Answer to another "cool physics problem"

I having fun with these elementary physics problems. Here was the last one I posted:

Cool Physics Problem:

Suppose there is a hole through the center of the earth from the North Pole to the South Pole. Now suppose you fell in. Neglecting air resistance, assuming a spherical and uniform density earth, how long would it take to fall all the way through? How fast would you be going as you passed the center?

The answer: about 42 minutes, and around 18,000 miles per hour.

Here is the solution. Requires some familiarity with the harmonic oscillator problem from elementary physics.