What will the gravitational field inside the Earth be?

In summary, if someone were to dig a channel from the North pole of the Earth to the South pole of the Earth, they would release a ball into this channel from the North pole, and the ball's motion would be oscillating from pole to pole. The force would change linearly with the distance from the centre, and assuming uniform density, the gravitational pull of Earth would actually increase almost half the way to the center before beginning to drop.
  • #1
Negi Magi
10
0
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
 
Physics news on Phys.org
  • #2
Ignoring friction and air resistance, etc., it would oscillate from pole to pole.
 
  • #3
With a pole to pole length of time of about 42 minutes!
 
  • #4
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).

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

[tex]F=\frac{4}{3}πr ρGm[/tex]
[tex]a=r \frac{4}{3} πρG[/tex]

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.
 
  • #5
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.
 
  • #6
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.
 

1. What causes the gravitational field inside the Earth?

The gravitational field inside the Earth is caused by the mass of the Earth itself. As a planet, the Earth has a large amount of mass that creates a gravitational pull towards its center.

2. How does the gravitational field change as you go deeper into the Earth?

The gravitational field inside the Earth increases as you go deeper into the Earth. This is because the mass of the Earth is concentrated towards its center, resulting in a stronger gravitational pull.

3. Is the gravitational field uniform throughout the Earth?

No, the gravitational field inside the Earth is not uniform. It varies slightly depending on the location and density of the Earth's materials. For example, areas with higher concentrations of dense materials will have a slightly stronger gravitational field.

4. How is the gravitational field inside the Earth measured?

The gravitational field inside the Earth can be measured using a device called a gravimeter. This device measures the slight variations in the strength of the gravitational pull at different locations on the Earth's surface.

5. Can the gravitational field inside the Earth change over time?

Yes, the gravitational field inside the Earth can change over time due to various factors such as changes in the distribution of mass within the Earth (e.g. movement of tectonic plates) or external influences such as the gravitational pull of other celestial bodies.

Similar threads

Replies
6
Views
852
Replies
11
Views
566
  • Other Physics Topics
Replies
9
Views
3K
  • Other Physics Topics
Replies
8
Views
1K
  • Other Physics Topics
Replies
17
Views
2K
Replies
3
Views
526
  • Electromagnetism
Replies
6
Views
492
  • Other Physics Topics
Replies
11
Views
2K
  • Introductory Physics Homework Help
2
Replies
40
Views
712
  • Astronomy and Astrophysics
Replies
3
Views
1K
Back
Top