What will the gravitational field inside the Earth be?

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Discussion Overview

The discussion revolves around the gravitational field inside the Earth, particularly focusing on the motion of a ball released in a hypothetical channel dug from the North Pole to the South Pole. Participants explore the implications of gravitational forces at varying depths and the resulting motion of the ball.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that the ball would oscillate from pole to pole if released, assuming no friction or air resistance.
  • Another participant estimates the time for a pole-to-pole oscillation to be approximately 42 minutes.
  • It is proposed that the gravitational force changes linearly with distance from the center of the Earth, based on the shell theorem, which states that there is no net gravitational force acting on a body inside a uniformly dense sphere.
  • A participant explains that as the ball descends, the gravitational force is influenced by the mass beneath it, which decreases with the volume of a sphere, while the attraction to the center increases inversely with the square of the distance.
  • There is a discussion about the assumption of uniform density, with one participant noting that density is likely higher towards the center, which could affect the gravitational pull experienced by the ball.
  • Another participant mentions that the gravitational pull actually increases until about halfway to the center before beginning to drop, referencing a graph from Wikipedia to illustrate this effect.

Areas of Agreement / Disagreement

Participants express differing views on the uniformity of Earth's density and its implications for gravitational force. While some agree on the linear relationship of force under the assumption of uniform density, others challenge this assumption, suggesting that density variations complicate the scenario.

Contextual Notes

The discussion includes assumptions about uniform density and ignores factors such as Earth's rotation and friction, which may not reflect real-world conditions.

Negi Magi
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There is a question: If we dig a channel from the North pole of the Earth to the South pole of the Earth, then we release a ball into this channel from the North pole, what will be the motion of this be like?

I think this question is related to the inner gravitational field of the Earth
 
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Ignoring friction and air resistance, etc., it would oscillate from pole to pole.
 
With a pole to pole length of time of about 42 minutes!
 
The force will change linearly with the distance from the centre.

This is connected with the shell theorem, i.e. there is no net force acting on a body inside an uniformly dense sphere.

As the ball gets deeper under the surface, the layers above it stop exerting gravitational force, and all that matters is the mass underneath.

That mass gets smaller with the third power of distance(volume of a sphere) as the ball goes down, but at the same time it is getting closer to the centre of mass attracting it, which force is inversely proportional to the distance squared(Newton's law of gravity).

F=\frac{GMm}{r^2}
M=ρV
V=\frac{4}{3}πr^3
F=\frac{\frac{4}{3}πr^3 ρGm}{r^2}

F=\frac{4}{3}πr ρGm
a=r \frac{4}{3} πρG

G is constant and we can assume the density of Earth ρ to be constant as well, so we have a linear relationship between acceleration and distance.

So the ball starts falling by being accelerated by g=9,81 m/s2, then the acceleration falls to 0 in the centre of the Earth just as the velocity reaches maximum.
Then it gets slowed down more and more the farther away from the centre it gets. As it reaches the surface on the other side, the velocity is again 0 and the acceleration is again g.
 
Bandersnatch said:
The force will change linearly with the distance from the centre.
That does assume uniform density, whereas the density is sure to be higher towards the centre. But on that assumption (and ignoring the spin of the Earth), the linear relationship would make it simple harmonic motion.
 
haruspex said:
That does assume uniform density, whereas the density is sure to be higher towards the centre.
It most certainly is. Which results in gravitational pull of Earth actually increasing almost half the way to the center, and only then beginning to drop.

Wikipedia has this graph that illustrates the effect.
 

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