# Naked Science Forum

## Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: thedoc on 22/08/2012 21:30:01

Title: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: thedoc on 22/08/2012 21:30:01
Ellen asked the Naked Scientists:

Is gravity stronger or weaker at the bottom of a mine shaft?

What do you think?
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: CliffordK on 22/08/2012 23:01:06
I believe that because you have the crust above you, and thus pulling upward, the deeper you go, the lower the gravity, until you reach the center of the Earth which would effectively have zero gravity (but tremendous pressure from the weight of everything above it).

The Total Recall film re-make was very odd with the portrayal of gravity.  Presumably if one made an elevator through the center of the Earth (which would be difficult because of the heat, pressure, and liquid core)...  anyway, one would likely operate it with free fall so one would get effectively zero-G (or low-G due to friction) from the launch until the arrival.  Or, perhaps accelerate it just above free fall, so the gravity reversal would occur twice, at both ends of the trip.

If the elevator was operated at constant velocity rather than free fall, then one would experience a gradual reduction in gravity until one reaches the core, then a gradual increase again.

Note, from comments below, the density of the crust is lower than the density of the core, causing a slight increase of the gravity as one tunnels through the crust and mantel, followed by a decrease in the gravity as one passes into the core.
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: distimpson on 22/08/2012 23:52:28
yes, weaker, less force the deeper you go into the Earth (or any solid body). Gravity increases linearly with the distance from the center (r) inside a solid body and decreases by r squared when outside. You would be weightless when dead center and r=0.

At least according to theory, not sure what experiments have been done. Earth radius is about 6370km, a quick search gave TauTona mine as one of the deepest at 3.9km, a fractional decrease of only 0.9994! My bathroom scale won't look any better.
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: damocles on 23/08/2012 01:04:32
Quote
At least according to theory, not sure what experiments have been done. Earth radius is about 6370km, a quick search gave TauTona mine as one of the deepest at 3.9km, a fractional decrease of only 0.9994! My bathroom scale won't look any better.

But it probably will, Di! the heat and stuffiness at the bottom of such a mine will surely dehydrate a kilo or two from you!
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: CliffordK on 23/08/2012 05:03:25
So I would expect to weigh a few more pounds in a deep well.

So, what would be your weight in a well that was 6,378 km deep?
Which direction would your weight vector be?
Assuming you could get that deep without being cooked alive.

Note, from comments below, the density of the crust is lower than the density of the core, causing a slight increase of the gravity as one tunnels through the crust and mantel, followed by a decrease in the gravity as one passes into the core.
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: imatfaal on 23/08/2012 17:12:30
Shamelessly posting an answer to this question from another forum because to paraphrase DH's excellent answers would be sacrilege and plagiarism all in one

DH is explaining why the assumption that gravity varies linearly with depth (which is the correct ideal solution) is very wrong on our far from ideal home planet

Quote
That is assuming a uniform density Earth. That's a nice simplifying assumption for a freshman physics kind of problem, but not so good assumption of the real Earth.

This assumption results in gravitational acceleration dropping linearly with increasing depth. This is not what happens inside the Earth. The D" layer (the core/mantle boundary) is 2890 km below the surface. Per this simplifying assumption, gravitational acceleration at that depth would be a bit over half of surface gravity. That is off by a factor of two! Gravitational acceleration is 10.6823 m/s^2 at that depth.

A better model is that gravitational acceleration is constant at 10 m/s^2 from the surface down to halfway to the center of the Earth and then drops linearly from that point inward. ...

.
Quote
..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.
...
Thus gravity increases with decreasing r (increasing depth) if the local density is less than 2/3 of the mean density . 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!

Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: distimpson on 23/08/2012 23:57:29
If this image uploads OK (my first attempt) it is a graphical representation of the same.

The graph is from http://commons.wikimedia.org/wiki/File%3AEarthGravityPREM.jpg (http://commons.wikimedia.org/wiki/File%3AEarthGravityPREM.jpg) By AllenMcCloud. (Own work) [CC-BY-3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons.

(Another first for me, hope to have the copyright info correct here, I'm not an attorney.)
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: CliffordK on 24/08/2012 03:46:25
Thank you.

(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fwww.thenakedscientists.com%2Fforum%2Findex.php%3Faction%3Ddlattach%3Btopic%3D45321.0%3Battach%3D16736%3Bimage&hash=e35aa94474fafab8893fe89d4e99bf7b)

I had forgotten the density of the crust (various oxides) is much lower than the core (mostly iron),  So, you get a slight increase in gravity as one tunnels through the crust, and the gravity doesn't drop until one reaches the iron core.

Ahhh,
Here's the discussion associated with the image.
http://en.wikipedia.org/wiki/Structure_of_the_Earth
http://en.wikipedia.org/wiki/Gravity_of_Earth
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: evan_au on 24/08/2012 12:17:18
The above discussion has been focusing on "weight", or "the acceleration due to gravity".

There is another measure of gravity: How far you are down a "gravitational well". This can be characterised by an Einstein red-shift in the energy of photons which are emitted out of the well.

I expect that the center of the earth is the lowest part of earth's gravitational well, and will show an Einstein red-shift in photons emitted towards the surface in any direction. In this sense, gravity gets stronger the further you go down the well, in terms of producing a greater Einstein red-shift.
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: imatfaal on 24/08/2012 15:41:24
distimpson - nice graph and I think you nailed the copyright stuff!  this question comes up all the time and I will be sure to keep a link to that graph

evans-au - gravitational redshift increases as move away from the centre of mass of the gravitating object; by definition the red-shift of a photon emitted and measured at the centre of the earth will be non-existent.  you can always think of gravitational red-shift in terms of acceleration (equivalence) - and at centre of the earth there will be no acceleration and thus no grav red-shift.  the centre of the earth is indeed the bottom of the gravity well - it is place where there is no potential, ie no energy is there to be exchanged by moving to a different spot - you need to put energy in to the system to move anywhere else.
Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: evan_au on 24/08/2012 23:25:21
I guess you could measure the acceleration due to gravity with a pendulum.

To measure relativistic effects as you move down the mine you need pretty sophisticated equipment - a pair of calibrated atomic clocks, or equipment to count the beat frequency between very stable sources (probably cryogenically cooled). Difficult, but possible.

The part that is really beyond us at the moment is descending more than 20km into the Earth's crust.

Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: CliffordK on 25/08/2012 05:10:36
Geezer went on one of his rants about Micro-Gravity and Zero-Gravity here (http://www.thenakedscientists.com/forum/index.php?topic=43452.msg383448#msg383448).

In space one feels weightlessness due to the constant free fall of the orbiter towards the planet, and the effect of it trying to continue in a straight line (Newton's first law), and the centrifugal force associated with the constant curvature of the orbiter's path.

Wikipedia has a page about Micro Gravity (http://en.wikipedia.org/wiki/Micro-g_environment#Absence_of_gravity).  The gravity at low earth orbit (where the International Space Station is), is about 9 m/s2, just slightly below the 9.8 m/s2 at the surface of the Earth.  Of course, there is also gravity associated with the Sun, Milky Way, and etc.

When the orbiter is unpowered, the astronaut is actually feeling zero-G weightlessnes with respect to the orbiter.  But, the actual characteristics of the orbit are determined by the gravity of the Earth, Sun, and etc.

Title: Re: Is gravity stronger or weaker at the bottom of a mine shaft?
Post by: imatfaal on 26/08/2012 23:39:22
Topic Split - for discussion outside OP please go to new theories