(c) Extra credit – why this period differs from the rotation period of the Earth

The period of inertial oscillation $T=2\pi/\vert f\vert$ is not the same as the rotation period of the reference frame (the Earth) which is $T_{\rm Earth}=2\pi/\Omega_{\rm Earth}\approx 24~\mathrm{h}$. Recall $f=2\Omega_{\rm Earth}\sin\phi$, so

$$T_{\rm inertial} = \frac{2\pi}{2\Omega\sin\phi} = \frac{T_{\rm Earth}}{2\sin\phi}.$$ (24)

Thus the inertial period depends on latitude $\phi$; at the poles ( $\sin\phi=1$) the inertial period is half a day, and it increases toward infinity as $\phi\to 0$ (the equator). Physically, inertial oscillations arise from an initial horizontal velocity in a rotating frame: the Coriolis force provides the restoring acceleration and the parcel executes circular motion in the rotating frame. The rotation period of the frame is the “background” mechanical rotation rate of coordinates, while the inertial oscillation is the natural frequency of a parcel responding to Coriolis restoring forces. See Durran (1993) and other references for detailed geometric interpretations (in particular the distinction between motion seen in inertial vs rotating frames and how a straight-line inertial motion in the inertial frame becomes a circle in the rotating frame).