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Gravitation Inside A Uniform Hollow Sphere
The gravitational force inside a hollow sphere shell of uniform areal mass density is everywhere equal to zero, and may be proved by the following argument:
Let the sphere have a radius a. Place a point P inside the sphere at a distance r from the center where r < a; i.e., r is strictly less than a. Draw a line through P to intersect the sphere at two opposite points. Call these points and. Let the distance from P to be r1, and the distance from P to be r2.
Now place a differential area dA at , and project straight lines through P to acquire its image dA at . These two areas subtend a solid angle d at P. Let the sphere have areal mass density (kg/m2). Then the net differential attraction dF of dA and dA at P directed toward is just
dF = ( dA /r12 - dA/r22).
But dA = r12 d, and dA = r22 d by definition of the solid angle. Thus,
dF = ((r12 d)/r12 - (r22 d)/r22) = 0.
This result is true
for all choices of dA
The gravitational force within the sphere is everywhere equal to zero.
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