If there was a hole through the middle of the Earth...

[OneOfOne]

and you fell into it....would you stop in the centre? I was thinking about this problem a couple of days ago and another question about orbital distances revived my curiosity.

Ignoring the fact that the core is molten, or just select a body that is completely solid. If you were to drill a hole through the centre of gravity of this object and jump in, you would fall all the way through to the centre without hitting anything. As you pass the centre of gravity you would now be falling up and drop back again. This oscillation would continue to dampen until you eventually stopped in the centre. The gravitational attraction from all sides would be equal, so there would be no net force to move you in any particular direction, so you would just float there. However, as the object will be in some sort of orbit, there will be some net movement of the body in one direction...but you will be moving at the same time, so this shouldn't effect the result? Would it?

As you fall, you would initially accelerate at the same rate as the surface gravity, but as you fall further, the increasing gravitational effect of material above you will begin to reduce the net effect of the gravity pulling you down, so half way you may accelerate at something closer to half the normal rate. So you would be falling at an ever increasing velocity, but accelerating at an ever decreasing rate...until you reach the centre when the effect will swap.

What do you think? Pretty cool experiment. Thoughts?

[Nesti]

You're pretty much correct, but there are two ways to look at that question. The second may shock you a little, until you see the mathematics and get the picture, then it's quite obvious.

Lot's of people played with this idea, including Newton and Gauss. I think Gauss did the really big work on why it is so. For me, it's Gauss and Riemann who were the really big thinkers on gravitation (spatial curvature).

1. Assuming the Earth had a uniform density and a vacuum tunnel connecting two opposing sides, then a falling body would commence acceleration toward the centre. The greatest expression of gravitation at the surface, the least (zero actually) at the very centre (core). The opposite would happen within the opposing hemisphere, until you came to that same height at the surface from which you started and you would accelerate back in again. This cycle (oscillation) would continue on, possibly forever.

2. Assuming the Earth was just an empty shell, with material only at the thin surface (say 1 meter thick) and the inner volume was a vacuum. Then a falling body would accelerate from the surface to the inner surface (1 meter down). The greatest expression of gravitation at the surface, the least (zero actually) at the inner shell. The entire inner volume of the Earth would be complete zero G; no expression of gravitation whatsoever!!!

If you watch the first hour of this he covers exactly what you've asked.
http://www.youtube.com/watch?v=s8UrYIZhm60&feature=channel