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Free fall down (Chord) Tunnel through Earth

  1. Dec 20, 2014 #1
    1. The problem statement, all variables and given/known data

    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.


    2. Relevant equations

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

    I believe this involves trig and calculus.

    3. 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.
     

    Attached Files:

  2. jcsd
  3. Dec 20, 2014 #2
    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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