Originally posted by: silverpig
So yeah, this has been asked several times and the answer is quite complicated. The simple answer is they would oscillate from surface to opposite surface over and over again. If you add in air resistance, then the person would oscillate back and forth, but never regain the altitude from the previous fall, and would quite quickly come to a stop at the center of the earth. If you take into account the pressure gradient, the air would probably liquify under the intense pressure of the air above it, and you would fall into this liquid and possibly float.
I've spent countless hours on this problem in the past. If you could magically maintain a vacuum inside the pipe, the object bang off the sides, gradually slowing it while it oscillated back and forth (thanks to the correolis effect.) To avoid this, drill through the axis of rotation. Silverpig's correct - you aren't making it to the center if you take into account the air pressure gradient. No matter; you'd be crushed to death due to the air pressure long before you came close to the center.
If, somehow magically, you could maintain a 1 atmospheric pressure throughout the entire column, you would not oscillate back and forth. You would just barely reach the center... eventually. It's going to take a LONG time to get there though. Remember, the F_net due to gravity is going to decrease as you get closer to the center of the earth. IIRC, assuming a uniform density, the gravitational acceleration would decrease at a linear rate with distance. Just barely making it there also assumes that the drag is proportional to the square of the velocity; not that this matters much, I believe proportional to the velocity (unsquared) would also result in just barely making it there.
Think of it this way: terminal velocity occurs when the force due to gravity is equal to the air resistance. Well, half way to the center of the earth, you're only going to have 1/2 (or less) the force due to gravity. This means your terminal velocity 1/2 down the hole is going to be half of what it is near the surface. And, 3/4's of the way down, you'll only have 1/4 the velocity. Of course, the velocity won't change instantaneously. However, the drag forces have a relatively long time to slow the object. (Think about it; the fastest you're going to go is 120 mph... think of the distance involved - roughly 4000 miles -and how many hours it's going to take to get to the center, or rather, days.)
edit: and this assumes that the tube is able to insulate from the extremely high temperatures, radiation, etc.