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

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

in the

International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...

(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. It is defined such that one radian is the angle subtended at ...

per

second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...

is also the SI unit of

angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...

(symbol ''ω'', omega). The radian per second is defined as the angular frequency that results in the

angular displacement The angular displacement (symbol θ, , or φ) – also called angle of rotation, rotational displacement, or rotary displacement – of a physical body is the angle (in units of radians, degrees, turns, etc.) through which the body rotates ( ...

increasing by one radian every second.

Relation to other units

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

of one

hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...

(1 Hz), or one

cycle per second The cycle per second is a once-common English name for the unit of frequency now known as the ''hertz'' (Hz). Cycles per second may be denoted by c.p.s., c/s, or, ambiguously, just "cycles" (Cyc., Cy., C, or c). The term comes from repetitive p ...

(1 cps), corresponds to an angular frequency of 2 radians per second. This is because one cycle of rotation corresponds to an angular rotation of 2 radians. Since the radian is a dimensionless unit in the SI, the radian per second is dimensionally equivalent to the hertz—both can be expressed as reciprocal seconds, s⁻¹. So, context is necessary to specify which

kind of quantity A physical quantity (or simply quantity) is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a '' numerical value'' and a '' ...

is being expressed, angular frequency or ordinary frequency. One radian per second also corresponds to about 9.55

revolutions per minute Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or r⋅min−1) is a unit of rotational speed (or rotational frequency) for rotating machines. One revolution per minute is equivalent to hertz. Standards ISO 80000-3:2019 de ...

(rpm). ''Degrees per second'' may also be defined, based on degree of arc, where 1 degree per second (°/s) is equivalent to rad⋅s⁻¹. :

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) is a standard system of Quantity, quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This ...

(SI) and the

, widely used in

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

and

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

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

torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

''τ'' applied to the shaft: . When coherent units are used for these quantities, which are respectively the

watt The watt (symbol: W) is the unit of Power (physics), 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 quantification (science), quantify the rate of Work ...

, the radian per second, and the

newton-metre The newton-metre or newton-meter (also non-hyphenated, 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 fr ...

, 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 t ...

, turn/min, and pound-foot), an additional factor will generally be necessary.

References

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

Relation to other units

Coherent units

See also

References