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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
$$20 s$$
B
$$50 s$$
C
$$100 s$$
D
$$10 s$$
Solution
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#### Correct option is B. $$20 s$$$$A = A_0 e^{-\gamma t}$$$$A = \dfrac{A_0}{2}$$ after 10 oscillations$$\because$$ After 2 seconds$$\dfrac{A_0}{2} = A_0 e^{-\gamma (2)}$$$$2 = e^{2 \gamma}$$$$\ell n 2 = 2 \gamma$$$$\gamma = \dfrac{\ell n 2}{2}$$$$\because A = A_0 e^{-\gamma t}$$$$\ell n \dfrac{A_0}{A} = \gamma t$$$$\ell n 1000 = \dfrac{\ell n 2}{2} t$$$$2 \left(\dfrac{3 \ell n 10}{\ell n2} \right) = t$$$$\dfrac{6 \ell n 10}{\ell n 2} = 1$$$$t = 19.931 sec$$$$t \approx 20 \, sec$$

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