Is earth doing work on the moon?

[Checkfate]

The definition of work states that the object must move in the same direction as the force. W=Fd. I am contemplating whether or not the Earth is doing "work" on the moon. My answer would be no, because no energy is being transferred, and the moon stays at around the same height (we are assuming a perfectly circular orbit). But... then I think that the moon IS constantly falling towards the earth, but it keeps missing it due to it's velocity... I am confused. lol, thanks.

 

[neutrino]

The definition of work is W = F.d, where the '.' refers to the scalar product (or dor product) - http://hyperphysics.phy-astr.gsu.edu/hbase/vsca2.html#vsc4 .

So, assuming a circular orbit for the moon, in which it moves with the same linear velocity, the Earth does no work on the moon.

From the conservation of energy, the potential and kinetic energies are constant at all times, and thus we arrive at the same conclusion.

 

[Checkfate]

thanks neutrino

 

[SGT]

In a simplified way neutrino's answer is correct. But we must remember that the moon causes tides on Earth (and the Earth on the moon). Since the Earth turns, the tides cause friction in the bottom of the ocean. This causes a loss of energy. The Earth's rotation is becoming slower. In order to conserve momentum, the moon is getting farther from earth.
Since there is loss of energy, there is work being done.