## Is the gravitational force exerted by the Earth on the Moon equal to the centripetal force acting on the Moon? (Intermediate)

The moon is revolving around the earth! Then,the gravitational force exerted by earth on moon should be equal to the centripetal force(the force acting inwards when an object is moving in circular path). Is that right?

Thank you for your question, what you say is true, 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:

ag=G(MEarth+MMoon)/r2

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

Which implies:

r3/T2=G(MEarth+MMoon)/4 pi2=const.

This is the so-called third Kepler law, that states that the cube of the radius of the orbit is proportional 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 radius. I hope that this answers your question.

This page was last updated on July 18, 2015.

### About the Author

#### Valerio Carruba

Valerio is currently a Professor at the São Paulo State University in Brazil (UNESP), and he mostly works with asteroid dynamics.  He went to college in Italy at the University "La Sapienza", took his Ph. D. in Qstronomy at Cornell University, and then went to Brazil in 2004 for various pos-docs that then "evolved" into his current permanent position.

More information on Valerio and his research is available at: www.feg.unesp.br/~vcarruba/

## Our Reddit AMAs

AMA = Ask Me (Us) Anything