Does Gravity Increase with Depth Inside the Earth?

[Doc Al]

The pull would be outward, in all directions, toward the great mass.

Actually, at the center there would be no outward pull at all.

 

[rossphysics]

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

 

[D H]

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".

 

[rossphysics]

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]

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

 

[Dadface]

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.

 

[Dadface]

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.

 

[Dadface]

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.

 

[quietrain]

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]

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.

 

[mikeph]

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 canceled 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]

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.

 

[Dadface]

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.

 

[Dadface]

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 Earth's 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?

 

[Dadface]

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?

 

[mikeph]

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?

 

[Dadface]

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?

Have you got it all wrong?Don't know I am trying to visualise the set up you describe.As I understand it there is zero force at all points within a concentric spherical shell in an idealised spherical world .How do we know this to be the case since we can't enter such a space to do the measurements?We can use theoretical predictions by summing the contributions of all points from the surrounding shell but when we come to other shaped cavities the integration can become more complex.To make it even more complicated we are dealing with extended objects the various parts of which attract each other gravitationally.

 

[rossphysics]

Doesn't anyone like Einstein any more? General Relativity says that space/time is pushing from the out, in. The center of the Earth would be at maximum pressure.

 

[DaveC426913]

Doesn't anyone like Einstein any more? General Relativity says that space/time is pushing from the out, in. The center of the Earth would be at maximum pressure.

Space-time does not push. Space-time simply curves.
Einstein has nothing to say about pressure.

 

[rossphysics]

I thought that the Earth displaced space/time and made it more dense . . . you're right, if it were hollow, it would displace the same as if it were solid.

 

[DaveC426913]

I thought that the Earth displaced space/time and made it more dense . . . you're right, if it were hollow, it would displace the same as if it were solid.

Made what more dense? Earth?