Does the law of gravitation suggest the earth is hollow?

[bwana]

classically, acceleration towards the Earth's surface is described by
a=G*m/r^2

usually we think of using this formula to calculate the force of gravity as we rise away from the Earth's surface. but consider what would happen if if you traveled through a tunnel to the center of the earth.
Assuming constant density of the earth, m~volume~r^3
so a=G*r
This means, as you travel inwards, the gravitational force is proportional to the distance from the center of the earth. At the center, there is no gravity. But the Earth is spinning and generating centripetal acceleration. Therefore, as in a centrifuge, the denser objects will move away from the center. The core of the Earth could thus be a giant gas bubble. Can this be experimentally examined?

 

[DaveC426913]

classically, acceleration towards the Earth's surface is described by
a=G*m/r^2

usually we think of using this formula to calculate the force of gravity as we rise away from the Earth's surface. but consider what would happen if if you traveled through a tunnel to the center of the earth.
Assuming constant density of the earth, m~volume~r^3
so a=G*r
This means, as you travel inwards, the gravitational force is proportional to the distance from the center of the earth. At the center, there is no gravity. But the Earth is spinning and generating centripetal acceleration. Therefore, as in a centrifuge, the denser objects will move away from the center. The core of the Earth could thus be a giant gas bubble. Can this be experimentally examined?

While it is true that gravity is zero at the centre of the Earth, that does not mean that the pressure isn't still huge - there are untold megatons of rock in the 4000 miles overhead bearing down.

Think of this: You're lying under a tall stack of six hundred crates.The top two hundred are filled with bricks, the middle two hundred are filled with balsa wood, the bottom two hundred are filled with feathers. Do you feel no pressure?

 

[russ_watters]

Also, the centripetal acceleration is tiny enough on the surface and decreases as you go toward the center.

 

[bwana]

While it is true that gravity is zero at the centre of the Earth, that does not mean that the pressure isn't still huge - there are untold megatons of rock in the 4000 miles overhead bearing down.

Think of this: You're lying under a tall stack of six hundred crates.The top two hundred are filled with bricks, the middle two hundred are filled with balsa wood, the bottom two hundred are filled with feathers. Do you feel no pressure?

well, my reply might be the following -if i were a gas bubble would i not float into the feathers? or to be less pedantic, you are describing the effects of density at the Earth's surface and that is not valid at the center of the earth. at the center of the earth, there is NOTHING bearing down. to use your logic, i might say that the mass of the Earth above me is pulling me up. it is only the spherical symmetry that negates this.

What might happen though if there were a perturbation in my location?- suppose i wandered a little bit off center? then the massive gravitational pull would be stronger towards that surface. Heavier elements would accelerate faster towards the surface leaving the lighter elements in the center. Wouldnt it be funny if the center of the Earth had a hydrogen core? Wouldnt it be even funner to discover a little fusion reaction going on at the Earth's center and that's what keeps the Earth hot and the rock molten?

 

[DaveC426913]

you are describing the effects of density at the Earth's surface and that is not valid at the center of the earth. at the center of the earth, there is NOTHING bearing down.

No. I'm describing the effect at the bottom of the 4000 miles of rock. But fair enough, the analogy is too loose to make the point effectively, so let's drop it. (It is however, valid.)

What I'm trying to point out is this:

If you are standing at the centre of the Earth, you are under hundreds of miles of rock that are virtually weightless, granted, and so contribute little to the weight above you. But those rocks are under hundreds (thousands) of miles of rock that are NOT weightless, and DO bear down upon you.

I know this may seem counter-intuitive. I know it may seem like if you're weightless you're also not being crushed by 4000 miles of rock, but in fact, you are. The pressure at the centre of the Earth is very, very high, and it is because of the weight of the rocks pressing down on you. Really.

It's easy to pretend that there could be a bubble there at the centre in which you could float, but you've got to realize that, in order for that bubble to stay a bubble, its walls would have to withstand trillions of tons of pressure.

Don't confuse lack of weight with lack of pressure.

 

[DaveC426913]

What might happen though if there were a perturbation in my location?- suppose i wandered a little bit off center? then the massive gravitational pull would be stronger towards that surface.

No. Remember first that, inside a hollow sphere the gravity is zero. Everywhere. So if you drift off-centre, toward a wall, you still feel no pull. You could hug the wall of your bubble and you will still feel no pull.

Heavier elements would accelerate faster towards the surface leaving the lighter elements in the center.

No. The pull is always toward the centre. It's just that the pull gets weaker as you near the centre. Heavier elements still sink to the centre, at least, in principle, they will just sink very slowly. (Bascially, they won't separate into layers because the gravity is too weak to delineate them).

However, no gravity still does not mean no pressure. It is still under millions of tons of pressure.

 

[bwana]

No. Remember first that, inside a hollow sphere the gravity is zero. Everywhere. So if you drift off-centre, toward a wall, you still feel no pull. You could hug the wall of your bubble and you will still feel no pull.


No. The pull is always toward the centre. It's just that the pull gets weaker as you near the centre. Heavier elements still sink to the centre, at least, in principle, they will just sink very slowly. (Bascially, they won't separate into layers because the gravity is too weak to delineate them).

However, no gravity still does not mean no pressure. It is still under millions of tons of pressure.

your statements are contradictory. In one paragraph you say

>
inside a hollow sphere the gravity is zero. Everywhere. So if you drift off-centre, toward a wall, you still feel no pull. You could hug the wall of your bubble and you will still feel no pull.
>

and in the next you say

>
The pull is always toward the centre. It's just that the pull gets weaker as you near the centre. Heavier elements still sink to the centre, at least, in principle, they will just sink very slowly.
>


A simple experimental solution would be to find a planet made entirely of liquid that we could investigate. We could look into its center with a simple probe. How does one go about identifying a liquid planet?

The theoretical answer to the problem of course are the equations i outlined above.
a=Gm/r and that is what a creature living in an entirely liquid planet would deduce for its law of gravitation. I suppose the pressure would be great and ultimately squeeze any gases out until they met a solid layer.

 

[gmax137]

If I put my finger between the jaws of a vise and then turn the handle, there is no net force on the finger, but still, there will be crushing pressure...

 

[Varnick]

your statements are contradictory. In one paragraph you say

>
inside a hollow sphere the gravity is zero. Everywhere. So if you drift off-centre, toward a wall, you still feel no pull. You could hug the wall of your bubble and you will still feel no pull.
>

and in the next you say

>
The pull is always toward the centre. It's just that the pull gets weaker as you near the centre. Heavier elements still sink to the centre, at least, in principle, they will just sink very slowly.

The first applies to a hollow sphere, the second to a non-hollow sphere, such as the earth.

 

[Researcher X]

There's no such thing as zero anyway.

 

[DaveC426913]

your statements are contradictory. In one paragraph you say

>
inside a hollow sphere the gravity is zero. Everywhere. So if you drift off-centre, toward a wall, you still feel no pull. You could hug the wall of your bubble and you will still feel no pull.
>

and in the next you say

>
The pull is always toward the centre. It's just that the pull gets weaker as you near the centre. Heavier elements still sink to the centre, at least, in principle, they will just sink very slowly.
>

They are not contradictory, as Varnick points out.

What's happening is that we're now talking about two different situations:
a hollow sphere (where you talk about forces experienced within the cavity) and
a solid sphere (where you tlak about what the materials of the Earth will do, those materials are necessarily part of the Earth, not in the cavity)
and they're each wrong in their own way.

In the hollow sphere you introduced, there is no net gravitational force experienced anywhere in the cavity (presuming it is at the centre). In a more realistic sphere where it's solid right to the core, gravity is always towards the centre.


That being said, these two situations are the same, they are only different in the degree to which the hollow bubble varies from arbitrarily large to arbitrarily small. (i.e. A solid sphere is merely a hollow sphere with a cavity of negligible size). The point is, how gravity behaves is consistent in both.

One can write a generalized rule:

Inside a sphere, the force of gravity will have a direction that ALWAYS points toward the centre of mass. And it will have a magnitudethat is proportional to how much gravitationally-significant material is "under" the point of reference. (i.e. if there is no mass "under" you (because you are in a hollow bubble) then the magnitude of the force will be zero.)

 

[bwana]

there is not weightless ness everywhere inside a hollow planet. imagine a planet that is hollow to half its radius. if you stand at the edge of the cavity, the center of gravity is still the same AT THE CENTER. Another way to look at it is this--there is more mass 'below you than above you' so there will be a gravitational pull towards the center.

The 'hollowness' that i was wondering about spontaneously forming has to do with the different 'law of gravitation' that would exist in these 2 cases.

in a homogeneous liquid planet, Force(weight)= Gm/R
but what happens when you reach the hollow part as you travel towards the center of the earth? Is the rate of change of your weight different? I would guess not since the geometry is no different. I imagine it would be equivalent of traveling through a liquid homogeneous planet of lesser density. So to answer my own question, no the Earth cannot be hollow on the basis of gravity.

 

[Vanadium 50]

Another way to look at it is this--there is more mass 'below you than above you' so there will be a gravitational pull towards the center.

That's not so. The mass below you is farther away so it cancels. There is no gravitational force inside a hollow spherical shell.

 

[gmax137]

Vanadium50 is correct - I remember we did this calc in a physics class, solving the triple integral to sum up all of the attractions between a point inside the hollow sphere and each infinitesimal volume of solid. It all cancels out leaving no net force. We did this after first doing the other problem - summing up the attraction on a point outside a solid sphere - doing which 'proves' that the net force in that case is equal to concentrating all of the planet's mass at its center.

 

[Varnick]

As Vanadium said, there is no force. As they are analagous, consider a hollow charged sphere. What is the E-field inside? By Gauss' law we can determine it to be 0.

 

[DaveC426913]

bwana, please look into Newton's shell theorem.
http://en.wikipedia.org/wiki/Shell_theorem
"If the body is a spherically symmetric shell (i.e. a hollow ball), no gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell."