The radian per second (symbol: rad⋅s⁻¹ or rad/s) is the unit of

angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

in the

International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...

(SI). The

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

per second is also the SI unit of

angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...

, commonly denoted by the Greek letter ''ω'' (omega). The radian per second is defined as the angular frequency that results in the

angular displacement Angular displacement of a body is the angle (in radians, degrees or revolutions) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed as ...

increasing by one

every

second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...

. The angular frequency of one radian per second corresponds to a

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

of 1/(2)

hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...

(Hz), or cycles per second. This is because one cycle of rotation corresponds to an angular rotation of one turn (360 degrees), which equals 2 radians. Since the radian is a

dimensionless unit A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

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 Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines. Standards ISO 80000-3:2019 defines a unit of rotation as the dimensionl ...

. :

Coherent units

A use of the unit radian per second is in calculation of the power transmitted by a shaft. In the

International System of Quantities The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underlie ...

(SI) and the

, widely used in

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

and

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

, the power ''p'' is equal to the rotational speed ''ω'' multiplied by the

torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...

''τ'' applied to the shaft: . When

coherent units A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, inc ...

are used for these quantities, which are respectively the

watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...

, the radian per second, and the

newton-metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...

, and thus , no numerical factor needed when performing the numerical calculation. When the units are not coherent (e.g.

horsepower Horsepower (hp) is a unit of measurement of power, or the rate at which work is done, usually in reference to the output of engines or motors. There are many different standards and types of horsepower. Two common definitions used today are the ...

, turn/min, and

pound-foot A pound-foot (lbf⋅ft) is a unit of torque representing one pound of force acting at a perpendicular distance of one foot from a pivot point. Conversely one pound-foot is the moment about an axis that applies one pound-force at a radius of one ...

), an additional factor will generally be necessary.

References

SI derived units Units of angular velocity {{physics-stub

Coherent units

See also

References