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  1. #1
    Platinum Lounger
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    Just watched the old movie version of this classic with my young nephew. He was able to dispel the magnetic pole phenomenon as bull:

    "They eventually encounter a subterranean ocean, and make a raft (made from the stems of giant mushrooms) to cross it. Somewhere in the middle of the ocean, they pass through the center of the earth and their raft begins circling in a mid-ocean whirlpool. The professor deduces that must be the center of the earth, because the magnetic forces from north and south meeting there are strong enough to snatch away even gold (in wedding rings and tooth fillings). "

    I was very impressed to hear him, even at his tender age, state emphatically that "gold isn't even magnetic!" But he did ask me if there would be a whirlpool as suggested in the film. He also noted that the "gravity on the people" seemed to stay the same throughout their journey and asked me if that's the way it would really be. Can you figure it out?

    Alan

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    OK, I'll stick my neck out to start the ball rolling.

    I think Professor Lindenbrook may have had to do some more research prior to make such a sweeping statement in my opinion. If he was to state that they were at the centre of the Earth what were his assumptions?

    1. The Earth structure was perfectly uniform
    2. The Earth was a perfect Sphere (bulges at the equator)
    3. The Earth had no mountains and was perfectly smooth
    4. Was the Earth's axis was fixed
    5. No other gravitational effects from orbiting bodies (the moon and the tidal movement)


    Be it that none of the above are correct we can assume that the actual "Centre of Gravity" will vary. If the above assumptions(and I am sure there are more I have missed) were true then gravitational force would reduce linearly until it reached the centre but it in fact increases sharply at the core mantel boundary and will then dip down again.

    My tu'pence worth.
    Jerry

  3. #3
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    <span title="Spoiler: Highlight to reveal" style='color:#000000;background:#000000;padding:0p x 0.2em;'>[color=#666;margin-right: 0.2em]SPOILER:[/color]then gravitational force would reduce linearly until it reached the centre but it in fact increases sharply at the core mantel boundary and will then dip down again.</span>

    I came to the same conclusion with the first part, but can't see how you deduced the second bit. How did you figure this?

    Alan

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    There have been a number of papers written on the subject of the core-mantle boundary, it is suggested that it is not clearly defined. Boffins have done seismic surveys of the core and discovered core-rigidity zones or small patches of rigid rock within the fluid core. These zones apparently may explain why the Earth wobbles on its axis.

    As the Earth's core cools its inner section is solidifying iron and therefore the outer core has a higher concentration of lighter elements as they reach saturation point they will settle out and rise to the top (or outer edge) at the core-mantle boundary. The high density region causes the gravitation increase slightly but once past it, it will decrease again.
    Jerry

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    [quote name='Jezza' post='777570' date='30-May-2009 21:34']The high density region causes the gravitation increase slightly but once past it, it will decrease again.[/quote]

    I can't agree with that bit. I believe that the gravitation must always be on the decrease as one heads towards the centre. It would decrease at a more rapid rate when passing through denser regions, but could never increase as the total amount of underlying mass is decreasing.

    Alan

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    [quote name='AlanMiller' post='777664' date='31-May-2009 15:56']I can't agree with that bit. I believe that the gravitation must always be on the decrease as one heads towards the centre. It would decrease at a more rapid rate when passing through denser regions, but could never increase as the total amount of underlying mass is decreasing.

    Alan[/quote]

    Re-reading my answer that bit doesn't make sense does it?

    I agree with your correction
    Jerry

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    WS Lounge VIP sdckapr's Avatar
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    Here is some Formula/Discussion on the topic and here is a nice video of what is happening.

    Both seem to presume that the density of "sphere of the Earth" is relatively constant. Here is some (very gory) details of the change in gravity/density based on radius (page 8) shows the decrease is not linear and increases at one point but is relatively constant thru the mantle (about 1/3rd of the trip to the "core")...

    Steve

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    [quote name='Jezza' post='777570' date='30-May-2009 19:34']These zones apparently may explain why the Earth wobbles on its axis.[/quote]

    'Wobbles on its axis' is an example of the law of gyroscopic precession in action. Liquor has the same effect on humans, for a different reason

    Here’s a pretty picture of PRECESSION

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    [quote name='AlanMiller' post='777463' date='29-May-2009 22:42']Just watched the old movie version of this classic with my young nephew.[/quote]

    If you'd like to try the longer route next, you could read Following the equator: a journey around the world, By Mark Twain.

    Incidentally, the equatorial bulge is called the equatorial bulge.

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    [quote name='sdckapr' post='777725' date='01-Jun-2009 09:45']details of the change in gravity/density based on radius (page 8) shows the decrease is not linear and increases at one point[/quote]

    I'm quite bewildered by this supposed increase. Basic physics tells me that g should always be decreasing with distance, approaching the centre. I have no clue with regards to those complex models, but in the immortal words of one famous Scots engineer, "Ya canna change the laws of physics". Do the modelers realize that, or am I missing something... again?

    Alan

  11. #11
    WS Lounge VIP sdckapr's Avatar
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    My (limited) understanding is if the the earth was a solid sphere of of matter containing the same density throughout, that g would only be dependent on the distance. But the density of the earch varies depending on the radius and thus the gravitational constant will vary both with the density and the radius causing the "anomalous" appearing dependency.

    I don't know if the models take into account (or if it even matters) whether the "gravitational source" is a point at the center of mass or the true shape. I recall (and it was a number of years ago) from physics, that we always treated it as a point source, and while this works alright if we are outside of the sphere, I don't have a clue what effect this has on the model when we are inside so that some of what we are modeling as a point source is actually outside of where we are. (so the mass above us should be pulling us away from the center and towards the surface.

    I imagine that the people who created the model in the pdf file understands the ideas better than I do and includes the radius, the density gradient, not using a point source (as well as other factors) to get the change in g, so even though it is not intuitive to me (my intuition leads me to the linear change towards zero like we see after we are within the radius of 4000) I can "accept" (though not completely understand) how my simple "intuition model" may not be enough.

    It would be a different case if we were comparing the g on "models" of the earth, (eg asteroids of various radii with earth density) where as we go down we would ignore the material above us. I think in this case, the models would be more like our "intuition".

    Steve

  12. #12
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    The way I figured it was to start with a hollow sphere of mass (thin walled shell if you like). I recalled Gauss's Theorem when applied to such a body carrying a uniform static electric surface charge, and how it can be shown that the electric field inside such a sphere is zero at all points. The application to a gravitational field is analogous i.e. zero gravity everywhere inside a spherical shell of mass.

    If you're somewhere inside a solid sphere of mass (Earth) then you can visualize yourself standing on a small sphere of mass, and within a series of concentric spherical shells. Each shell, regardless of its individual density, produces zero gravity within. Regarding the gravity you feel, it's as if there were no mass above you at all, since superposing a series of zero fields results in a net zero field.

    This means that the g you experience will be entirely due to the mass beneath you. Said mass must always be decreasing with decreasing size (radius) - possibly at varying rates but always decreasing - so I find it impossible to see how g could undergo any increase as one travels towards the centre. Note that the above does assume a spherically symmetrical density distribution, but certainly also accomodates any density changes at any radius.



    Alan

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    It's true that each shell you go through will have no effect on you gravitationally. Only the shells beneath you will. But if you are going through a very NON dense layer and there are very dense layers beneath you, you can get an increase. For example, as you get closer to the earth with a spacecraft, you are going through layers of atmosphere. So you are ignoring the mass of each shell of atmosphere that you just went through. Yet the gravity increases. A rather extreme example.
    Suppose I have a large sphere made up of two layers, a core with density 10 and radius r, and an outer layer with density 1 and radius 2r. The volume of the outer layer would be seven times that of the core, but the mass would only be .7 that of the core. Standing on the outer layer, the gravity would be proportional to mass/r^2, or 1.7/4=.425. Standing on the inner layer (we ignore the outer layer now) the gravity would be proportional to mass/r^2, or 1/1=1. The gravity actually went up as we went down.

  14. #14
    WS Lounge VIP sdckapr's Avatar
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    Perhaps this spreadsheet will help with what I (as well as pi-eater) are trying to get across. I have approximated the density vs radius from the pdf chart (my chart is in the worksheet) and from these "volume bands" of different radii, calculated the mass changing of each segment (it is off a little as my total mass does not exactly equal the earth's total mass, but I leave it to you if you want to cut smaller radii slices to try and improve this...)

    My plot of "gravity" as function of radius matches the pdf file, even showing the increase from mantle to outer core and the constant gravity through the mantle.

    As a comparison, I copied the worksheet and just entered the average density for all locations, which is what I would consider "intuitive thought" ...

    Steve
    Attached Files Attached Files

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    Thanks to both Pi and Steve for taking my blinders off. I couldn't see the forest for the trees... or was I so busy looking behind me that I failed to notice what might be in front? And do I also overdo the cliches sometimes? The tellingly obvious example of a space vehicle approaching Earth makes the situation... well, obvious. Anyhow, my fundamental "thought error" was to conclude that just because mass must be ever-decreasing, then so must g.

    I guess the definitive stamp of truth though, must come from Steve producing an XL spreadsheet solution - these are now always assumed to be axiomatically correct!

    Alan

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