JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Elasticity & SHM

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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 it will take to drop to

\frac{1}{1000}

of the original amplitude is close to:

(A)10 s
(B)100 s
(C)50 s
(D)20 s✓ Correct

Answer: (D)

Solution

A damped oscillator's amplitude decays as

A = A_{0} e^{- γ t}

. At

5

oscillations per second,

10

oscillations take

2 s

, and in that time the amplitude halves:

\frac{A _{0}}{2} = A_{0} e^{- 2 γ}

, so

ln 2 = 2 γ

and

γ = \frac{l n 2}{2}

. To fall to

\frac{1}{1000}

of the start we need

ln 1000 = γ t

, giving

t = \frac{l n 1000}{γ} = \frac{2 l n 1000}{l n 2} = \frac{6 l n 10}{l n 2} \approx 19.9 s

, which is close to

20 s

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Single correct option

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