(894) |

(895) |

(896) |

for , and

for . Here, is the total mass of the sphere.

According to Equation (898), the gravitational potential *outside* a uniform sphere of mass is the same as that generated by a point mass located
at the sphere's center. It turns out that this is a general result for *any*
finite spherically symmetric mass distribution.
Indeed, from the
previous analysis, it is clear that
and
for a spherically symmetric mass distribution. Suppose that the mass
distribution extends out to . It immediately follows, from Equation (893),
that

(899) |

According to Equation (897), the gravitational potential *inside* a uniform
sphere is *quadratic* in . This implies that if a narrow shaft were
drilled though the center of the sphere then a test mass, , moving in this
shaft would experience a gravitational force acting toward the
center which scales *linearly* in . In fact, the
force in question is given by
. It follows that a test mass dropped into the shaft executes simple
harmonic motion about the center of the sphere with period

(900) |