Derivation of the equation for determining the earth-moon distance from the pendulum measurement.

The basic underlying physical reason that the period of the pendulum and the period of the moon's orbit are related is the gravitational field of the earth. This field is generated by all the mass within the earth. Assuming the earth is perfectly spherical it follows that the strength of the field can only depend on the distance away from the center of the earth. The period of the pendulum for small oscillations is given by
                            T_p = (2*pi)(L/g)^1/2.

With L being the length of the pendulum, and g the gravitational acceleration at the earth's surface. This acceleration is given as

                 g = GM/R_e², with G being Newton's gravitational constant and
R_e the radius of the earth.  At the moon's orbit the acceleration due to the earth's gravitational field is given by

               a = GM/R_m² = gR_e²/R_m² .
 

 
For a perfectly circular orbit with M>>m the speed of the moon in its orbit is given by

                                                    v = (2*pi)R_m/T_m,  with T_m the moon's period.

The centripetal acceleration keeping the moon on a circular orbit is

      a = v²/R_m. This centripetal acceleration is provided by the earth's gravitational field.

If we substitue into these different equations we  can write

        ((2*pi)R_m/T_m)²/R_m = gR_e²/R_m² .

Finally when we substitute from the pendulum measurement our determination of g we obtain

           R_m = { (R_e²LT_m²)/T_p² }^1/3 .