William C. Keel wrote:
> Nope, I had quite forgotten about Chandrasekhar's work on this (and
> given that review, it doesn't sound worth my time to go through it).
> However, it does seem that second-guessing Newton's derivations
> is a more common hobby than I would have thought. Getting to his
> result about the gravity of a spherical shell cancelling throughout
> its interior is a classic - John Faulkner found a calculus-free
> way to do it involving the equivalent of image charges.
I'm not sure I understand why this is hard: Given any point P inside a
spherical shell centered on C, draw the line PC, which intersects the
shell at points A and B. Next, extend two tiny and similar cones whose
fulcra are at P, and whose bases are centered at A and B.
The area of the base at A is approximately to the area of the base at B,
as PA^2 is to PB^2. Its mass is therefore in the same ratio. But the
attenuation of the force due to distance also goes as PA^2 to PB^2,
cancelling out the factor in mass. Thus, the forces exerted on point
P by the bases at A and B balance out. We may tesselate the shell
completely with such pairs of cones.
The balancing out isn't perfect, admittedly, as long as the cones have
some positive size. But in the limit, as the cone bases shrink toward
zero area, the balance is perfect and the proposition is demonstrated.
Is there something I'm missing?
Brian Tung <brian@xxxxxxx>
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