Gravity at the center of the Earth?

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>I wonder how far from the center of a planet the zero G extends and does the gravity increase linearly with distance or increase steeply only a short distance from the center.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Remember the uniform spherical shell theorem? Consider a point a distance x from the center. All planet-centered spherical shells with radius greater than x contribute nothing to the force on x. All planet-centered spherical shells with radius less than x act as if all their mass were concentrated at the center. So x feels an acceleration of Gm/x^2, where m is the mass in all shells interior to x.

If these all had the same density rho, then their combined mass would be m = rho (4/3 pi x^3) [the product of density and volume]. That makes the force on x equal to 4/3 G pi rho x. It goes up linearly with x until the density starts to drop off significantly. -|Tom|-

OK did the math and sketched it out (my brain is visually orientated) I've got it now. Thanks for the help.



Ben

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Remember the uniform spherical shell theorem? <hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
I understand this theorem and I see that it is widely used in astronomy, but sorry for my ignorance, was that theorem proved?
Will a limited range for gravity break that theorem?

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>[jacques]: I understand this theorem and I see that it is widely used in astronomy, but sorry for my ignorance, was that theorem proved?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

Yes. It is proved rigorously by setting up volume integrals and evaluating them. It is an easy proof if one knows integral calculus.

<BLOCKQUOTE id=quote><font size=2 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Will a limited range for gravity break that theorem?<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>

No. The limited range is so large compared to the size of stars or planets that the effect is negligible. For the structure of galaxies, limited range of gravity is significant, and the uniform spherical shell theorem no longer holds. -|Tom|-

Thank you!

As everyone except myself is convinced the pressure at the mass center of a sphere increases as the distance from the mass center is reduced maybe I should forget this and not ask, but after reviewing the past posts on this topic I'm puzzled about several details. First how does the model of inceasing pressure keep the pressure from going to infinity as I said early on? If the pressure goes higher because the area of the sphere is less and less it seems to me at the mass center the area gets so small as the radius of the sphere goes to a very small size. The pressure at a millimeter radius must be very great(too much for anything I would suggest). The other points are hard to put in words.