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

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SUMMARY

This discussion focuses on calculating the time it takes to fall through a hole in the Earth, specifically when the hole is not aligned with the center but along a chord. The problem requires the application of calculus to account for mass distribution and gravitational forces. Additionally, the conversation explores variations in Earth’s density, including non-uniform density distributions and the implications of modeling the Earth as an oblate spheroid. The fundamental conclusion is that the time of fall remains consistent regardless of the hole's placement, as long as the mass distribution is uniform.

PREREQUISITES
  • Understanding of classical mechanics and gravitational theory
  • Proficiency in calculus, particularly in integration techniques
  • Familiarity with concepts of uniform and non-uniform density distributions
  • Knowledge of Earth’s geometric models, including spherical and oblate spheroid shapes
NEXT STEPS
  • Research the mathematical modeling of gravitational forces in non-uniform density distributions
  • Study the effects of Earth's shape on gravitational calculations, focusing on oblate spheroids
  • Explore advanced calculus techniques for solving physics problems involving mass distribution
  • Investigate historical and contemporary solutions to the classic tunnel through the Earth problem
USEFUL FOR

Physics students, educators, and enthusiasts interested in gravitational theory, calculus applications in physics, and the complexities of Earth's physical properties.

Kavorka
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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.
 
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