Conference dypss1::brain_bogglers

>	What would the gravitational pull be, and in which direction, if
>	you could stand at the centre of the Earth?

The net gravity would be zero and you'd be weightless.  There'd be plenty
of gravitational pull but it would be equal in all directions so it
cancels out.  Obviously you are in a capsule of chilled air down there,
right? ;-)

Gravity essentially peaks at the earth's surface. If you go up and away
from the center of the earth then gravity decreases.  If you go down and
toward the center of earth then you have a "topography (terrain) effect"
which reduces gravity; part of the earth's mass is attracting you away
from the earth's center.

Chuck
(who observed a Pacific Ocean earthquake's effects from Framingham with
a LaCoste & Romberg gravimeter once)

    Some related questions prompted by the "chilled air" comment:
    
    o	The center of the earth is molten, right ?  Is this molten because
    	it hasn't cooled off "yet" in 5 billion years ?  Is it molten
    	because somehow the sun's energy continues to accumulate there ?
    	Is it molten because of some sort of pressure from the mass of
    	the earth compressing hard ?
    
    o	How much does the molten center contribute to our weather ?  i.e.
    	if it weren't molten, would the average temps be similar on surface
    	as they are now ?
    
    o	Here in Boston area, "code" for putting in cement pillars when
    	building a back deck of a house says dig the pillars in 48 inches,
    	which is considered "below frost line".  Is that because below 48
    	inches, the molten center is heating up the earth ?  Or is it
    	because below 48 inches, the heat from previous summers still
    	lurks?
    
    /Eric

re .2:

The outer core of the earth is molten.  The inner core is solid. (earthquake
data is how this is known)

The heat of the inner earth comes from two sources: Radioactive decay of
thorium, uranium and potassium-40, and the heat of crystallization of the
inner core.  I think both contribute about 50% to the heat.
The heat of crystallization of the inner core simply means the solid part is
growing at the expense of the outer core, and this releases energy as heat.
The core is mostly iron and nickel.


The "frost line" is where there is enough leftover heat from the previous
year so as to never freeze. Caves are often a constant temperature (in the 50s)
year round.  Deep mines will increase in temperature a small amount every
100' down you go so there is some small effect from the earth's inner heat.

-Mike

    Re .0:
    
    Not only is gravity zero at the center of a sphere like the Earth, it
    is zero at all points inside a spherical shell of uniform density. 
    That is, if you hollowed out a sphere, you could float around inside it
    without being attracted to the shell.  When you are near part of the
    shell, the larger gravitational attraction of the material near you is
    exactly balanced by the lesser gravitational attraction of the greater
    amount of material further away from you, in other directions.
    
    
    				-- edp
    
    
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To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.

As was mentioned earlier, gravity "peaks" at the surface of the earth
(or any similar, nearly homogeneous spherical gravitational body).
As you travel towards the center of the sphere, the net force of gravity 
towards the center decreases linearly, to zero at the center.
As you ascend from the surface, the force of gravity decreases by the 
inverse square law, measured by the distance to the center of mass of the body.

If the earth were denser (smaller radius for the same mass), then the 
gravitational attraction at ~4000 miles from the center would remain one "gee"
(9.8 meters per second per second) but the surface gravity (wherever the
surface was) would be higher, again by the inverse square law.

- tom]

    
    Is a sphere the only shape one would "float around in" ?  I seem
    to recall that the Faraday cage principle talks of zero electric charge
    inside a "cage" and in the case of electric charge the shape of the
    cage wasn't so critical.  (perhaps this is why "they" claim that we're
    safe in a car during a thunderstorm?).  Is the shape critical for the
    gravity case ?
    
    As for frostline being "the point below which previous year's warm
    weather still prevails", how much would the frostline change if the
    center of the earth weren't generating any of its own heat ?
    
    /Eric

    Re .6:
    
    There is no electric field inside a volume surrounded by a conductor
    because the electrons in the conductor move so as to cancel any
    electric field.  E.g, if you bring a positive charge near the cage, it
    attracts the negative charges of electrons.  Electrons move toward it. 
    When enough electrons have moved closely enough to the positive charge
    that they balance out the field at the conductor, electrons stop
    moving.
    
    The same is not true of gravity and mass.  If you brought a positive
    mass near a container, it will attract any parts of the container that
    are free to move.  This will increase the local gravitational field,
    not decrease it.
    
    
    				-- edp
    
    
Public key fingerprint:  8e ad 63 61 ba 0c 26 86  32 0a 7d 28 db e7 6f 75
To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.

    
    Sorry, edp, that wasn't what I was trying to ask.
    
    My question is, is a spherical shell the only shape shell in which we
    would experience "floating around" inside it.
    
    I'm fully aware that another mass near a shell would affect the gravity
    of something inside it.
    
    /Eric

    Well, you can only "float" at the exact center of the hollowed-out
    sphere, since the attraction from any part of the mass is balanced
    by an equivalent attraction from the opposite direction.
    
    Note that this isn't a stable position.  If you move even a little bit
    away from the center, you'll be a little closer to one side, and hence
    be more attracted to it.  This will cause you to move closer to that
    side until you eventually fall into it.
    
    With this in mind, it seems pretty clear that in any regular
    polyhedron (where the shell is of uniform thickness and density), the
    center is a zero-gravity point, since the attraction from any direction
    will be balanced by the attraction from one or more other directions.
    
    I believe you could extend this to show that in any convex polyhedron
    there must be some zero-gravity point (not necessarily the center.)
    This might be a bit harder to prove.
    
    Martin
    

re .9:

Reread .4 again.  When inside the shell and not at the center, being closer
to one part of the sphere is exactly counterbalanced by the fact that "more"
of the sphere is on the opposite side of the close point.  It really is zero
gravity inside of a uniform spherical shell.

This wouldn't be true for any arbitrary closed shape (unlike a Faraday shield,
as explained by .7) but there might be a family of closed shapes where this
is true.

    Yes indeed.  
    
    .4 from edp points out to us that *anywhere* in the shell, you'd float,
    *not* just at the exact center !
    
    /Eric

    Re .10:
    
    > This wouldn't be true for any arbitrary closed shape (unlike a
    > Faraday shield, as explained by .7) but there might be a family of
    > closed shapes where this is true.
    
    We can see that there are some shapes for which this is true:  There is
    no net gravity inside the intersection of a collection of spherical
    shells.  For example, inside a small shell inside a larger shell, even
    if the smaller shell is off-center in the larger shell.  Or inside the
    intersectio of two shells that have been pushed together.
    
    From the Faraday cage, I think we can construct a variety of regions
    with no net gravity.  Designate an arbitrary boundary.  Construct a
    spherical shell such that the desired boundary fits completely between
    the inner and outer surfaces of the shell.  Figure out where electrons
    would move to if the boundary were a conductor.  Everywhere you have
    figured there would be a negative charge, scoop a little matter out of
    the shell.  Everywhere you figure there would be a positive charge,
    compress some extra matter into the shell.  (Fabricating this in three
    dimensions is left as an exercise for the reader.)
    
    If you "subtracted" the spherical shell from this construction, it
    would leave behind volumes of positive and negative mass -- positive
    where you added mass, negative where you scooped it out.  From the
    Faraday cage calculation, we know the field inside this volume is zero,
    since electric fields and gravitational fields obey the same
    inverse-square law.  When we "add" back the spherical shell, we have
    its net zero gravity plus the cage's net zero gravity, for a grand
    total of zero gravity.
    
    
    				-- edp
    
    
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To find PGP, read note 2688.4 in Humane::IBMPC_Shareware.

    
    Ah, it is so nice to see other people who would scoff at the "hollow
    earth civilizations" stories that Sci-Fi tried to pawn off thirty years
    ago.
    
    Marc