Gravity at the centre of the earth

[Fozzie]

I'm sure this has been asked before, but I can't find the answer right now. If you stand on the moon and face the earth, you will feel the attraction of the earth's gravity (as well as the moon's). If you then move towards the earth, the force of attraction will increase until you reach the earth's surface when the force will be 1G.

 

If you then burrow into the earth, the force should increase above 1G because you are closer to the total mass of the earth. If you then continue downwards, at some stage the force will start to decrease because it must tend to zero at the centre because it must then increase in the opposite direction as you ascend the "other side" until you reach the surface when the force will again be 1G, but 180 degrees in direction to what it was at the "other surface".

 

If you then plot a graph of the strength of the force against distance from the centre of the earth, what shape would the graph be?

I can only see this making sense if you take the direction into account and so plot the force and direction together - ie as a vector.

Any ideas?

[Cap'n Refsmmat]

The force would not increase above 1G inside the Earth. As you tunnel deeper, more of the Earth's mass is above you, and less of it is below. That means the overall attraction to the center becomes less.

[Klaynos]

And at the centre of the earth the net force exerted on you is infact 0. This is quite trivial to show using Guass's law for gravity.

[Martin]

The force would not increase above 1G inside the Earth.

 

I don't think that is quite right. I think it depends on the density of the outer layer that you tunnel down thru.

 

The average density of the earth IIRC is 5 times water.

 

but there are places where the surface of the earth is ice and you can go down a kilometer or more thru ice. I think in such places as you tunnel down that gravity would get stronger.

 

(but only for a little while!, soon it would start getting weaker just as the O.P. surmised, and eventually it would go down to zero around earth center)

===================

 

this is easy to show with a simplified numerical example. suppose the outer 1 percent of the planet has density 1 (like water or approx like ice) and suppose the rest has density 5

 

then if you dig down 1 percent of the way from the surface to the center------so you still have 0.99 of the radius left----then the equation that says gravity down there is stronger is simply the equation

 

1 + 4(0.99)³ < 5 (0.99)

 

1 + 3.881 < 4.95

 

4.881 < 4.95

 

In other words at the bottom of the tunnel gravity you feel will be stronger than surface gravity as 4.95 is bigger than 4.88

and we have to conclude that if the outer layer you tunnel thru is significantly less dense than the average then

gravity at the bottom of the hole WOULD be greater than 1 gee.

 

Fozzie's intuition seems right:

If you then burrow into the earth, the force should increase above 1G because you are closer...

closer to the center of mass

[Fozzie]

OK, so if I plot a graph with gravity up the side and distance to the centre of the earth along the bottom, I would get a line falling from 1g to zero g. The question is, would that be a straight line, or would it be curved?

[insane_alien]

if the earth was a uniform density it would be a straight line, but thats not reality, you would get a curved line if you actually went and measured it.

 

i think the curved line would be above the straight line since the earth is denser near the centre but i would get a second opinion on that.

[Martin]

OK, so if I plot a graph with gravity up the side and distance to the centre of the earth along the bottom, I would get a line falling from 1g to zero g. The question is, would that be a straight line, or would it be curved?

 

not quite right, the line might not start off going down.

 

it might, if you were going down thru low-density material, go up first, to a little above 1 gee.

 

then it would start sloping down to zero gee.

 

It would definitely not be a straight line (the earth is far from uniform density, that would be totally unrealistic to assume)

[Sisyphus]

if the earth was a uniform density it would be a straight line, but thats not reality, you would get a curved line if you actually went and measured it.

 

i think the curved line would be above the straight line since the earth is denser near the centre but i would get a second opinion on that.

 

This is correct, it would be above the straight line, a shallower gradient in the less dense regions, and a steeper one in the more dense. This is because, in the less dense regions, the effect of putting more mass "overhead" is weaker compared with the effect of getting closer to the much denser core. As Martin said, this might even result in gravity increasing at first, similarly to how it would increase while descending through Earth's upper atmosphere, where the overhead mass is trivial compared with the approach to the center.

 

I would also like to add that I don't think it would be a smooth curve at all, since the density would change in fairly well-defined layers instead of continuously. If I had to guess, I would say that it would look like a series of distinct, nearly flat (but still slightly downward curving) regions of progressively increasing steepness.