# Gravity inside the Earth

Thanks for this prompt reply after 4 years of lying dormant!

The cumulative building up of pressure while going down, even under diminishing gravity, sounds very plausible.

The forces are not linear because most of the heavier materials have probably moved to the center causing the distribution of mass in the sphere to be non-linear.

So as a thought experiment, if we would try to measure the weight of all mass surrounding us at the center of the earth, we would not measure any weight. But because of their mass - not their weight - the molecules immediately surrounding the center, exert an attracting force on the molecules that are further to the outside. The accumulated effect of these forces causes the pressure to be the highest at the center of the sphere.

If there is no gravity when you are at the center of the earth, then the rocks just above you would have next to no mass, so I would guess that pressure would also be zero.

Janus
Staff Emeritus
Gold Member
If there is no gravity when you are at the center of the earth, then the rocks just above you would have next to no mass, so I would guess that pressure would also be zero.
The rocks just above you have mass, they just have little weight. However, they are supporting the weight of the rock above them which weigh a little more and which in turn support the weight of the rocks above them, etc. So the pressure at the center is due to the weight of all the rock above you and is quite large.

Try to think of it like this: Force is a vector, so the net force could be said to be 0N. However, pressure is a scalar - the forces pile up; its sort of like pressing against a balloon from multiple directions - it will eventually pop. Also note that we're not referring to point particles, so a net force of 0N doesn't necessarily mean there is no deformation (though it does mean the acceleration is 0m/s^2).

In Newtons gravity the center of the earth would be like a piece of iron surrounded by magnets. The pull would be outward, in all directions, toward the great mass. The greatest pressure would be half way between the surface and the center of the earth.
In Einstein's world it would be more like being in a car that was in a car crusher. General relativity pushes from all directions inward.

Doc Al
Mentor
The pull would be outward, in all directions, toward the great mass.
Actually, at the center there would be no outward pull at all.

Doesn't Newton say that a mass is pulled to another mass?

D H
Staff Emeritus
Newton also derived an equation for the gravitational force on an object inside a spherical shell of mass. It is a very simple equation.

Google "Newton's shell theorem".

If we divided the earth into cubic meters, would the man in the middle be drawn to each of the cubes according to the distance to those meters?
And, I don't think that two planets is a good way of thinking because they are not one inside the other.

DaveC426913
Gold Member
Doesn't Newton say that a mass is pulled to another mass?
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And, I don't think that two planets is a good way of thinking because they are not one inside the other.
If you have a planet on your left and a planet on your right, which way are you pulled?
Nowhere. You are weightless.

If you have half a planet on your left and half a planet on your right, which way are you pulled?
Nowhere. You are weightless.

If we divided the earth into cubic meters, would the man in the middle be drawn to each of the cubes according to the distance to those meters?
Yes. They would all balance out, resulting in zero weight.

Doesn't Newton say that a mass is pulled to another mass?

Newton does say that but according to the theory every single point is pulled equally in all directions(assuming a spherical earth of uniform density) the result being that the forces cancel exactly at each point and the resultant force is zero.

If you have a planet on your left and a planet on your right, which way are you pulled?
Nowhere. You are weightless.

If you have half a planet on your left and half a planet on your right, which way are you pulled?
Nowhere. You are weightless.

Yes. They would all balance out, resulting in zero weight.

I like the planet analogy but in this example I think that the person experiences a stretching force.Something that seems odd is that inside the earth there is no stretching force due to the forces acting on every point of infinitessimally small size cancelling.I have come across this somewhere else on this forum and I must admit I feel very uncomfortable with this concept.

A.T.
stretching force
What is a "stretching force"? There are just forces, and if you apply them nonuniformly to some body it can cause stretching, compressing, bending, etc. If you are big enough, you would actually fell compression in the center of the earth.

What is a "stretching force"? There are just forces, and if you apply them nonuniformly to some body it can cause stretching, compressing, bending, etc. If you are big enough, you would actually fell compression in the center of the earth.

Of course there are just forces but stretching force is a commonly used and accepted term as are the terms resultant force, compressive force, breaking force,frictional force,surface tension force ......the list goes on.My use of the term stretching force is appropriate to the situation being described in that the gravitational force can have a stretching effect.
In your last sentence do you mean that "you would actually feel compression in the centre of the earth"?If so what causes this compression?

A.T.
In your last sentence do you mean that "you would actually feel compression in the centre of the earth"?If so what causes this compression?
Gravity applied non uniformly to your body. I'm thinking about a very very tall person in a tunnel resting at the center.

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erm, what if we are inside the earth but not exactly at the centre, then the forces experienced would be of different magnitudes? then the summation of forces will not be 0?

Doc Al
Mentor
erm, what if we are inside the earth but not exactly at the centre, then the forces experienced would be of different magnitudes? then the summation of forces will not be 0?
Assuming the usual spherical symmetry, the gravitational force you'd feel at some distance D from the center of the earth will be due to the mass below you (r < D) and will be directed towards the center. The mass above you (r > D) will exert no force on you.

There is only one force, the net gravitational force (weight), and that always points towards the centre of the Earth, increasing linearly inside the radius, and decreasing like 1/r^2 outside.

If you are for example 1km away from the centre, you experience a force identical to when you are standing on a 1km radius planet of the same density as the core. All mass "above" you is cancelled and this is true for any spherically symmetric distribution.

edit- darn.. Doc Al has said exactly what I said in fewer words and 3 minutes earlier. At least we agree which is a good sign of my understanding!

D H
Staff Emeritus
There is only one force, the net gravitational force (weight), and that always points towards the centre of the Earth, increasing linearly inside the radius, and decreasing like 1/r^2 outside.
That is a simplistic, and in this case, overly simplistic model of gravity inside the Earth. Moving from the center of the Earth outward, gravitational acceleration is zero at the center of the Earth. The gravitational acceleration increases non-linearly inside the core and reaches a value 10.7 m/s2 at the core/mantle boundary. From there outward the gravitational acceleration decreases.

Gravity applied non uniformly to your body. I'm thinking about a very very tall person in a tunnel resting at the center.

But when you do the calculations you find that the gravitational effects due to the hollow shell surrounding that person cancel out.

A.T.
But when you do the calculations you find that the gravitational effects due to the hollow shell surrounding that person cancel out.
Only if the cavity is spherical. That is why I said: 'a very very tall person in a tunnel' would feel compression. My point was, there is no way you would feel any streching at the center on the earth.

If there was a tunnel through the earth then,assuming uniform density, spherical symmetry and ignoring other forces each point on the person would experience a force which is roughly directly proportional to its displacement from the earths centre the result being that the person would keep moving from one side of the earth to the other with motion that is approximately simple harmonic.
Yes,with spherical cavities there is no stretching at the centre of the earth because all forces from the surrounding shell cancel.It does seem odd though.

A.T.
Yes,with spherical cavities there is no stretching at the centre of the earth because all forces from the surrounding shell cancel.It does seem odd though.
Assuming uniform density, which type of cavity would cause stretching by gravity inside the planet?

Assuming uniform density, which type of cavity would cause stretching by gravity inside the planet?

Nice question but it is something I have never really thought about.My initial feeling is that there is no such cavity.Take your tunnel as an example.As you said if a man comes to rest in the middle he can experience compression.What if he is held in an off centre position his feet pointing towards the centre?If he is fixed in position at his feet he can experience compression and if he is fixed in position at his head he can experience stretching.Fix him at any other point and there can be both stretching and compressing.Of course this is cheating because of the fixing forces.Can you think of a suitable cavity?
I hope I haven't given the impression that I believe there are stretching forces,since theory suggests otherwise.What I have said is that it seems odd and that it is a concept I feel uncomfortable with.Are you familiar with feelings like this ,when there is something niggling but you can't quite pin down what it is?

How about an oblate spheroid cavity at the centre of a sphere? All mass from the maximum radius of the spheroid and above cancels, but there are still two masses above and below you, so your head experiences slight attraction upwards and your feet slightly downward.

Or have I got this all wrong?