Angular Velocity and Angular Frequency

In two steps: first divide the rotational speed by 60 to get the frequency in revolutions per second (Hz), then multiply by 2π, because each revolution corresponds to the angle 2π in radians. Compactly: ω = 2π·n/60 with n in rpm. Drill example: 3000 rpm gives f = 50 Hz and ω = 2π × 50 ≈ 314 rad/s. As a rough rule of thumb ω ≈ n/10 (more precisely n × 0.1047). The detour through radians is necessary because all rotation formulas, such as v = ω·r or a = ω²·r, are only valid with ω in rad/s.

The frequency f counts full revolutions or oscillations per second and is measured in hertz. The angular frequency ω instead measures the swept angle per second in radians and is larger by the factor 2π: ω = 2π·f. Both describe the same pace, only in different counting schemes: one in laps, one in angle. The factor 2π is not a cosmetic detail but computationally decisive: in v = ω·r, in the centripetal acceleration a = ω²·r and in the oscillation law x(t) = A·sin(ωt) it must always be ω. Whoever inserts f there is off by a factor of 6.28, or its square.

Through the radius: v = ω·r. The angular velocity describes the rotation pace and is the same for every point of a rigid body; the orbital speed describes how fast a specific point actually travels through space and grows linearly with the distance from the axis. On a carousel with T = 4 s, ω = 2π/4 ≈ 1.57 rad/s everywhere, but a seat at r = 3 m moves at 4.7 m/s while a child at r = 1 m moves at only 1.6 m/s. The same principle explains why the blade tips of a wind turbine exceed 200 km/h although the rotor looks leisurely.

Radians are defined so that the angle equals the ratio of arc length to radius: φ = s/r. Only with this definition does the relation s = φ·r become a simple multiplication, and its time derivative directly yields v = ω·r without extra conversion factors. Working in degrees would drag the factor π/180 into every formula. A full circle corresponds to 2π rad ≈ 6.283 rad, and one radian is about 57.3°. The derivative rules for sin and cos also hold only in radians. Therefore: switch the calculator to RAD mode for rotation and oscillation problems, and back to DEG for geometry tasks stated in degrees.

The Earth needs one sidereal day of 86,164 s per revolution, so ω = 2π/86,164 ≈ 7.29×10⁻⁵ rad/s (using 24 h: 7.27×10⁻⁵). That sounds tiny, but the large Earth radius turns it into an equatorial orbital speed of v = ω·r ≈ 465 m/s, more than 1600 km/h. With geographic latitude the orbital radius shrinks (r·cos φ); in central Europe it is still about 300 m/s, at the poles zero. Consequences of this rotation: launch sites are preferably placed near the equator to collect the speed bonus, the centrifugal effect reduces effective gravity at the equator by about 0.3 %, and the Coriolis force deflects large-scale winds and ocean currents.