Math & Science Home | Proficiency Tests | Mathematical Thinking in Physics | Aeronauts 2000 |
||||
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 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
and dA.
The gravitational force within the sphere is everywhere equal to zero. |
||||
Please send suggestions/corrections to: Web Related: David.Mazza@grc.nasa.gov Technology Related: Joseph.C.Kolecki@grc.nasa.gov Responsible NASA Official: Theresa.M.Scott (Acting) |