> Suppose there was a hole through the center of the eartr clear to the other side. Assume comfortable temperatures and pressure all the way through. We are concerned only with motion here. If you fell into the hole, aerodynamic drag would limit your speed and you would fall past the center but not all the way up to the other side. After passing center, gravity would stop working for you and would begin to work to slow you down. Somewhere short of the other side, you would slow down, stop, and then fall back the other way. After falling a shorter distance, you would pass center at a slower rate and therefore would not go as far past center. Sooner or later you would come to a stop at the center and remain there. A 4000 foot ladder would be very handy after that.
>
> But the question is, how many times would you pass center? Further, how long would all of this take? And more important, who cares?
>

This scenario, with no air resistance, is a relatively common classical mechanics problem. It turns out (after you assume the Earth has constant density and calculate the gravitational acceleration at every point along the path) that the back-and-forth motion through the center of the earth is a simple harmonic oscillator.

Also, the lack of air resistance keeps you from getting stuck 4000 miles underground with only a 4000 foot ladder.

^v^:)^v^
F"of course, it's much easier to just use a pendulum"B