Coulomb's Law and simple harmonic motion

[mickellowery]

Homework Statement

Two identical particles, each having a charge of +q are fixed in space and separated by a distance d. A third particle with charge -Q is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance x from the midpoint between the two fixed charges. Show that if x is small compared with d, the motion of -Q is simple harmonic along the perpendicular bisector. Determine the period of that motion. How fast will the charge -Q be moving when it is at the midpoint between the two fixed charges if it is initially released at a distance a << d from the midpoint?

Homework Equations

F_e= k_e$$\frac{(q_1)(q_2)}{r^2}$$
-kx=ma_x I'm not completely sure if I need this one, but the problem wants me to show that -Q is simple harmonic so I was thinking that I might need to set these equal to each other somehow, but I'm absolutely lost as to where to go.

The Attempt at a Solution

 

[collinsmark]

F_e= k_e$$\frac{(q_1)(q_2)}{r^2}$$
-kx=ma_x I'm not completely sure if I need this one, but the problem wants me to show that -Q is simple harmonic so I was thinking that I might need to set these equal to each other somehow, but I'm absolutely lost as to where to go.

I believe your interim goal is to show that the net force is proportional to x, for small x. Because if that's the case, the force resembles that of a spring (force being proportional to x).

I suggest first finding the equation for the net force on -Q. (First without making any assumptions.)

Secondly, once you have your net force equation, make the assumption that x is small compared to d. See what happens.

 

[kuruman]

See also

https://www.physicsforums.com/showthread.php?t=409063