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Q.

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 ( in s) it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to

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answer is 20.

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Detailed Solution

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Time for 10 oscillations is $\frac{10}{5} = 2 s$

Since, $A = A_{0} e^{- k t}$

$\Rightarrow \frac{1}{2} = e^{- 2 k}$

$\Rightarrow l n 2 = 2 k$

Also $10^{- 3} = e^{- k t}$

$\Rightarrow 3 ℓ n 10 = k t$

$\Rightarrow t = \frac{3 \ln 10}{k} = \frac{3 \ln 10}{\ln 2} \times 2$

$\Rightarrow t = 6 \times \frac{2.3}{0.69} \approx 20 s$

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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 ( in s) it will take to drop to 11000 of the original amplitude is close to