Question

Show that the fractional change in frequency ω of a weakly damped simple harmonic oscillator
is ≈ 1/(8Q^2)

Answer 1

I think that this is about the LINEWIDTH of the resonance of a weakly damped harmonic oscillator, with implications for MAGNETIC RESONANCE Excellent q. Just wish I could solve it quickly ...

Answer 2

\[\omega^{\prime}=\sqrt{\omega^{2}-b^2}\]

Answer 3

where omega primed is the damped frquency

Answer 4

\[b=\frac{ \gamma }{ 2m }\]

Answer 5

where gammaq is the restrictive constant.

Answer 6

\[\frac{ \omega \prime }{ \omega }=1-\frac{ \gamma^{2} }{ 4m^2\omega^2 }\]
If
\[\sqrt{1-\frac{ \gamma^2 }{ 4m^2\omega^2 }}\approx1-\frac{ \gamma^2 }{ 4m^2\omega^2 }\]
for very small gamma

Answer 7

so,
\[\frac{ \omega }{ \omega }-\frac{ \omega \prime }{ \omega }=\frac{ \Delta \omega }{ \omega }=\frac{ \gamma^2 }{ 4m^2\omega^2 }=\frac{ 1 }{ 4Q^2 }\]
since
\[Q=\frac{ \omega m }{ \gamma } \]

Answer 8

This where I'm stuck. I don't know why I don't get 8 in the denominator.

Answer 9

@Z4K4R1Y4 in the third post from the bottom there's what LOOKS like a BINOMIAL expansion of a square root, an attempt at it. But, the 1/2 power going into the expansion seems to be missing. 1/2x1/4 = 1/8 ????
That MIGHT answer the q.

Answer 10

@Z4K4R1Y4 In my tangles with this sort of stuff, I've come to expect algebraic/arithmetic hurdles. Some time ago I got into one when looking at an explanation for the design of a Weiss magnet. It is impressive looking at what you have done and I wonder what the motivation behind it is. I've often thought that a big key to trying to understand physics is to crack the algebra first, line by line and symbol by symbol. Once I have the confidence that I've done that correctly, then I try to get hold of the physics/reason behind what's being done.
I've done a post in the maths section which sort of shows this.
The way books and other things tend to miss out steps is something I find very frustrating. So, good on you for the persistence ... now what does it mean in terms of the physics that you are presumably looking at ?

Answer 11

\[\sqrt{1-\frac{ \gamma^2 }{ 4m^2\omega^2 }}\approx1-\color{red}{\dfrac{1}{2}} \dfrac{ \gamma^2 }{ 4m^2\omega^2 } + \mathcal{O} \left(\dfrac{ \gamma^2 }{ 4m^2\omega^2 }\right)^2\]

Generalised Binomial: \[(1+x)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}\]

Answer 12

@IrishBoy123 excellent graphic.

Answer 13

Thank you @IrishBoy123 that was really helpful.