# Velocity of earth due to the rotational motion of the earth moon system about the center of mass

I calculated the location of the centre of mass of Earth-Moon system using $$\frac{\sum m_ix_i}{M} = \frac{mx}{M}$$ where m is the the mass of the Moon $$= 7.35\times10^{22} kg$$, x is the distance between centres $$= 3.844\times10^8 m$$ (taking the centre of the Earth as origin), $$M$$ is the sum of the Earth mass and the Moon mass $$= 5.97\times10^{24} + 7.35\times10^{22} kg$$.

Now I want to calculate the velocity of the Earth due to the rotational motion of the earth moon system about the center of mass that I evaluated from above to be $$= 4675km$$

I was wondering if I could use the orbital speed formula here? That is, $$\sqrt{\frac{GM}{R}}$$ where $$M$$ is the sum of masses of the Earth and the Moon and $$R$$ is the evaluated $$4675km$$.

When I do this, the answer is different to when, for example, I try using:

$$V=\omega R = \frac{2\pi R}{T}$$ where $$T = T_{moon} = 30\times24\times3600$$

(I understand that the moon rotation period is not exactly one month, but this is an approximation since I just want to understand the concept).

I was also thinking that I could possibly calculate it using energy conservation, but I can't get my head around it.

Any help/hint would be greatly appreciated!

We know that if two frames are inertial the velocity of the observed object can be written as, $$V_{earth} = V_{CM}+V'_{earth}$$ Where $$V_{earth}$$ is the velocity of earth respect to the moon and $$V'_{earth}$$ Velocity of earth respect to the center of mass.