Simple Harmonic Motion 5 Question 2

2. 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 11000 of the original amplitude is close to

(2019 Main, 8 April II)

(a) 20s

(b) 50s

(c) 100s

(d) 10s

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Answer:

Correct Answer: 2. (a)

Solution:

  1. Given, frequency of oscillations is f=5oscs1

Time period of oscillations is T=1f=15S

So, time for 10 oscillations is =105=2s

Now, if A0= initial amplitude at t=0 and γ= damping factor, then for damped oscillations, amplitude after t second is given as

A=A0eγt

After 2s,

A02=A0eγ(2)2=e2γγ=log22

Now, when amplitude is 11000 of initial amplitude, i.e.

A01000=A0eγtlog(1000)=γtlog(103)=γt3log10=γtt=2×3log10log2t=19.93s or t20s

[using Eq. (i)]



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