A Ball Through The Center of the Earth

[joesniper100]

Hello everyone,
I will begin by saying if this question is a repeat, I am sorry. I searched and only found similar questions but not any discussion on this specific point. With that that:

Consider an ideal model of Earth: uniform density, spherical, solid mass.

The classic question is if a hole is drilled through the center along the axis of rotation, and a ball is dropped, what will be it's behavior. I know this means that (neglecting air resistance) the ball will oscillate from one pole to the other with the same period as a revolution around the circumference of the earth.

Case 2: The ball is now dropped from the equator through a hole drilled through the center to the other side. Will this ball hit the side of the hole?

Please see attached image for my work. Because omega works out to be constant this should mean that the ball never hits the side.

What are your thoughts?

 

[rcgldr]

Angular momentum of the ball and Earth are preserved since there are no external forces. As the ball falls towards the center of the earth, in order to preserve angular momentum, the rate of rotation of Earth and ball would have to increase. This can only happen if the ball collides with the side of the hole and generates impulses or a continuous force on the side of the hole in order to increase the Earth's rate of rotation the tiny amount it takes to preserve angular momentum as the ball approaches the center of the earth.

Once past the center, as the ball falls "outwards", then the rate of rotation ball and Earth has to decrease.