Time to fall through a hole in Earth, but not through center

[Kavorka]

It is a classic physics problem to calculate how long it would take to fall through a hole that passes through the center of the Earth to the other side, assuming Earth is a sphere with uniform density. I also remember being posed a problem for if you were falling through a hole that passed not directly through the center of the Earth, but say along a chord from one part of the Earth's surface to another with a distance A from the chord's bisect to the Earth's center.

My question is how to calculate the trip of the fall in the chord situation (I know you'd need calculus to approach the mass distribution).

Also, how would you approach these types of problems using a non-uniform density distribution for the Earth? What if you considered the Earth to be an oblate spheroid and not a sphere? I'm interested in messing around with these problems.

 

[BobG]

The same way you solved the tunnel through the center. The answer is the same regardless of where you place the hole.

https://www.physicsforums.com/threads/tunnel-through-the-centre.56323/