radian per second

The radian per second is the speed every rotational formula is written in, and a few anchors make its magnitudes readable: a clock's second hand turns at $2\pi/60 \approx 0.105$ rad/s, a 3,000 rpm motor at 314 rad/s, and a large wind turbine rotor at only 1 to 2 rad/s. That last number explains a common surprise, because $v = \omega r$ over a 50 m blade puts the tip of that slow-looking machine at 80 m/s or more. Power follows the same way: $P = \tau\omega$ is how the rotational-mechanics solvers on this site turn a nameplate speed and a torque into watts.

One value in this unit is a constant of the planet. The Earth turns at $\Omega = 7.292 \times 10^{-5}$ rad/s, and the figure looks slightly wrong if you compute $2\pi$ over the 86,400 s day, because the true rotation period is the sidereal day of 86,164 s, the Earth's turn relative to the stars rather than the Sun. That $\Omega$ is the ingredient in the Coriolis parameter that weather models and long-range ballistics corrections are built on, and it is why cyclones spin opposite ways in the two hemispheres.