The gravitational force exerted by the Earth on the Moon has to be equal to the centripetal force. An interesting application of this principle is that it allows you to determine a relation between the period of an orbit and its size. Let us assume for simplicity the Moon’s orbit as circular (it is not, but this is a good approximation for our purposes).
The gravitational acceleration that the Moon experience due to the gravitational attraction from the Earth is given by:
Where G is the gravitational constant, M stands for mass, and r is the radius of the orbit. The centripetal acceleration is given by:
acentr=(4 pi2 r)/T2
Where T is the period. Since the two accelerations have to be equal, we obtain:
(4 pi2 r) /T2=G(MEarth+MMoon)/r2
This is the so-called third Kepler law, that states that the cube of the radius of the orbit is propprtional to the square of the period. This has interesting applications. In the Solar System, for example, if you know the period and the radius of one planet orbit, by knowing another planet’s period you can determine its orbit radis. Edited from Cornell