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Question

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 $$\dfrac{1}{1000} $$ of the original amplitude is close to :-


A
100s
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B
20s
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C
10s
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D
50s
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Solution

The correct option is B $$20 s$$
$$A = A_0 e^{-\gamma t}$$
$$A = \dfrac{A_0}{2}$$ after 10 oscillations
$$\because $$ After 2 seconds
$$\dfrac{A_0}{2} = A_0 e^{-\gamma (2)}$$
$$2 = e^{2 \gamma}$$
$$\ell n 2 = 2 \gamma $$
$$\gamma = \dfrac{\ell n 2}{2}$$
$$\because A = A_0 e^{-\gamma t}$$
$$\ell n \dfrac{A_0}{A} = \gamma t$$
$$\ell n 1000 = \dfrac{\ell n 2}{2} t$$
$$2 \left(\dfrac{3 \ell n 10}{\ell n2} \right) = t$$
$$\dfrac{6 \ell n 10}{\ell n 2} = 1$$
$$t = 19.931 sec$$
$$t \approx 20 \, sec$$

Physics

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