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Something I've always wondered

  1. May 29, 2009 #1
    If you dug out a hollow sphere in the centre of the earth, ignoring heat or pressure or any chance that the hole would collaspe in on itself, then would I just float there if I was in it? Or would i be stretched out in every direction?
     
    Last edited: May 29, 2009
  2. jcsd
  3. May 29, 2009 #2

    atyy

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    Just float.
     
  4. May 29, 2009 #3
    Float and stretched :D
     
  5. May 30, 2009 #4
    Actually, I think he would just float no matter where he was located within the sphere and no matter how large the sphere.
     
  6. May 31, 2009 #5
    your body is not perfectly sphere(not a sphere in any sense), and the gravity is zero at a central point only(a point you know, really unimaginable), but your body lies over a point(considerabaly large) I think your body receives unequal force and your body will be broken disaster.
     
  7. May 31, 2009 #6

    HallsofIvy

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    No. Turtle Meister is correct. If you cut a spherical hole in the center of the earth the mass outside the hole cancels at any point inside that hole. As long as the hole was large enough to hold your entire body, you would feel no gravitational pull on any part of your body.

    Consider a spherical surface inside the matter. From any point inside the hole, imagine a cone from that point to the spherical surface in both directions. The gravitational force on that point due to the part of the spherical surface inside that cone is proportional to 1/r2, where r is the distance from the point to the spherical surface but the amount of surface inside the cone is proportional to r2. The two cancel out.
     
  8. May 31, 2009 #7
    I'm not so sure that is correct.At any point in the hole the net gravitational pull is zero because the attraction in any one particular direction is balanced by an equal pull in the opposite direction the result being a stretching force as suggested above.
     
  9. May 31, 2009 #8

    atyy

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    Ok, I correct myself. Considering only the force from the sphere, you'd just float. But since your finger will attract your toe gravitationally, there will be some stretching or compression there. Of course, your finger will attract your toe even if you are in completely empty space.
     
  10. May 31, 2009 #9
    If there was a 'stretching force' that would mean there would have to be a net force on your body. Since the net gravitational force everywhere inside the sphere is zero, there is no such force, and there is no stretching.
     
  11. May 31, 2009 #10
    Since the gravitational force varies linearly with distance from the center of the Earth for r<R (radius of Earth), you will feel essentially no gravity. If you were in a truly inertial reference frame, stabilized by gyroscopes, you will find your enclosure rotating around you.
     
  12. May 31, 2009 #11
    But you are pulled equally in all directions and this is why the net force is zero.
     
  13. May 31, 2009 #12
    Yes. Every particle in your body is pulled on equally in all directions; or equivalently: there is no pull at all.

    You seem to be mixing this up with being pulled by your feet and your hands for example. If you are being pulled by your hands and feet, in opposite directions, then yes, you will feel that. However, that is not the same! The net force in that example is still zero, but only if you take your whole body as 'the system'. If you are in the center of the earth in a hollow sphere, every particle in your body feels no net force. So there is no pull. It's a different scenario!
     
  14. May 31, 2009 #13
    Imagine an object with a single force on it .The object will accelerate in the direction of that force.Now imagine the same object with the same original force on it but now with a second force of equal magnitude but acting in the opposite direction .As an example the object could be a length of wire in equilibrium supporting a weight, the wire itself being supported at the top.The net force is zero but the wire will have stretched its molecules having been pulled further apart..
    Of course the object I have referred to is macroscopic but you want to go into the microscopic i.e into particles so lets go there.Firstly we know that collections of molecules can be separated by forces.Next, if we take a single molecule we know that it can be stretched eg in an electric field it can be polarised.Getting smaller still can protons and electrons etc be stretched or distorted in some way?Remember that a macroscopic object is made of real separate microscopic particles as is the wire in my example above,it is not made of point particles.
    It's a great question though Nick and I may try to think about it some more.I may come back and change my mind.
     
  15. May 31, 2009 #14
    The fact that particles may or may not be able to be stretched is beside the point.

    In the example of the rope you use, you did not say on which part of the wire the forces are acting. If one force is acting on one end of the wire, and the other force on the other end, then yes it will be stretched. But, in this case, the forces act on the same point of the wire! So the wire is not stretched.

    Of course, there are forces acting on one end of the wire, and other forces on the other end. But if you look at one point on the wire (it doesn't even matter if we take that point to be an atom, or a proton, or a hypothetical position in space), the net force on that point will be ZERO.

    This can easily be proven using Newton's laws of gravitation, and is valid for any point inside a hollow sphere. So, every point on your wire will have NO net force acting on it, and subsequently, it will not be stretched.



    EDIT
    Here's a picture that will hopefully clarify it for you:
    2sb7ckx.jpg

    In (a), there is no net force on the chain / wire. But the chain is still stretched, because the two equal but opposite forces don't act on the same position in the chain.

    In (b), there is still no net force on the chain / wire. But now, the chain is not stretched. Each particle / point in space feels two equal but opposite forces. There is no net force on any particle / point in space.


    In a hollow sphere, you have situation (b) (except that the forces acting on your are uniformly around you in a sphere, instead of just two forces horizontally; but that's a bit hard to draw ;) ).
     
    Last edited: May 31, 2009
  16. May 31, 2009 #15

    Doc Al

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    Nick89 is correct. In order for an object within the hollow to be stretched due to gravity (aka, experience a tidal force), there would need to be a variation of gravitational field strength (force per unit mass) with position within the hollow. But the gravitational field within the hollow is uniformly zero. Not only is there no tidal force, there's no gravitational force at all.
     
  17. May 31, 2009 #16
    Nice one Nick, thank you very much for clarifying things.
    :wink:
    And thank you Doc Al
     
  18. May 31, 2009 #17
    I agree with Nick89. See for example Hecht's textbook on introductory physics. There is a chapter about gravity, and this "theorem" is demonstrated. I'm not giving a page number because I have a french version.

    This is also true in General Relativity.
     
  19. May 31, 2009 #18
    Of course, this is all neglecting any outside forces such as gravitational force from the sun and other planets. Usually they can be safely ignored because they are so small, but in the absence of the earth's gravitational pull I'm not so sure...
     
  20. May 31, 2009 #19
    But perhaps not quite so easily proven. If I recall correctly, Newton himself delayed publishing the result for many years, because he was not satisfied that his proof (requiring a volume integral) was good enough.
     
  21. May 31, 2009 #20
    The original OP was a hollow inside the Earth.
    The Earths crust and it's oceans are not uniform so there would be variations of the gravitational field strength.
    Not to mention that the Earth rotates so you would be flung to the side of the hollow by variations in the field strength and centrapetal effects.
     
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