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A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of [[mass]]. This can be seen as follows: take a point within such a sphere, at a distance <math>r</math> from the center of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass <math>m</math> is proportional to <math>r^3</math> (because it is based on volume), and the gravitational force exerted on it is proportional to <math display="inline">\frac{m}{r^2}</math> (the [[inverse square law]]), so the overall gravitational effect is proportional to {{nowrap|<math display="inline">\frac{r^3}{r^2} =r</math>,}} so is linear in {{nowrap|<math>r</math>.}}
These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with [[calculus]]. (
In addition to [[gravity]], the shell theorem can also be used to describe the [[electric field]] generated by a static spherically symmetric [[charge density]], or similarly for any other phenomenon that follows an [[inverse square law]]. The derivations below focus on gravity, but the results can easily be generalized to the [[electrostatic force]].
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