Why Does Displacing a Charge Lead to Simple Harmonic Motion?

[CrypticWeirdo]

Homework Statement

A charge +q of mass m is free to move along the x axis. It is in equilibrium at the origin, midway between a pair of identical point charges, +Q, located on the x axis at x = +b and x = -b. The charge at the origin is displaced a small distance x << a and released. Show that it can undergo simple harmonic motion with an angular frequency

omega=(4kqQ/(mb^3))^(1/2)

Homework Equations

E=k_e(q/r²)
(1+c)ⁿ is approximately equal to 1+nc

a=x(omega)^2

The Attempt at a Solution

Well, I'm not really asking for a solution per se. I get the question, got the correct answer, how it was done; what I want to know is why my method is wrong.

I got it by first using Coulomb's law to set up a force comparison, between the point-charge in the origin, and one of the point charges next to it. So...

F=kqQ/b²=ma

Where I substituted a for x(omega)^2.

Solving for omega got me close to the correct answer, but my TA could not explain why my method was wrong...so I'm curious why.

My answer was omega=(kqQ/(mb^3))^(1/2)

Any takers?

 

[zorro]

I got it by first using Coulomb's law to set up a force comparison, between the point-charge in the origin, and one of the point charges next to it. So...

The charge is in equilibrium at the origin, the forces of repulsion due to two other charges cancel out. You cannot compare with only one other charge. The force on it is due to the other 2 charges.