First slide

Damped and forced oscillations

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 (1/1000) of the original amplitude is close to

Moderate

A

50 s
B

36 s
C

20 s
D

45 s

Solution

$a m p l i t u d e o f d a m p e d o s c i l l a t i o n = A = A_{0} e^{- b t} - - - (1) a f t e r 10 s e c o n d s A = \frac{A_{0}}{2} \Rightarrow \frac{A_{0}}{2} = A_{0} e^{- b 2} \log 2 = 2 b b = \frac{\log 2}{2} - - - (2) f r o m e q u a t i o n (1) \log \frac{A_{0}}{A} = b t s u b s t i t u t e e q n (1) \log 1000 = \frac{\log 2}{2} t t = 2 (\frac{3 \log 10}{\log 2}) t = 19.93 ≃ 20 s e c o n d$

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