Tunnel in NON-Uniform Charge Density Object, and Drop in an Electron: Huh??

In mechanics there is a classic 'what if' problem for gravity, where one asks what happens if you were to jump into a tunnel drilled through the center of the earth, all the way to the other side?  Conceptually you could guess you would fall in, accelerate, overshoot the center, and stop at the other side, and then fall back in and repeat.  This is precisely what would happen (minus air friction and the molten iron core burning you up, etc) if the earth had a uniform mass density.

Now, change this around electrostatics style.  With a uniformly, positively charged insulating sphere, drill a tunnel and drop an electron in.  What happens, and in addition, how fast will it be moving when it gets to the center?  How do we do this mathematically, so we have all the details?  Check here for the uniform charge density problem.

We want to complicate matters a bit more in this problem!  What if the insulating sphere has a NON-Uniform charge density, and you drill a tunnel? 

Check out this video to see how it works.  It is a Gauss's law problem to find the E-field inside.  Then, to find the speed, we need energy since this is a non-constant force problem.  We will find a strange E-field and force, which is not simple harmonic any more, and that from the E-field we can integrate to find the potential difference, and then find the change in potential energy (which becomes kinetic energy).  Lots of concepts involved, but a doable problem that is fun to think about!