Help understanding centre of mass

Question

So basically the question I posed myself a few days ago was for a torus shaped planet, what would the gravitational field be like inside the torus? After viewing some answers on the internet, (Gravity on a doughnut-shaped/Möbius planet, and also having my friend code http://ideone.com/s9tlyM for me to check), it seems that anywhere inside the torus, you would experience a force towards the .centre (ie the centre of mass). TORUS RING misinterpreted results oops

So what does the centre of mass really mean for an object, if when I'm in the torus planet, and I'll fly to the edges, instead of being attracted to the centre of mass (which seems logical to me) Or does centre of mass have no relation to gravity at all? http://en.wikipedia.org/wiki/Barycentric_coordinates_(astronomy) As best as I understand this article, it says that a system of stars will orbit around their centre of mass, so if you treat me and the torus planet as two stars shouldn't I be attracted to the centre of mass?

Thanks for any help clearing this up.

Answer 1

When considering the gravitational field at a point, you need to sum the effects of all mass based on the quantity, direction, and distance of those masses. In the general case, there may be no simplification for this summation.

But there are several cases where simplifications are possible. In particular, spheres and shells with radial symmetry can have their contribution replaced by a point mass of equal mass at their center of gravity for regions outside their surface. Because many real bodies such as stars and planets are a very close fit for this distribution, center of mass is used quite often. The barycenter article that you linked to is using the assumption that the masses involved are radially symmetric.

But as you've seen in your torus, it cannot be used as a simplistic replacement in all situations. Even for a spherical shell of mass, objects inside the shell are not attracted to the center.