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)

 

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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