Determination of the distance to the Moon Using Newton's Laws of Motion

The Ancients, up to the time of Galileo even, believed that the laws governing celestial motion were different from the laws governing terrestial motion. The stuff of the heavens and its motions were unique to that realm, except perhaps for some crude semblance on earth of the celestial motions. With the detailed scientific study of motion inaugurated by Galileo, and brought to fruition in the dynamical equations of Isaac Newton it was possible to see that the laws of nature were indeed universal. Simple laboratory measurements on earth are relevant for understanding the motions seen in the heavens. In this exercise you will calculate the distance to the moon using three experimental measurements and applying Newton's laws of motion and the law of universal gravitation.

Measurements needed:

1) radius of the earth ( either from your earlier result or using Re = 3959 miles.
2) siderial period of the moon around the earth (either you measure it or use Tm=27.3 days)
3) period of a simple pendulum Tp of length L.(You must measure this as shown below).

 
A convenient way to measure the period is by suspending a small dense object from a fishing line attached to a door frame. For example, measure the time it takes for ten complete swings(T10) of the pendulum and divide that time by ten, Tp = T10/ten. From an application of Newton's laws of motion it can be shown that the distance to the moon, Dm is given by  (derivation)
                                                                                                                          (distance to moon)
   Dm = { (Re2LTm2)/(Tp2)}1/3.

When you do this calculation be sure to use consistent units! For example, all distances should be rewritten in meters and all times rewritten in seconds.

Assumptions used in the calculation

Your result is approximate because of several assumptions made in the calculation.

1)  The mass of the earth is >> mass of the moon. Actually  (mass of moon)/(mass of earth) = .0123.

2) The moon's orbit is a perfect circle.( Actually the eccentricity of the moon's orbit is about 0.055.)

3) The sun's gravitational field does not affect the orbit of the moon about the earth.

Despite these assumptions your result should come to within a few percent of the average earth-moon distance if you are careful in your measurements.