Harmonic oscillator with 3 charged particles

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1. Jan 23, 2017

Jean-C

1. The problem statement, all variables and given/known data
I got an alpha particle (charge 2+) fixed at x=0 and an electron fixed at x=2. I then add a fluor ion (charge 1-) to the right of the electron and we note his position xeq. The question is to find the constant spring (k) relative to the harmonic oscillation made by the fluor atom for small movements around its equilibrium.

2. Relevant equations
The electric force between two charges :
F=kecqQ/r2
where r is the distance between the two charges.
Note : I used kec so It is clear it is different from the spring constant k.

The force in an harmonic oscillator :
F=-kx
where k is the constant I'm looking for.

3. The attempt at a solution
It is easy to see that the motion is that of an harmonic oscillator but I can't demonstrate it mathematically. I start off by finding the equilibrium position by finding where the resulting force is 0 :
F=kec(2*-1)/(xeq-0)2 + kec(-1*-1)/(xeq-2)2
With that I found out the the equilibrium position was at 2(2+sqrt(2)).
But then I have no idea how to find the relative constant k since the equation isn't linear in x like F=-kx but rather like F=k/x2.

My first idea is to transform the equation to a linear one and then isolate k, but I can't seem to find how. If anyone can give me a cue as to how to do that or to find k I'd be very grateful!
Thanks!

2. Jan 23, 2017

BvU

Hello again, Jean-C,

Much better !
Not exactly: in these equations x means the deviation from equilibrium, so (your $x)-x_{\rm eq}\$. You want to find the first order coefficient in that F. Effectively you approximate F(x) as a straight line at $x_{\rm eq}$ to find $k_{\rm\; 'spring'}$

3. Jan 23, 2017

Jean-C

Thanks for the fast answer I really appreciate it! And sorry for the last post, I was very tired...
I understand that I need to observe small deviations from equilibrium, but I must be doing something wrong : do I replace xeq by x-xeq and then use the Taylor approximation for the first order (since F(0) is 0 at equilibrium)?

4. Jan 23, 2017

BvU

Yes. It's called differentiation, I think . Mind you, your $x$ is already defined. The game is now to rewrite F(x) in terms of $x'= x - x_{\rm eq}$. So you don't want to replace $x_{\rm eq}$ .

5. Jan 23, 2017

Jean-C

Alright, thanks to you I have figured it out! Thanks a lot!

6. Jan 23, 2017

BvU

You're welcome. Nice exercise.
warning: make sure you have the units right.