Rotational frequency (also known as rotational speed or rate of rotation) of an object
rotating around an axis is the
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 rotation of the object. Its unit is
revolution 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 ...
(rpm),
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 ...
(cps), etc.
The symbol for rotational frequency is
(the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
lowercase letter
nu).
Tangential speed ''v'', rotational frequency
, and
radial distance
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...
''r'', are related by the following equation:
:
:
An algebraic rearrangement of this equation allows us to solve for rotational frequency:
:
:
Thus, the tangential speed will be directly proportional to ''r'' when all parts of a system simultaneously have the same ''ω'', as for a wheel, disk, or rigid wand. The direct proportionality of ''v'' to ''r'' is not valid for the
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s, because the planets have different rotational frequencies.
Rotational frequency can measure, for example, how fast a motor is running. ''Rotational speed'' is sometimes used to mean
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 ...
rather than the quantity defined in this article. Angular frequency gives the change in
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
per time unit, which is given with the unit
radian per second
The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omega) ...
in the SI system. Since 2π radians or 360 degrees correspond to a cycle, we can convert angular frequency to rotational frequency by
:
where
*
is rotational frequency, with unit cycles per second
*
is angular frequency, with unit radian per second or degree per second
For example, a
stepper motor
A stepper motor, also known as step motor or stepping motor, is a brushless DC electric motor that divides a full rotation into a number of equal steps. The motor's position can be commanded to move and hold at one of these steps without any posi ...
might turn exactly one complete revolution each second.
Its angular frequency is 360
degrees per second (360°/s), or 2π
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 (2π rad/s), while the rotational frequency is 60 rpm.
Rotational frequency is not to be confused with
tangential speed, despite some relation between the two concepts. Imagine a merry-go-round with a constant rate of rotation. No matter how close or far you stand from the axis of rotation, your rotational frequency will remain constant. However, your tangential speed does not remain constant. If you stand two meters from the axis of rotation, your tangential speed will be double the amount if you were standing only one meter from the axis of rotation.
See also
*
Angular velocity
*
Radial velocity
*
Rotation period
References
{{DEFAULTSORT:Rotational frequency
Kinematic properties
Temporal rates