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Damping constant and angular frequency

  • Thread starter bacon
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While discussing [itex]ω^{'}[/itex], the angular frequency of a damped harmonic oscillator, given by:
[itex]ω^{'}=\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}[/itex]
where k is the "springiness", m is the mass, and b is the damping constant,
my book, Halliday, Resnick and Walker, says if b is small but not zero,[itex]b<<\sqrt{km}[/itex] then [itex]ω^{'}\approxω[/itex]. [itex]ω=\frac{k}{m}[/itex], the undamped frequency.

If I say that [itex]\frac{k}{m}>>\frac{b^{2}}{4m^{2}}[/itex]and go through the algebra to get the relation in the book, I get [itex]b<<\sqrt{2km}[/itex]
Is this a meaningful difference when talking about a quantity that is much, much less than another?
Thanks for any help.
 

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  • #2
Redbelly98
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The two statements, [itex]b \ll \sqrt{km} \text{ and } b \ll \sqrt{2km}[/itex], are essentially considered to be equivalent statements.
 

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