 x  Encyclopedia
The radian per second is the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

unit of angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

, commonly denoted by the Greek letter ω (omega). The radian per second is also the unit of angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

.

The radian per second is defined as the change in the orientation
Orientation (rigid body)
In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it is in....

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s, every second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

.
Angular frequency ω (Ordinary) frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency... 2π radians per second exactly 1 hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

(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
Revolutions per minute
Revolutions per minute is a measure of the frequency of a rotation. It annotates the number of full rotations completed in one minute around a fixed axis...

Note that because the radian is a dimensionless unit, the radian per second is dimensionally equivalent to the hertz. 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
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

, widely used in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, the power, p, delivered to the shaft is given by the product of ω (in radians per second) times the torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

, τ, in newton-meters applied to the shaft. Thus, p = ω • τ, and the unit is the watt
Watt
The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...

, with no numerical coefficient needed.

In other systems, the calculation is somewhat more complicated, because if one multiplies angular velocity in revolutions per minute
Revolutions per minute
Revolutions per minute is a measure of the frequency of a rotation. It annotates the number of full rotations completed in one minute around a fixed axis...

(r.p.m.) times the torque in pound-feet
Pound-foot (torque)
A pound-foot is a unit of torque . One pound-foot is the torque created by one pound force acting at a perpendicular distance of one foot from a pivot point....

, then a multiplicative constant is needed to give the result in horsepower
Horsepower
Horsepower is the name of several units of measurement of power. The most common definitions equal between 735.5 and 750 watts.Horsepower was originally defined to compare the output of steam engines with the power of draft horses in continuous operation. The unit was widely adopted to measure the...

.

• Order of magnitude (angular velocity)
• Cycle per second
Cycle per second
The cycle per second was a once-common unit of frequency.With the organization of the International System of Units in 1960, the cycle per second was officially replaced by the hertz, or reciprocal second—i.e. the cycle in 'cycle per second' was dropped...

• Normalized frequency
Normalized frequency (digital signal processing)
In digital signal processing, the normalized frequency of a periodic signal is its frequency expressed in units of cycles per sample, rather than in the usual SI units of hertz... 