Free fall down (Chord) Tunnel through Earth

[Agrasin]

Homework Statement

I'm trying to show that any tunnel through the Earth (not necessarily through the center) will have a free-fall time that is the same. I heard this was true somewhere.

Homework Equations

acceleration of free-fall = GM / r^2 where r is changing

I believe this involves trig and calculus.

The Attempt at a Solution

I attempted a solution (attached-- please bear with me. I'm sorry if it's not very clear or if my process is weird).

I considered the acceleration of free-fall for a particle at an arbitrary position in a tunnel x that makes an arbitrary angle theta with the radius of the Earth. I eventually solve for the acceleration as a (complicated) function of x and theta.

Not sure what to do after that... If it involves integration, I feel like it's hopelessly complicated.

 

[Agrasin]

I caught my mistake. It's in the "Solving for r" stage. I foolishly assume d is an arc instead of a flat chord.

Admins, please feel free to delete this thread. Not sure how to.