Radian per second

Radian per second
Angular frequency ω (in radians per second), is smaller than frequency ν (in cycles per second, also called Hz), by a factor of 2π, because 1 rad/s corresponds to 2π Hz.
General information
Unit system	SI
Unit of	rotational speed
Symbol	rad/s, rad⋅s−1

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

The angular frequency of one radian per second corresponds to a frequency of 1/(2π) hertz (Hz), or cycles per second. This is because one cycle of rotation corresponds to an angular rotation of one turn (360 degrees), which equals 2π radians. Since the radian is a dimensionless unit in the SI, the radian per second is dimensionally equivalent to the hertz—both are defined as s⁻¹.

One radian per second also corresponds to about 9.55 revolutions per minute.

Quantity correspondence

Angular frequency ω	Frequency $\nu =\omega /{2\pi }$
2π rad/s	1 Hz
1 rad/s	≈ 0.159155 Hz
1 rad/s	≈ 9.5493 rpm
0.1047 rad/s	≈ 1 rpm

A use of the unit radian per second is in calculation of the power transmitted by a shaft. In the International System of Units, widely used in physics and engineering, the power p is equal to the rotational speed ω (with the unit radian per second) multiplied by the torque τ applied to the shaft, in newton-metres. Thus, p = ω ⋅ τ, and the unit is the watt, with no numerical coefficient needed. In other systems, an additional factor may be necessary. For example, if one multiplies rotational speed, which has the unit revolution per minute (rpm), by the torque in pound-feet, then a factor is needed to convert the result to units of horsepower.

See also

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