Turn (angle)

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The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to Template:Math radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c)<ref name="Fitzpatrick_2021" /> or to one revolution (symbol rev or r).<ref name="IET_2016" /> Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm).Template:Efn The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. Template:Xref

Because one turn is <math>2\pi</math> radians, some have proposed representing <math>2\pi</math> with the single letter tau (<math>\tau</math>).

Unit symbolsEdit

There are several unit symbols for the turn.

EU and SwitzerlandEdit

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: {{#invoke:Lang|lang}} 'full angle') for turns.<ref name="German_2013"/><ref name="Kurzweil_1999"/> Covered in Template:Ill (October 2010), the so-called {{#invoke:Lang|lang}} ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU<ref name="EWG_1980"/><ref name="EG_2009"/> and Switzerland.<ref name="Einheitenverordnung_1994"/>

CalculatorsEdit

The scientific calculators HP 39gII and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to newRPL for the HP 50g in 2016, and for the hp 39g+, HP 49g+, HP 39gs, and HP 40gs in 2017.<ref name="Lapilli_2016"/><ref name="Lapilli_2018"/> An angular mode TURN was suggested for the WP 43S as well,<ref name="Paul_2016"/> but the calculator instead implements "MULTemplate:Pi" ([[multiples of π|multiples of Template:Pi]]) as mode and unit since 2019.<ref name="Bonin_2019_OG"/><ref name="Bonin_2019_RG"/>

DivisionsEdit

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Many angle units are defined as a division of the turn. For example, the degree is defined such that one turn is 360 degrees.

Using metric prefixes, the turn can be divided in 100 centiturns or Template:Val milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″.<ref name="Hoyle_1962" /><ref name="Klein_2012" /> A protractor divided in centiturns is normally called a "percentage protractor". While percentage protractors have existed since 1922,<ref name="Croxton_1992" /> the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962.<ref name="Hoyle_1962" /><ref name="Klein_2012" /> Some measurement devices for artillery and satellite watching carry milliturn scales.<ref name="Schiffner_1965" /><ref name="Hayes_1975" />

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of Template:Sfrac turn. The binary degree, also known as the binary radian (or brad), is Template:Sfrac turn.<ref name="Savage_2007" /> The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into Template:Math equal parts for other values of Template:Mvar.<ref name="Hargreaves_2010" />

Unit conversionEdit

File:2pi-unrolled.gif

The circumference of the unit circle (whose radius is one) is Template:Math.

One turn is equal to <math>2\pi</math> = <math>\tau</math> ≈ Template:Val<ref name="OEIS2C_A019692" /> radians, 360 degrees, or 400 gradians.

Conversion of common angles
Turns	Radians		Degrees	Gradians
0 turn	0 rad		0°	0^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	5°	Template:Sfrac^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	15°	Template:Sfrac^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	22.5°	25^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	30°	Template:Sfrac^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	36°	40^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	45°	50^g
Template:Sfrac turn	1 rad		Template:Circa 57.3°	Template:Circa 63.7^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	60°	Template:Sfrac^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	72°	80^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	90°	100^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	120°	Template:Sfrac^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	144°	160^g
Template:Sfrac turn	Template:Sfrac rad	Template:Pi rad	180°	200^g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	270°	300^g
1 turn	Template:Tau rad	2Template:Pi rad	360°	400^g

In the ISQ/SIEdit

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In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number of revolutions:<ref name="ISO80000-3_2019" />

N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:
<math>N = \frac{\varphi}{2 \pi \text{ rad}}</math>
where Template:Varphi denotes the measure of rotational displacement.

The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time),<ref name="ISO80000-3_2019" /> and adopted in the International System of Units (SI).<ref name="SIBrochure_9" /><ref name="NISTGuide_2009" />

Rotation count or number of revolutions is a quantity of dimension one, resulting from a ratio of angular displacement. It can be negative and also greater than 1 in modulus. The relationship between quantity rotation, N, and unit turns, tr, can be expressed as:

<math>N = \frac \varphi \text{tr} = \{ \varphi \}_\text{tr}</math>

where {Template:Varphi}_tr is the numerical value of the angle Template:Varphi in units of turns (see Template:Slink).

In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by Template:Mvar:

<math>n = \frac{\mathrm{d}N}{\mathrm{d}t}</math>

The SI unit of rotational frequency is the reciprocal second (s⁻¹). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

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Template:Anchor The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one",Template:Efn which also received other special names, such as the radian.Template:Efn Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.<ref name="ISO 80000-1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> "Cycle" is also mentioned in ISO 80000-3, in the definition of period.Template:Efn