Question

An undamped oscillator has period ro= 1.000 s, but I now add a little damping so that its period changes to r i= 1.001 s.

Answer 1

This Question has mistakes.Correct question is

An undamped oscillator has period  τo=1.000s , but I now add a little damping so that its period changes to  τ1=1.001s.

What is the damping factor β? By what factor will the amplitude of oscillation decrease after 10 cycles? Which effect of damping would be more noticeable, the change of period or the decrease of the amplitude?

Answer:

β=0.28 /sec

Amplitude=0.0606

Explanation:

$T_{o}=2\pi /w\\T_{1}=2\pi /\sqrt{w^{2} -\beta^{2} }\\ As\\T_{o}=1 \\T_{1}=1.001\\From T_{o}\\w=2\pi \\So\\T_{1}=w/\sqrt{w^{2} -\beta^{2} }\\\beta =0.00447w\\\beta =0.00447(2\pi )\\\beta =0.28sec^{-1}\\ Amplitude=Ae^{-\beta t}\\ after\\ t=10T_{1}\\so\\Ae^{-\beta(10t_{1}) }=e^{-\beta(10t_{1}) }\\=e^{-\(0.28)(10)(1.001) }\\Amplitude=0.0606$