# Angular Displacement

If something is rotating about a point and it covers a complete circle, should we take its angular displacement as 360 degree or 0?

Please give link to some established material on this subject which defines your answer, whether it is that it should be taken as 0 or 360 degrees.

Question that led to this problem : The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. 1. What is its angular acceleration, assuming the acceleration to be uniform, 2. How many revolutions does the wheel make during this time.

In the solution to above question the author(s) for solving part 2 have used equation of motion and mentioned that they have obtained angular displacement for it, it basically implied that the equations of motion used for rotation provide angular displacement which does not become zero on returning to original point which is not followed when we consider it analogous to displacement of linear motion.

Addendum : Even here ( Angular displacement and the displacement vector ) the selected answer says that on completing the circle, angular displacement is 360 degrees, is there some established text to support this ?

Similarly here ( http://www.ask.com/question/calculating-angular-displacement ) something else is said in the answer, is then angular displacement ambigous and hence has no correct definition, if there is please guide me to some established text.

• If going the full circle brings the system back to where it started, it doesn't matter. If the motion is spiral, then you need to take 360 degrees – Pranav Hosangadi Nov 19 '13 at 8:19
• Why does it matter? If the angle is $\theta$ or $\theta + n\, 2\pi$ for any integer $n$ it is still the same angle. – John Alexiou Nov 19 '13 at 13:14
• I guess I want more details on the specifics of the problem to understand what sparked the question. – John Alexiou Nov 19 '13 at 16:02
• @rijul gupta as pointed by others if you replace $\theta$ from 0 to $2\pi$ the trigonometric equations will not change but if you say what is value of $\theta$ after a rotation then it is technically wrong to say :$\theta=0$. – user31782 Nov 27 '13 at 6:47

In most cases when we're dealing with angles we are using the trigonometric functions, and since these are periodic in angle with period $2\pi$ it doesn't matter whether you use zero, $2\pi$ or any multiple of $2\pi$ as your equations will give the same result.

Alternatively you could be describing some object moving in a circle in an external field e.g. a gravitational field, and again most of the time tracing one circle is the same as tracing any number of circles. This is true of all conservative fields.

The exception is in electrodynamics e.g. when you're a charged object moving in a circle, because in that case you will be generating a magnetic field and each revolution of the circle puts energy into the magnetic field. In that case how many times you go round the circle does matter.

Re the edit to the question:

Aha, you're mixing up two different concepts. The angle can mean the position or it can mean the total angle moved. Let me attempt to given example. Suppose you walk 1 km north then turn round and walk 1 km south back to where you started. Then your position in space hasn't changed, but you have still walked 2 kms. Likewise if you rotate an object by $2\pi$ its angle hasn't changed, but it has still travelled through $2\pi$ radians.

In the question you cite the total angle moved is just the integral of angular velocity wrt time, just as in linear motion the total distance moved is the integral of velocity wrt time.

Basically there are three different definitions assigned to the term angle as explained here. I am not familiar with the notations used in the link that I've given. So I cannot comment on it.
Your question is completely legitimate because in some circumstances the angle is taken as 0 degree if something is been rotated one complete circle(e.g. in case when we are talking of angle between vectors.)
When we use the term angle in the definition of Angular displacement$^{}$ we are infact referring to the term rotated angle for the term angle. So in the situation that you have depicted in your question the definition of an angle as given in the established-text$^{}$ is applicable for the term angle.
From the definition of an angle given in the referred established text it is clear that if something is rotating about a point and it covers a complete circle, we should take its angular displacement as 360 degree not 0 degree.

"If something is rotating about a point and it covers a complete circle, should we take its angular displacement as 360 degree or 0?"
We should take its angular displacement as 360 degree not 0 degree because of its(angular displacement's) definition.

You also say "Is angular displacement ambiguous?"
No, suppose something covers one rotation in anticlockwise direction then we say it is rotated through an angle $+2\pi$ but if it is suddenly rotated back in clockwise direction we say it has covered an angle $-2\pi$ now so the net angle it covers is $0$. this is what displacement is all about i.e net change in parameters which is now $\theta$.

${}^{}$Definition of Angular displacement:
Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis.

## References

A Paragraph written in the Book: Fundamentals of Physics (8th Edition)
"we do not reset $\theta$ to zero with each compelete rotation of reference line about the rotation axis. if the reference line completes two rotations from the zero angular position then the angular position of the line is $\theta = 4\pi$ rad."

  Established text


"Unified algebra and trigonometry (Addison-Wesley mathematics series) article 3-5"

${}^{}$3-5 Angles.
In geometry an angle has usually been defined as the configuration consisting of two half-lines (rays) radiating from a point. However in trigonometry we generalize the definition by stating that an angle thus defined by two half lines has a measure which corresponds to the amount of rotation required to move a ray from the position of one of these lines to the other. consider the figure$^{}$ with two lines $m$ and $n$ intersecting at $o$ and lying in a plane prependicular to our line of vision. if we consider $m$ as initial line and $n$ as terminal side of the angle $o$ as its vertex, there are two possible directions of rotation of the initial side $m$. The angle is said to be positive if the rotation is counter clockwise, but negative if clockwise. A curved arrow will indicate the direction of rotation.
let us now consider a ray m which issues from the origin of a rectangular coordinate system and coincide with the positive x-axis(Fig 3-9). As this ray rotates, any point $P$ on $m$ will trace out part or all of the circumference of circle of radius $OP$. In fact the circumference may be traced several times.
After the rotation $OP$ will be in some position $OP'$, where the circular arc $\stackrel \frown {PP}^{'}$ denoted by $s$, may be used to measure the $POP^{'}$ is said to be standard position, and to be in quardrant in which its terminal side $OP^{'}$ is located .

The most logical units for measuring the magnitude of an angle $POP^{'}$. An angle would seem to be the number of revolutions due to the rotation from the initial to the terminal side of the angle. since the number of revolutions of any angle is determined by the ratio of the intercepted circular arc length $s$ to circumference of the circle we define, magnitude of an angle in revolutions as

Angle in revolutions$= \frac{s}{2\pi r}$
For example if $P$ traces out an arc half the circumference, the corresponding angle is one of one-half a revolution. Likewise if arc is twice the circumference, the angle measure is two revolutions.
Consider the two coencentric circles at $o$, in Fig.3-10, with $\stackrel \frown {PP}^{'}$ as arc of length $s$ on the circle of radius $r$, and $QQ^{'}$ an arc of length $s^{'}$ on the circle of radius $r{'}$. By using the theorem that similar triangles have propotional sides, and recalling the definition of arc length from article 3.4 it can be proved that
$$\frac{s^{'}}{r^{'}} = \frac{s}{r},$$ and therefore,

$$\frac{s^{'}}{2\pi r^{'}} = \frac{s}{2\pi r}$$
The magnitude of any angle is thus independent of the length of its initial or terminal side. Although the use of revolutions is the most natural method for measuring angles, there are other more convenient systems.
The system most commonly used in elementary work such as survyeing and nevigation is sexagesimal, in which the degree is the fundamental unit. In this system one revolution = $360^0$, $1^0 = 60^0$(minutes) and $1^{'} = 60^{''}$(seconds)
Angle in degrees = (revolutions) $360^0$
For example, one-half a revolution is $180^0$, or an angle of two revolutions is $720^0$.
The other important system used in ... ...
Angle in radians = (revolution) $2\pi$
__________________________

${}^{}$I could not add the figure. Please refer to the original context.

• This answer is confusing and it is hard to find the answer to the question inside a maze of quotes. While citing your sources is good, it is not very helpful to simply provide a wall of (pretty convoluted) text upfront. An alternative course of action, which may be easier to read, is to provide a clear answer that concisely answers the question, and which is visually easy to recognize, visibly separated from your sources. Additionally, it would help if each of your references was visually separated from the rest. The paragraphing is also very confusing and disorienting. – Emilio Pisanty Mar 14 '14 at 12:09
• @EmilioPisanty The quotes are written because the OP asked for the definitions(with references) in his comments. I have edited my answer. If it is still confusing then tell me. – user31782 Mar 14 '14 at 12:47
• Please re-read my comment. Yes, the answer is still confusing. You need to reference your sources, not make an answer that is made up entirely of quotes. The answer should be at the top, and you should then proceed to back it up with references. – Emilio Pisanty Mar 14 '14 at 12:49
• Try, for example, this type of approach. – Emilio Pisanty Mar 14 '14 at 13:43
• This answer is complete in itself. I have written whole of the text of the reference article because it is not available on internet. The quoted established material is the fundamental answer to the question. There is no need of further explanation. References were asked - references have been given. Definition was asked - it has been given. This answer has some formatting errors which will be corrected tomorrow, I am not allowed to edit more today. – user31782 Mar 14 '14 at 14:11

An angle is a ratio. The ratio of arc length to radius. So if you want to consider zero arc length or full circle is really up to you. It depends on the situation. If you are using the angle for orientation it does not matter, but if you are using it for angular displacement it depends on the initial conditions.

• '-1' your answer is ambiguous. angle is not always the ratio it is expressed as an inverse trigonometric function which are multivalued when we talk about polar coordinates. can you first give a proper definition of arc length and then show from the definition it is zero. also can you elaborate when we use angle for orientation. – user31782 Dec 13 '13 at 13:11