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/snip/ I'd be interested in knowing how a distribution that places
most of the mass in one object at arbitrarily large distances (not
to mention in directions that cover a solid angle of nearly 2*pi)
from any given portion of the other object could end up giving a
larger net force than a distribution that keeps all of the mass
within a small distance *and* a smaller range of solid angles.
John provided this most interesting argument for the optimality of
two spheres each of constant mass m whose proximal surfaces are
separated by a modest distance x, as compared with two disks each
of mass m and separated on their proximal flat faces by a
distance x in respect to maximizing gravitational
attractive force.
I never claimed it as an argument for optimality. It is nothing
more than a devastatingly clear argument (IMNSHO) for the *vast*
superiority of the spherical configuration over the laminar
alternative.
I wonder if he would help me evaluate the following configuration:
two hemispheres each of mass m separated on their flat faces
by a distance x.
I'll think about that; maybe someone else will beat me to the
punch. In the meantime, perhaps it would help if I simply
presented the details of the two sphere vs. two lamina comparison:
=============
First consider two spheres, both having volume V, both having mass
M, with centers as close as possible, (i.e. 2R where R is
determined from V.)
Force of attraction from Newton's law of gravity:
F_spheres = G M^2/(4 R^2)
Volume:
V = (4/3) Pi R^3
Eliminating R from the expression for F:
F_spheres = (Pi/6)^2/3 [G M^2/V^(2/3)]
=============
Next consider two thin lamina, ("thin" means having a thickness
d << V^1/3), both having volume V, both having mass M, separated
by a distance similar to their thickness.
Gravitational field produced by either disk in vicinity of the
other from Gauss' Law:
g = 2 Pi G M/area
Force of attraction
F_disks = M g = 2 Pi G M^2/area
Volume:
V = area * d
Eliminating area from the expression for F:
F_disks = 2 Pi d/V^(1/3) [G M^2/V^(2/3)]
=============
Finally, find the ratio of the two forces:
F_disks/F_spheres = (288 Pi)^(1/3) d/V^(1/3) = 9.67 d/V^(1/3)
This number becomes arbitrarily small as d -> 0.
John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm