Is gravity affected by mass on the earth?

1. Sep 7, 2009

jeryst

Lets say that you have a valley 6 miles deep. Next to it, is a mountain 6 miles high. Assume that you have a measuring device that is capable of measuring infinitesimal differences in gravity. The entire area is made of the same material, from the top of the mountain right down to the core of the earth. Assume that all other conditions are also equal.

You take one measuring device and place it in the bottom of the valley, and you place another measuring device at the top of the mountain.

Would gravity measure more at the top of the mountain or the bottom of the valley? Or would it be exactly the same at both places?

2. Sep 7, 2009

tony873004

It would be more in the valley, as you are closer to the center of the Earth.

3. Sep 7, 2009

It would be less in the valley than the surface because the mass would be less too. Remember, the mass is a function of the radius.

$$F = \frac {G m M(r)}{r^2} = Gm\frac{4}{3}\rho \pi r$$

So as you descend you get less and less force (compared to the surface).

Now when you go on a mountain the amount of extra mass you get under you is negligible. So climbing a mountain would also lower your force from gravity. A some point the two (the valley and the mountain) would be equal, just depends on how high/deep you consider. But they would each be less than the surface.

4. Sep 7, 2009

jeryst

If all matter exerts some amount of gravitational pull, why wouldnt the gravity be higher on top of the mountain, since there is more matter under the instrument on top of the mountain?

5. Sep 7, 2009

Nabeshin

Actually, owing to the fact that the density of the earth is not constant, gravitational force is predicted to increase up until about the mantle-core boundary.

The problem seems to me to depend on the specific material of the mountain. For a sufficiently dense mountain, g at the top could be higher than at the bottom. Since for in general the rocks that make up the upper layers as the earth, as distance away from the center of the earth increases, g decreases, one expects that if the mountain is made up of normal earth rock (not molten metals or anything), g will be lower at the top than at the bottom.

6. Sep 7, 2009

KnowPhysics

Last edited by a moderator: Apr 24, 2017
7. Sep 8, 2009

D H

Staff Emeritus
That would have to be an incredibly dense and large mountain to make that happen. The free air correction to Earth gravity is that the gravitational acceleration falls by 3.086×10-6 m/s2, or 0.3086 mGal, for every one meter rise above the surface of the Earth. For example, the gravitational acceleration toward the Earth felt by an airplane flying 5 km above the Earth's surface is 0.01543 m/s2 less than that felt by a person standing on the point on the surface of the Earth directly below the airplane.

There is a slab of air between the plane and the surface of the Earth. This slab of air has negligible mass (compared to the Earth). What if that slab was made of rock rather than air? The Bouguer correction assumes that an infinite slab of rock lies between some elevated point and the reference ellipsoid. For average surface rock (density = 2.67 g/cm3), the gravitational acceleration due to the slab alone increases by 0.11 mGal for every meter of increased slab thickness (i.e., elevation). The net effect of the free air and Bouguer corrections is that gravitational acceleration decreases by 0.20 mGal for every meter in surface elevation above the reference ellipsoid.

What if the rock was denser? For the free air and Bouguer corrections to cancel, the rock density needs to be about 7.4 g/cm3. Granite is about 2.7 g/cm3. Basalt, a very dense rock, is about 3.0 g/cm3. Hematite, an iron ore, about 5.1. Gummite (uranium ore): Up to 6.4. Solid iron: 7.874 g/cm3.

So, it would take an infinite slab of iron between you and the nominal surface of the Earth to make gravity increase with increasing slab thickness. The Bouguer correction is pretty good for vast stretches of terrain at more or less the same elevation (think Great Plains). It isn't so good for mountains. Vast chunks of the slab need to be chopped off to make the slab into mountains, and that of course decreases gravity. In other words, even a mountain made of pure iron wouldn't do the trick.

Finally, a mountain of pure iron couldn't exist. It would sink into the Earth. Mountains are essentially thick slabs of light material afloat on the plastic (but not molten) mantle rock. Gravity almost alway falls off with increased elevation.

8. Sep 8, 2009

jeryst

I am just trying to determine, in the purest theoretical sense, if the mass of the mountain would contribute to the pull of gravity, making the instrument on top of the mountain measure more gravity than the one in the valley.

Maybe I should change the location to some unknown planet. The planet and the mountain are made of lead. The planet is the same size as earth, and does not rotate. There is no atmosphere.

Would the gravity measurement at the top of the mountain read higher than the one in the valley due to the added mass of the mountain underneath it?

9. Sep 8, 2009

f95toli

Gravimeters have been used for many years to look for e.g. mineral deposits and they work by detecting the change in gravity as the density of the rock changes.
There are gravimeters that can detect changes in the local g of the order of 10^-11 to 10^-12
meaning they can quite easily e.g distinguish a mountain that contains a lot of minerals (because of the density) from a mountain mostly composed form low-density rock.

10. Sep 8, 2009

uart

It's an intersting question jery, but I don't think it has an easy answer. My hunch is that it will depend on both the size and shape of the mountain. The problem is that once you bring the mountain into the gravity calculations then you no longer have spherical symmetry and that makes it a *lot* harder.

I would guess that if the mountain is in the form of a tall narrow spire then the gravity would be less at the top. On the other hand if the mountain were large enough (think for example of the "mountain" covering a large part of the planets surface) then the gravity will certainly be greater at the top. That implies that there must be an intermediate case where the two are equal. So the answer really is : "it depends".

11. Sep 8, 2009

D H

Staff Emeritus
:grumpy: A theoretic approach is exactly what I did in post #7. Did you read it? Bottom line: A mountain of pure iron would not do the trick. This is complicated by the fact that a mountain of pure iron can't exist (for long).

A shield volcano of pure iron coupled with a valley over a salt dome might do the trick -- but only briefly. A pure iron mountain of that size would sink into the mantle in very short time, geologically speaking.

12. Sep 8, 2009

Nabeshin

Which is the long, specific justification for my closing sentence that g would decrease with height for any realistic scenario, thanks for that D H.

13. Sep 12, 2009

jeryst

To me, this suggests that the answer to my question is yes, since the gravity measured over denser deposits of minerals is different from measurements over less dense deposits.

The reason that I want to know this, is because of the theory that gravity is a result of the curvature of space around a planet. If that were true, wouldnt the gravity measured on that object be totally uniform, and unaffected by the density of materials within the object?

14. Sep 12, 2009

D H

Staff Emeritus

15. Sep 12, 2009

jeryst

Yes I read it. Please explain, then, why gravimeters work as described.

16. Sep 12, 2009

D H

Staff Emeritus
Simple: The local variations in gravity due to variations in density are much smaller than the variations in gravity due to variations in altitude.

17. Sep 12, 2009

Subductionzon

To add onto DH's comment about gravity meters, they are used to find higher density rocks underneath a relatively low flat surface. I actually used one in a geophysics class in college many years ago. The U.S. almost split in two about 600 million years ago. We did a gravity and magnetic survey across it in a geophysics class. Gravity meters are incredibly sensitive and you have to set up a base area that you can return to so that you can correct for tidal motions of the Earth. As to the increase of gravity as you go down. If the Earth was of uniform density the force of gravity would start to decrease once you got below the surface. But the core is much denser than the mantle or the crust. As a result g goes from 9.8m/sec^2 at the surface to 10.8m/sec^2 at the core mantle boundary.

18. Sep 14, 2009

jeryst

Thank you for clearing that up for me.

If gravity is solely caused by a warping of space by the earth, then how can variations in gravity due to the density of relatively small local areas of the earth be explained? Wouldn't the interaction of the earth with space be uniform, thereby causing gravity to be uniform?

19. Sep 14, 2009

D H

Staff Emeritus
No.

You appear to be trying to run before you know how to walk. You need to learn how gravity works in a Newtonian sense before jumping to general relativity. You also have to learn a whole lot of hairy math before you can make that leap.

Bottom line: When it comes to a smallish mass like the Earth, the difference between what general relativity says versus what Newtonian gravity says is pretty dang small. If GR said otherwise it would be an incorrect model. Fortunately, it doesn't say otherwise.