|MadSci Network: Earth Sciences|
It is not possible to dig a hole through the center of the earth. Also, any attempt to even imagine such an undertaking would also to account for the difficulty in maintaining the hole after it was dug. But, let's assume you could dig such a hole and decided to jump in. Actually, you would also want to assume you dropped in and make a number of other very technical assumptions to ensure you did not strike the sides of the hole as you fell down.
Now, you are falling inside this imagined hole through the center of the earth and are wondering what happens next. It turns out, the gravitational acceleration you experience will keep you oscillating from one side of the planet to the other forever. More assumptions are needed. This time you must also assume there is no resistance to your falling due to either air or any other phenomena that may characterize the hole you are falling through.
Why would you oscillate from one side of the planet to the other? Because the gravitational acceleration you experience will vary from one 'g' to zero 'g' and back to one 'g'. One 'g' at the surface of the planet from which you first drop into the hole and that is one 'g' of acceleration toward the center of the planet, zero 'g' at the instant you pass through the center of the planet, and one 'g' of acceleration back toward the center of the planet as you reach the surface on the opposite side of the planet.
For references you will want to check your high school textbooks first. If the explanations there don't satisfy you, you next best bet is to look for a college physics textbook such as Halliday $ Resnick's 1966 Physics, there are of course later editions authored by Halliday & Resnick. There are also MadSci archived answers dealing with different aspects of this answer.
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