplitude in damped oscillation -

$A=A_{0}.e^{-\frac{bt}{2m}}$

$E=E_{0}.e^{-\frac{bt}{m}}$

- wherein

$A= Amplitude$

$E= Energy$

$A=A_0e^{-\lambda t}$

$\omega$ =5

so $T=\frac{1}{5}$

$10T=\frac{10}{5}=2s$

A=

$A=\frac{A_0}{2}=A_0e^{-\lambda t}$

So $\lambda =\frac{ln2}{2}$

now at time=t $A=\frac{A_0}{1000}$

$\frac{A_{0}}{1000}=A_{0} e^{-\lambda t}$

$e^{-\lambda t}=10^{-3}$

$2^{\frac{t}{2}}=1000$

$\frac{t}{2} \log 2 =3$

$t=\frac{6}{\log 2}\Rightarrow 20s$

Option 1)

50 s

Option 2)

100 s

Option 3)

20 s

Option 4)

10 s

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