Here is a problem someone asked me about recently.
I have come to find out that my solution is not the simplest possible one, but here it is anyway.
Assume that the middle of the string moves with constant speed. The position and velocity of the right ball is:
The kinetic energy , potential energy , and Lagrangian are:
Equation of motion:
Evaluating these equations results in:
Suppose the physical parameters are such that the oscillations are small. In this case:
This corresponds to simple harmonic motion with angular frequency . Writing out the general form for the solution:
Assume that the balls start at rest. Applying initial conditions and results in:
The maximum value of is:
The minimum distance between the charges is:
We can work this into an alternate form which matches the form expected in an answer key. Since the argument of the cosine function is small (by assumption), we can represent the cosine as a two-term Taylor expansion:
A bit more manipulation:
Since is presumed to be small: