Dropping a Ball Through the Center of the Earth

Suppose you drill a hole through the earth, starting at one surface and going straight through the earth's center to the opposite side. If you drop a ball in the hole, will it come out the opposite side?

The answer is yes (ignoring friction, wind resistance, and other side affects). Paraphrasing the explanation in the book, the ball would increase in speed until it reached the center of the earth, and then decrease in speed until it reached the opposite surface, at which point the speed would again be zero. The reason for this is that the gravitational force on the ball is proportional to its distance from the center of the earth. Due to the symmetry involved, the amount of force that increases the ball's velocity as it falls toward the center, is exactly equal to the amount of force that decreases the ball's velocity as it goes away from the center to the opposite surface. Ignoring friction, wind resistance, etc., there would therefore be no energy loss (or gain) by the ball, so once released, it would oscillate back and forth from one end of the hole to the other. It would be much like a pendulum swinging down and up repeatedly, with gravity alternately aiding and opposing the motion. If there were no friction or air resistance, the pendulum would oscillate indefinitely also. Incidentally, the time for the ball to go from one surface of the earth to the other would be about 42 minutes.

Its interesting that if the hole were to go through the earth from one surface point to another, but not go through the earth's center, the travel time would still be 42 minutes. The reason for this is that although the length of the hole is shorter in this case, the gravitational force along its path is also decreased compared to that of a hole that goes through the earth's center, which means that the ball would travel more slowly. Because the distance and the component of gravity decrease by the same factor, the travel time ends up being the same. A similar situation occurs with the pendulum in which the time required to make one down and back swing is the same regardless of how large an arc the pendulum swing makes. (This is the basis of the pendulum clock.) The physics in this problem and the bicycle problem are exactly the same. This leads to the conclusion that it would take the same amount of time for a bicycle to go down and up a valley road, as it would to go on a bridge across the valley.