Angular Velocity from Period

Anything that spins or orbits sweeps exactly $2\pi$ radians in one complete cycle, so the angular velocity and the period are locked together: $\omega = 2\pi/T$. This is the $\omega = \theta/t$ relation with one full turn substituted in, and it is the form to reach for whenever what you actually know is how long one revolution takes — which is most of the time for orbits, pendulums, alternating current and anything described by its cycle time.

The Earth turns once on its axis in one sidereal day, 23 h 56 min 4 s, which is 86 164 s. So $\omega = 2\pi/86\,164 \approx 7.292\times10^{-5}$ rad/s. That number is the input to every Coriolis calculation, every geostationary orbit computation, and the design of every inertial navigation system. At the other end of the scale, a crankshaft at 3000 rpm has a period of 0.02 s and $\omega \approx 314$ rad/s.

Because $T$ and frequency $f$ are reciprocals, this can equally be written $\omega = 2\pi f$, which is the form electrical engineering uses constantly: the 60 Hz mains supply corresponds to $\omega = 377$ rad/s, and that 377 turns up in every reactance calculation. The same relation ties the period of a pendulum, a mass on a spring, or an LC circuit to the angular frequency that appears inside the sine function describing its motion.

The trap here is that frequency and angular frequency are both often called "frequency" and they differ by a factor of $2\pi$. Mains power is 60 Hz and 377 rad/s; those are the same physical thing with different units, and substituting 60 where 377 belongs is wrong by 6.283. Watch the symbols, $f$ in hertz, $\omega$ in radians per second — and be suspicious of any source that writes "frequency" without saying which. The second point is subtler and specific to the Earth example. The day you live by is the solar day, 86 400 s, the time for the Sun to return to the same place in the sky. The Earth's actual rotation period is the sidereal day, about four minutes shorter, because the planet has moved along its orbit and has to turn a little extra to face the Sun again. Using 86 400 gives $7.272\times10^{-5}$ rad/s, a 0.3% error — negligible for a rough estimate and quite unacceptable for satellite work.