Free fall down (Chord) Tunnel through Earth

AI Thread Summary
The discussion focuses on proving that the free-fall time through any tunnel in the Earth is constant, regardless of the tunnel's position. The user is attempting to derive the acceleration of free-fall using the equation GM/r^2, where r changes based on the tunnel's angle and position. They express difficulty in progressing after calculating acceleration, particularly with integration complexities. A mistake was identified in their approach regarding the assumption of the tunnel's shape as an arc rather than a straight chord. The user is seeking guidance on how to proceed with the solution.
Agrasin
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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.
 

Attachments

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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.
 

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