Pendulum Gravity Determination -- from Eric Weisstein's World of Physics

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Pendulum Gravity Determination

Let the period of a pendulum be measured both on the surface of a spherical body, and at a depth h down a narrow shaft excavated the body. Let be the density of the outer shell. Let the radius and mass of the sphere be denoted R and M, then the gravitational acceleration is

where G is the gravitational constant. Therefore,

and using a series expansion of the numerator gives

Now Taylor expand the denominator using
The period of the pendulum at the surface is given by

At a depth h, the period is

Again using a Taylor expansion

so

Solving for M gives

If we know the average density of the sphere , then solving the above expression for gives

If the ratio is , the period of the pendulum in the shaft is shorter than the pendulum on the surface, so the shaft pendulum runs ahead.