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bacon
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While discussing ω^{'}, the angular frequency of a damped harmonic oscillator, given by:
ω^{'}=\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}
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,b<<\sqrt{km} then ω^{'}\approxω. ω=\frac{k}{m}, the undamped frequency.
If I say that \frac{k}{m}>>\frac{b^{2}}{4m^{2}}and go through the algebra to get the relation in the book, I get b<<\sqrt{2km}
Is this a meaningful difference when talking about a quantity that is much, much less than another?
Thanks for any help.
ω^{'}=\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}
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,b<<\sqrt{km} then ω^{'}\approxω. ω=\frac{k}{m}, the undamped frequency.
If I say that \frac{k}{m}>>\frac{b^{2}}{4m^{2}}and go through the algebra to get the relation in the book, I get b<<\sqrt{2km}
Is this a meaningful difference when talking about a quantity that is much, much less than another?
Thanks for any help.