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 ...
, 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
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 ...
per unit time (for example, in rotation) or the rate of change of the phase of a sinusoidal waveform (for example, in oscillations and waves), or as the rate of change of the argument of the sine function.
Angular frequency (or angular speed) is the magnitude of the
pseudovector quantity
angular velocity.
[(UP1)]
One
turn is equal to 2''π''
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 tha ...
s, hence
[
where:
*''ω'' is the angular frequency (unit: radians per second),
*''T'' is the ]period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
(unit: seconds),
*''f'' is the ordinary 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 ...
(unit: 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 o ...
) (sometimes ''ν'').
Units
In SI units
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...
, angular frequency is normally presented in 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 tha ...
s per second, even when it does not express a rotational value. The unit 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 o ...
(Hz) is dimensionally equivalent, but by convention it is only used for frequency ''f'', never for angular frequency ''ω''. This convention is used to help avoid the confusion that arises when dealing with quantities such as frequency and angular quantities because the units of measure (such as cycle or radian) are considered to be one and hence omitted in SI.
In digital signal processing, the frequency may be normalized by the sampling rate
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a value of the signal at a point in time and/or s ...
, yielding the normalized frequency.
Examples
Circular motion
In a rotating or orbiting object, there is a relation between distance from the axis, , tangential speed, , and the angular frequency of the rotation. During one period, , a body in circular motion travels a distance . This distance is also equal to the circumference of the path traced out by the body, . Setting these two quantities equal, and recalling the link between period and angular frequency we obtain:
Oscillations of a spring
An object attached to a spring can oscillate
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
. If the spring is assumed to be ideal and massless with no damping, then the motion is simple and harmonic with an angular frequency given by
where
*''k'' is the spring constant
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
,
*''m'' is the mass of the object.
''ω'' is referred to as the natural frequency (which can sometimes be denoted as ''ω''0).
As the object oscillates, its acceleration can be calculated by
where ''x'' is displacement from an equilibrium position.
Using "ordinary" revolutions-per-second frequency, this equation would be
LC circuits
The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
(''C'' measured in farads) and the inductance of the circuit (''L'', with SI unit henry
Henry may refer to:
People
*Henry (given name)
* Henry (surname)
* Henry Lau, Canadian singer and musician who performs under the mononym Henry
Royalty
* Portuguese royalty
** King-Cardinal Henry, King of Portugal
** Henry, Count of Portugal, ...
):[(LC1)]
Adding series resistance (for example, due to the resistance of the wire in a coil) does not change the resonant frequency of the series LC circuit. For a parallel tuned circuit, the above equation is often a useful approximation, but the resonant frequency does depend on the losses of parallel elements.
Terminology
Angular frequency is often loosely referred to as frequency, although in a strict sense these two quantities differ by a factor of 2.
See also
*Cycle per second
The cycle per second is a once-common English name for the unit of frequency now known as the hertz (Hz). The plural form was typically used, often written cycles per second, cycles/second, c.p.s., c/s, or, ambiguously, just cycles (Cy./Cyc.). The ...
* Radian per second
*Degree (angle)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle in which one full rotation is 360 degrees.
It is not an SI unit—the SI unit of angular measure is ...
*Mean motion
In orbital mechanics, mean motion (represented by ''n'') is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the ac ...
*Orders of magnitude (angular velocity)
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 ...
*Simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
References and notes
Related Reading:
*
{{DEFAULTSORT:Angular Frequency
Angle
Kinematic properties
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