Harmonic oscillator with 3 charged particles

In summary, Jean-Charles was trying to find the constant k for a harmonic oscillator, but was having difficulty doing so. He found that it was easy to approximate the equation as a straight line at equilibrium, but was unable to find the actual constant k.
  • #1
Jean-C
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Homework Statement


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.

Homework 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.

The Attempt at a Solution

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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!
 
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  • #2
Hello again, Jean-C, :welcome:

Much better !
Jean-C said:
isn't linear in x like F=-kx but rather like F=k/x2.
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
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
Yes. It's called differentiation, I think :smile:. 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
Alright, thanks to you I have figured it out! Thanks a lot!
 
  • #6
You're welcome. Nice exercise.
warning: make sure you have the units right.
 

FAQ: Harmonic oscillator with 3 charged particles

1. What is a harmonic oscillator with 3 charged particles?

A harmonic oscillator with 3 charged particles is a physical system that consists of three charged particles connected by springs and experiencing a restoring force that is proportional to the displacement from their equilibrium positions.

2. How does the motion of the particles in a harmonic oscillator with 3 charged particles differ from a simple harmonic oscillator?

The motion of the particles in a harmonic oscillator with 3 charged particles is similar to that of a simple harmonic oscillator in that it follows a sinusoidal pattern. However, the presence of three particles introduces a more complex motion with additional frequencies and amplitudes.

3. What factors affect the behavior of a harmonic oscillator with 3 charged particles?

The behavior of a harmonic oscillator with 3 charged particles is affected by the masses and charges of the particles, the spring constants of the connecting springs, and the initial conditions of the system.

4. What is the significance of studying harmonic oscillators with 3 charged particles?

Studying harmonic oscillators with 3 charged particles can help us better understand the behavior of complex systems in nature, such as molecules and crystals. It also has practical applications in fields such as physics, chemistry, and engineering.

5. How is the energy of a harmonic oscillator with 3 charged particles calculated?

The energy of a harmonic oscillator with 3 charged particles can be calculated using the equation E = (1/2)kx2, where k is the spring constant and x is the displacement from the equilibrium position. The total energy of the system is the sum of the kinetic and potential energies of the three particles.

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