Cyclotron frequency

Hi All,

For a hydrogen atom in the field B, torque on the electron is $\tau =\mu Bsin\theta =\omega_pJsin\theta$ => $\omega_p=gqB/(2m)$ If we only consider orbital motion (ignore spin), then g=1. So $\omega_p=qB/(2m)$. (It follows the steps from Feynman's lectures VII section 34-3)
For a single electron in the field, $qvB=v^2m/r$ => $\omega=qB/m$. I know these two cases are different, but why the results look the same (only a factor of 2 different). Is there any connection between them, or just coincidence?