Where is the error in this answer Newton's motion

Question

I am asked to find the distance between the Earth and the Moon given only the constant $G$, the mass of the Earth and that it takes the Moon 28 days to orbit the Earth once as assumed.

I said we have $$F=\frac{GM_eM_m}{r^2}=M_m\omega^2r$$ and that $$\omega^2=\frac{2\pi r}{28\times3600\times24}$$ using these two equations I subbed in $\omega^2$ into the equation for $F$ and then solved for $r$ getting $r=1.42\times10^5m$ according to google this is way off the true value, can anyone tell me why?

Thanks.

Answer 1

$\omega$ is the angular velocity, and should be: $$\omega=\dfrac{2\pi}{T}$$ with $T$ the period of the orbit.