Basic gravity questions

[HeyNow]

If gravity is produced from matter, why does it increase when you dig further down into the earth? Shouldn't there be less matter under your feet?

If you were able to dig very far down, shouldn't the matter above your head be pulling you up... Lowering the force under your feet?

If this isn't the case, does gravity's pull come from the center point of an object? Maybe because gravity is the lack of space. I think I've heard that before.

The force of gravity on the earth seems to always be pullilng me straight down. But the earth is also very wide. Am I also being pulled in all directions? What effect does this have on objects?

[insane_alien]

the deeper you go the less gravity you get.

 

although, for the earth that isn't true near the surface as the earth isn't a homogenous ball, the earth is denser at the centre. This means the effects of 1/r^2 over rule the effect of less mass beneath you.

 

and yes, you are being pulled in multiple directions. these all balance out to leave one net force i nthe down direction though. This does have effects on very large objects and is the cause of the tides (the moons gravity pulling the water sideways). you could notice it on something human sized near a blackhole or neutron star too.

[D H]

the deeper you go the less gravity you get.

Nope. That is assuming a constant density, and the Earth is anything but constant density. Gravity increases with depth down to the bottom transition zone (g=10.0143 m/s^2), then decreases to a local min of 9.9314 m/s^2 in the middle of the lower mantle, increases again to a global max of 10.6823 m/s^2 at the D" layer (core/mantle boundary), and then finally drops toward zero at the center of the Earth.

 

 

If gravity is produced from matter, why does it increase when you dig further down into the earth?

The answer is in the math. Gravitational acceleration increases with increasing depth if the local density at some depth is less than 2/3 of the mean density of all of the material deeper than than depth. Gravity decreases with depth only if the local density is 2/3 of the mean density or more.

 

Assume the Earth is a spherically symmetric body whose density is a function of distance from the center of the Earth. The gravitational acceleration at some distance r from the center of the Earth is

 

[math]g® = \frac{G M®}{r^2}[/math]

 

where [math]M®[/math] is the mass of that part of the Earth that lies within the sphere of radius r and origin at the center of the Earth. The part of the Earth outside of this sphere contributes nothing thanks to Newton's shell theorem (or Gauss' law of gravity). This interior mass [math]M®[/math] is given by the integral

 

[math]M® = \int_0^r 4 \pi \rho® R^2 dR[/math]

 

Differentiating [math]g®[/math] with respect to r,

 

[math]\frac{dg®}{dr} = 4\pi\rho®-\frac{2GM®}{r^3}[/math]

 

Define the mean density [math]\bar{\rho}®[/math] as the mass [math]M®[/math] divided by the volume of a sphere of radius r:

 

[math]\bar{\rho}® \equiv \frac{M®}{\frac 4 3 \pi r^3}[/math]

 

With this,

 

[math]\frac{dg®}{dr} = 4\pi G \left(\rho® - \frac 2 3 \bar{\rho}®\right)[/math]

 

Thus gravity increases with decreasing r (increasing depth) if the local density [math]\rho®[/math] is less than 2/3 of the mean density [math]\bar{\rho}®[/math]. Gravity decreases with depth only if the local density is 2/3 of the mean density or more.

 

Now to get back to the numbers I posted at the start of this post. The crust and upper mantle are of relatively low density material. The local density is less than 2/3 of the mean density, so gravitational acceleration initially increases with depth. The rock undergoes a phase transition to a different, more dense crystalline form in the transition zone. Local density now exceeds 2/3 mean density, so gravitational acceleration starts falling with increasing depth in the top of the lower mantle. The Earth's core is much denser than the mantle rock. Mean density increases much faster than does local density in the lower mantle. Local density becomes less than 2/3 of the mean density about 1600 km below the surface (4800 km from the center), so gravity once again starts rising. Gravity is at its greatest right at the core/mantle boundary, about 2890 km below the surface. That's almost halfway to the center of the Earth!

Edited January 18, 2012 by D H

[insane_alien]

Nope. That is assuming a constant density, and the Earth is anything but constant density. Gravity increases with depth down to the bottom transition zone (g=10.0143 m/s^2), then decreases to a local min of 9.9314 m/s^2 in the middle of the lower mantle, increases again to a global max of 10.6823 m/s^2 at the D" layer (core/mantle boundary), and then finally drops toward zero at the center of the Earth.

 

 

I did mention that, but i had no idea the maximum g was so low down. I thought it would have been within 1000km of the surface.

[D H]

I forgot to cite a reference for the Earth gravity data.

 

A.Dziewonskia and D.Anderson, "Preliminary reference Earth model", Physics of the Earth and Planetary Interiors, 25:4 (1981), 297-356

doi 10.1016/0031-9201(81)90046-7

Free reprint: http://www.gps.caltech.edu/uploads/File/People/dla/DLApepi81.pdf

Model data in tabular form: http://geophysics.ou.edu/solid_earth/prem.html

[Klaynos]

!

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