While exploring Netwon's gravitational discoveries, we calculated
g using the fact that the distance between
the mass *m* and the earth was the radius of the earth. In other words, we
assumed that all the mass of the earth is concentrated at its center. This
supposition may seem reasonable when we are far away from the earth (that is we
are at such a distance that the radius of the earth is negligible in
comparison), but it doesn't seem so good at all when we are at the earth's
surface. However, we will see that this assumption does hold exactly for any
body outside the surface of a gravitating sphere (to which the earth is a good
approximation). This is a profound result. It is a consequence of
superposition, the inverse square law, and the symmetry of a sphere.

The following theorem was proved by Newton in the *Principia*:

We will consider the gravitational attraction that such a shell exerts on a particle of massA spherical mass can be thought of as built up of many infinitely thin spherical shells, each one nested inside the other.

Figure %: A thin spherical shell.

To do this, consider cutting the shell into rings as shown in . Every point on the ring is a distance *l* from *m*, and the ring has
width *Rdθ* and radius *R* sin*θ*. The surface area of the ring equals
2*Π*× the area × the width = 2*ΠR*^{2}sin*θdθ*. The
total mass of the shell, *M*, is evenly distributed over the surface, so the
mass of the ring is given by the fraction of the total surface area (4*ΠR*^{2}):

M_{i} = M× = |

For infinitely thin rings, we can take the integral to find the total potential:

U = - |

But applying the law of cosines to the triangle with sides

U = - = dl |

For the ring closest to

U = dl = (2R) = |

This result mirrors the result we would receive if all the mass had been concentrated at the center of the shell. This similarity holds true for all shells, and since a sphere is composed of such shells, it must be true for a sphere too. The phenomenon holds even if the different shells are not of equal mass density--that is, if the density is a function of the radius. We can conclude that the gravitational force exerted by one planet on another acts as if all the mass of each planet were concentrated at its center.

Now let us consider the potential for a particle inside such a shell.

Figure %: Particle *m* inside thin shell.

U = dl = (2r) = |

Thus the potential inside the sphere is independent of position--that is it is constant in

Figure %: Forces exerted on a particle inside a solid sphere.