[litespeed (OP)]

The sum total of gravity acting on any object consists of the agregate vectors produced by mass existing in all directions. Accordingly, to achieve a zero gravity force in the center of the earth you would need two things. First, a hollow sphere that includes the accumulated intersection of the matter above. This includes, in small measure, both moon and sun.

Also, the earth is not of homogenous mass, accordingly, you will need to excavate a hollow sphere big enough to accomodate the mass variation above you in all directions. Still, the zero gravity point is much smaller then you, and changes as the moon and earth interact with the gravitational forces they expend. If you let go of the rope you would slowly move in the direction of the highest agregate gravity, which would change over time.

[LeeE]

Your only valid argument is that the Earth is not homogeneous.  If you were to consider a pure waterworld though, which would be homogeneous, then you'd find that gravity was still zero at the center.

The original question didn't specify a volume of space where gravity has to be zero, so the issue of size is only of practical interest, and there's no practicality to this scenario.

[litespeed (OP)]

Lee,

I don't understand how gravitational forces vectored from the sun and moon do not count?

[Bored chemist]

we are in freefall towards them.

[LeeE]

Lee,

I don't understand how gravitational forces vectored from the sun and moon do not count?

Weeell... the Moon might have some effect as it's large enough and close enough to us to distort the Earth.  The Sun though, is far enough away that there's little difference its gravitational gradient across the width of the Earth.

In both cases though, the Sun and Moon will be acting upon whole of the body, whether hollow or homogeneously solid, which is why the Earth orbits the Sun and the Moon orbits the Earth.  As BC says, both the hollow body and the object at its center will be in free fall towards them, but thanks to our velocity we orbit instead of simply falling.

The thing is, are we supposed to treat this as a practical scenario, where the Sun and Moon are factors?  If you're just looking at the theoretical properties of hollow bodies then the Sun and Moon shouldn't really come into it as the solution wouldn't work for a distant object where they weren't present.

[litespeed (OP)]

Yeah

A non-rotating hollow body would have a stable point of zero gravity. Just to be a smart ass I must point out that stable point might not stay centered in said human. At any given moment in time, his center of mass changes in one vector or another.

If the hollow body has a enough mass and the hollow is big enough, and he is not careful in breathing and arm motions, he could end up sitting on the ceiling

 

[fontwell]

These two post below have been moved from Geology and Paleontology and merged with this topic.
JimBob

Dave Lammel  asked the Naked Scientists:
 Since gravity would be pulling outward in all directions on the core, would not the very centre be filled with lighter material?  

No! At the exact centre there is zero gravity, definitely not being pulled in all directions. Mathematically, this is how you calculate the net local gravity but the result is the actual net local gravity i.e. zero. 

Apart from at the exact centre there is always some gravity. And this gravity is always pulling towards the centre. For any position inside the Earth, regardless of the absolute strength of local gravity, heavy (denser) material will still be under a greater force than lighter material in the same region. So heavier stuff will always tend toward the centre more than light stuff.

Also, regardless of local gravity, the actual pressure of all the stuff above you will ensure that anything near the middle is fairly dens. The very centre will be at zero gravity but under enormous pressure. If there were gas at the centre (I doubt it but I'm not a geophysicist) it would be gas that is as dens as molten iron and only remains gas due to the very high temperature, caused by the pressure.

[JimBob]

Dave Lammel  asked the Naked Scientists:
 Since gravity would be pulling outward in all directions on the core, would not the very centre be filled with lighter material?  

No! At the exact centre there is zero gravity, definitely not being pulled in all directions. Mathematically, this is how you calculate the net local gravity but the result is the actual net local gravity i.e. zero. 

Apart from at the exact centre there is always some gravity. And this gravity is always pulling towards the centre. For any position inside the Earth, regardless of the absolute strength of local gravity, heavy (denser) material will still be under a greater force than lighter material in the same region. So heavier stuff will always tend toward the centre more than light stuff.

Also, regardless of local gravity, the actual pressure of all the stuff above you will ensure that anything near the middle is fairly dens. The very centre will be at zero gravity but under enormous pressure. If there were gas at the centre (I doubt it but I'm not a geophysicist) it would be gas that is as dens as molten iron and only remains gas due to the very high temperature, caused by the pressure.

Then you can say unequivocally, without any reservation that the Sun, Venus, Mars, Jupiter, Saturn and all other bodies in the complete universe have absolutely NO effect, however slight, on the center of the earth. That is absolutely illogical. All Einstinian space is related. The earth does not exist in an isolated gravitational system.

[Geezer]

I must agree with the the Learned Member from Texarse:

There ainta no sanity clause, and there ainta no no gravity.

[JimBob]

Why, thank you sir.

[fontwell]

Damn! I made a post here and it seems to have gone, so I'll do another shorter and better one.

My post above was as a reply to a question about heavy versus light matter being at the centre of the Earth. Within the confines of the Earth, acceleration due to gravity is always experienced as being toward the centre of the Earth. As you get nearer to centre of the Earth, this gravity drops because the forces from all the mass inside the Earth cancel out to give a net vector of zero at the centre. Not pulled in all directions at the centre.

It is true that this whole situation takes place in the Sun's gravity field. However, this affects the whole Earth (near enough) in the same way.

Any observer inside a black box can't tell if he is accelerating due to a rocket or due to being held near a mass. Similarly, he can't tell if he is in free fall or at the centre of a mass. But he can tell that he is either accelerating or not. The pragmatic description of some versus zero gravity inside the Earth is what would an observer in that position (with respect to the Earth) measure. At the centre they would measure zero acceleration.

The way you measure this is to release a mass (like a stone) and see if it remains next to you or moves away from you.

The fact is that in relation to the centre of our local galaxy cluster, the path of an object released above the Earth is going to look like a corkscrew made out of a corkscrew made out of a corkscrew to the power of stupid. However, it still falls straight down to us because the whole Earth is doing this crazy path too.

[Farsight]

Litespeed: it's all rather hypothetical, because you can't in practice create a void in the centre of the earth. The real point is that that the force of gravity depends not on g_μv but on a derivative. Think of it as the "local slope". Take a horizontal slice across the middle of the earth, measure g_μv at every point, then plot it, and it looks like this:



Where the slope is steepest is where the force of gravity is strongest. There's not much slope at the bottom, instead there's a little "flat" bit. As to how big this is, or indeed whether it is infinitesimal, I don't know. See   
http://en.wikipedia.org/wiki/Gravitational_potential for more.

[graham.d]

Just to clarify a point from the figure in farsight's post: the figure represents the gravitational potential inside AND outside a spherical body. The surface of the body is at the point of inflection i.e. where the slope changes from getting steeper to getting less steep as you go towards the centre. Once inside the surface of the body the gravity (the force is the slope) always gets less, the maximum being at the surface, until it is zero at the centre. Note the flattish bottom means that the gravity is quite low over quite a distance from the centre.

I thought this was worth saying as it may not be clear to everyone.

[Farsight]

Good one graham.