A Ball Through The Center of the Earth

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SUMMARY

The discussion centers on the physics of a ball dropped through a hypothetical hole drilled through the center of the Earth, assuming a uniform density and solid mass. It concludes that the ball will oscillate between the poles without hitting the sides of the hole, as angular momentum is conserved. The analysis indicates that the ball's behavior is governed by the principles of angular momentum and gravitational forces, with the ball's trajectory being influenced by the Earth's rotation. The preservation of angular momentum suggests that any interaction with the sides of the hole would be minimal, allowing the ball to maintain its path.

PREREQUISITES
  • Understanding of classical mechanics, specifically Newton's laws of motion.
  • Familiarity with the concepts of angular momentum and gravitational forces.
  • Basic knowledge of oscillatory motion and harmonic oscillators.
  • Concept of uniform density in spherical objects.
NEXT STEPS
  • Study the principles of angular momentum conservation in closed systems.
  • Explore the effects of gravitational forces on oscillatory motion.
  • Investigate the mathematical modeling of harmonic oscillators.
  • Learn about the implications of uniform density in planetary bodies.
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Physics students, educators, and anyone interested in classical mechanics and gravitational dynamics will benefit from this discussion.

joesniper100
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Hello everyone,
I will begin by saying if this question is a repeat, I am sorry. I searched and only found similar questions but not any discussion on this specific point. With that that:

Consider an ideal model of Earth: uniform density, spherical, solid mass.

The classic question is if a hole is drilled through the center along the axis of rotation, and a ball is dropped, what will be it's behavior. I know this means that (neglecting air resistance) the ball will oscillate from one pole to the other with the same period as a revolution around the circumference of the earth.

Case 2: The ball is now dropped from the equator through a hole drilled through the center to the other side. Will this ball hit the side of the hole?

Please see attached image for my work. Because omega works out to be constant this should mean that the ball never hits the side.

What are your thoughts?
 

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Angular momentum of the ball and Earth are preserved since there are no external forces. As the ball falls towards the center of the earth, in order to preserve angular momentum, the rate of rotation of Earth and ball would have to increase. This can only happen if the ball collides with the side of the hole and generates impulses or a continuous force on the side of the hole in order to increase the Earth's rate of rotation the tiny amount it takes to preserve angular momentum as the ball approaches the center of the earth.

Once past the center, as the ball falls "outwards", then the rate of rotation ball and Earth has to decrease.
 

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