Thankfully I did not have to do any math for this question. And I was even able to find a website the question was posed and answered. All the way back in 2011 no less. The website can be found here. In laymans terms, the ball (in their example) would fall through the hole until reaching the other side. At the point the ball stops on the other side of the hole it would begin falling again. The ball would repeat this cycle until it finally stops at the center where gravity would be equal on all sides.
Now for an additional question. How long would it take the ball to reach the other side of the hole?
Again, I am not the first to pose this question. I did find two websites that present solutions to this question however. The websites are Hyper Physics and Live Science. According to Hyper Physics the trip would take around 42 minutes. According to Live Science the trip would take around 38 minutes (using a more realistic model of the earth). Either way, I would not want to take that ride. Especially since at top speed the ball (or person) would be traveling around 17,895 mph. It would be a hell of a ride though.
I think the trip would be like one of those dreams where you fall and as you are about to land you wake up. But in this case you would just start falling again unless you grabbed onto something when you reach the other side. Can you imagine a 7,900 mile water-slide though?! That would be an EPIC trip!
Ok, I admit. This post was not as exciting as I was expecting. So I'll add some Dinosaurs.
Oops. Wrong dinosaur...
As you fall you have to watch out for Dinosaurs. You would be a fast food delivery for some lucky creature.
Hopefully we will have teleporters soon as we wont have to worry about the Dinosaurs living at the center of the earth.