Pressure and gravity at the center of the earth

Question

I apologize if this sounds stupid, but as someone without a lot of physics training I was wondering...Regarding gravitational forces, since it is this that brings the dust and rocks together to form planets--- say you were able to place something in a void at the exact center of the Earth. Would it be crushed by the surrounding forces, or float since there is equal force on it from all directions, or both? It would appear to me that gravity is a function of mass, so my guess would be both; And how much would this force be affected by the speed of rotation of the earth?

Answer 1

The question, and many of the answers, are confusing two forces, pressure and gravity. Pressure is a surface force. By itself it does not cause motion, but if the pressure is not constant, then there will a force in the (negative) direction of the gradient. Gravity is a body force, and will always try to produce acceleration.

Think of the Earth as composed of shells, as suggested. Inside the Earth, if we not at the center, we are inside some of the shells, and these exert no gravitational force on us. But we are outside of other shells, which pull us toward their center, which is always the center of the Earth. There is therefore always an attraction to the center, except at the center itself, and the Earth has a tendency to collapse under its own weight. But it does not collapse, because there is a pressure gradient. If pressure depends only on depth, and increases with depth, there will be an outward force, essentially due to Archimedes principle, that just balances gravity. So the poor guy down there experiences no force tending to move him in any direction, but he will experience a pressure that crushes him with all of the weight of rock above.

ADDITION By pressure, I mean normal stress, such as would be present even if the interior of the Earth were liquid. If we assume a completely rigid material for the interior such that stress produces no strain, then the astronaut would be perfectly comfortable inside his little bubble. In fact all it would take would be a little ball of completely rigid material. However, at these pressures, things are usually liquid.

Answer 2

If the Earth is spherical, and has uniform [or at least purely radius dependent] density throughout, then according to the Shell theorem, the gravitational flux through the centre of this simplified Earth would be zero. [In reality though, the geographical centre of the earth isn't EXACTLY the gravitational centre].

A good explanation of the shell theorem here, without needing to go into any real mathematics, is to imagine that the simplified, spherical and uniform density earth is split into an infinite number of infinitely thin shells. Each point on a shell has a point directly opposite it, equidistant from the centre of the Earth. Since any hemisphere of a shell will cancel out with the hemisphere opposite, so the shell will result in zero gravitational force at the centre [in actual fact, ANYWHERE in that shell will result in zero F, due to the nature of gravitational attraction falling off at a square of the distance, but that takes some calculus to show].

So now we have the statement that for one shell, F inside = zero, so apply that to all the other shells that make up this simplified Earth, and we get $F_{net}$ at the core of the earth $=$ $0$.

Answer 3

If the materials in the earth are allowed to move, they will move towards you and crush you due to attractive gravitational force.

If the materials in the earth are not allowed to move due to its structure, then you will feel zero net gravitational force. But remember, there are still gravitational force between you and the materials but just that they are cancelled at your body (assume you are point mass and the Earth is spherically symmetric).

So, both are correct.

Answer 4

An object placed at the exact center of the Earth would feel no gravitationnal force at all. Indeed, because of the symetry of the situation, the sum of all the forces is null.

If the object is placed at the exact center of the Earth, it wouln't experience any force due to the rotation of the planet as well, since it would be on its axis of rotation. To understand this, you can imagine that you are on a carousel: the further you are from its center, the more force you undergo.

However, if the object isn't exeactly at the center of the planet, things change a little. One can show that any object placed inside a spherical shell won't experience any gravitational force (it's a consequence of Gauss's theorem). Let's call $r$ the distance between the object and the center of the Earth. We can see a ball as a stacking of very small spherical shells, so thanks to the previous result, the part of the Earth furthest from its centre than your object won't make your object move. It will be as if the object was at the surface of a ball of radius $r$, so it will be attracted.

Finally, the rotation of the Earth will also affect it, making it draw away from the axis of rotation.