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!