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m →Derivation of gravitational field outside of a solid sphere: Really this section just proves the first of the shell theorems. The next section covers the second of the shell theorems. |
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==Derivation of gravitational field outside of a solid sphere==
There are three steps to proving Newton's shell theorem (1). First, the equation for a gravitational field due to a ring of mass will be derived. Arranging an infinite number of infinitely thin rings to make a disc, this equation involving a ring will be used to find the gravitational field due to a disk. Finally, arranging an infinite number of infinitely thin discs to make a sphere, this equation involving a disc will be used to find the gravitational field due to a sphere.
The gravitational field <math>E</math> at a position called <math>P</math> at <math>(x,y) = (-p,0)</math> on the ''x''-axis due to a point of mass <math>M</math> at the origin is <math display="block">E_\text{point} = \frac{GM}{p^2}</math>
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