Radian per second

Radian per second
Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π, because 2π rad/s corresponds to 1 Hz.
Unit information
Unit system	SI derived unit
Unit of	Rotational speed
Symbol	rad/s or rad•s−1

The radian per second (symbol: rad·s⁻¹ or rad/s) is the SI unit of rotational speed (angular velocity), commonly denoted by the Greek letter ω (omega). The radian per second is also the unit of angular frequency. The radian per second is defined as the change in the orientation of an object, in radians, every second.

Angular frequency ω	(Ordinary) frequency $\nu = \omega/{2\pi}$
2π radians per second	exactly 1 hertz (Hz)
1 radian per second	approximately 0.159155 Hz
1 radian per second	approximately 57.29578 degrees per second
1 radian per second	approximately 9.5493 revolutions per minute (rpm)
0.1047 radian per second	approximately 1 rpm

Note that because the radian is a dimensionless unit, the radian per second is dimensionally equivalent to the hertz—both are defined as one s⁻¹. This means that great care must be taken to avoid confusing angular frequency ω and frequency ν.

One of the important uses of the unit radian per second is in calculation of the power transmitted by a shaft. In the International System, widely used in physics and engineering, the power, p, delivered to the shaft is given by the product of ω (in radians per second) times the torque, τ, in newton-meters applied to the shaft. Thus, p = ω • τ, and the unit is the watt, with no numerical coefficient needed.

In other systems, the calculation is somewhat more complicated, because if one multiplies angular velocity in revolutions per minute (r.p.m.) times the torque in pound-feet, then a multiplicative constant is needed to give the result in horsepower.

An angular frequency, ω = 1 rad/s, corresponds to an ordinary frequency, ν = 1/2π Hz = 0.159 Hz approx. which in turn corresponds to a frequency of rotation = 60/2π rpm = 9.55 rpm approx.

See also

• Cycle per second
• Normalized frequency
• Order of magnitude (angular velocity)