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Gravitation Inside A Uniform Hollow SphereThe 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 r_{1}, and the distance from P to be r_{2}. 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/m^{2}). Then the net differential attraction dF of dA_{} and dA_{} at P directed toward is just dF = ( dA /r_{1}^{2} - dA/r_{2}^{2}). But dA = r_{1}^{2} d, and dA = r_{2}^{2} d by definition of the solid angle. Thus, dF = ((r_{1}^{2} d)/r_{1}^{2} - (r_{2}^{2} d)/r_{2}^{2}) = 0. This result is true
for all choices of dA_{}
and dA_{}.
The gravitational force within the sphere is everywhere equal to zero. |
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