Free fall down (Chord) Tunnel through Earth

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SUMMARY

The discussion centers on proving that any tunnel through the Earth, regardless of its orientation, results in the same free-fall time. The key equation referenced is the acceleration of free-fall, expressed as GM/r², where r varies based on the position within the tunnel. The user attempted to derive the acceleration as a function of position x and angle theta but encountered difficulties, particularly in the integration process and miscalculating the geometry of the tunnel as an arc instead of a chord.

PREREQUISITES
  • Understanding of gravitational acceleration and the equation GM/r²
  • Basic knowledge of trigonometry and calculus
  • Familiarity with concepts of free-fall and motion in gravitational fields
  • Ability to visualize geometric shapes, specifically chords and arcs
NEXT STEPS
  • Explore the derivation of gravitational acceleration in tunnels using calculus
  • Study the properties of chords and arcs in circular geometry
  • Learn about harmonic motion and its relation to gravitational fields
  • Investigate the implications of the Shell Theorem in gravitational physics
USEFUL FOR

Students studying physics, particularly those focusing on gravitational theory and motion, as well as educators seeking to explain the principles of free-fall in non-standard geometries.

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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