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{{Classical mechanics|cTopic=Fundamental concepts}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[Image:angulardisplacement1.jpg|300px|right|thumb|Rotation of a rigid object ''P'' about a fixed object about a fixed axis ''O''.]]


'''Angular displacement''' of a body is the [[angle]] in [[radian]]s ([[degree (angle)|degree]]s, [[turn (geometry)|revolutions]]) through which a point or line has been rotated in a specified sense about a specified [[rotation|axis]]. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity and acceleration at any time (''t''). When dealing with the rotation of an object, it becomes simpler to consider the body itself rigid.  A body is generally considered rigid when the separations between all the particles remains constant throughout the objects motion, so for example parts of its mass are not flying off.  In a realistic sense, all things can be deformable, however this impact is minimal and negligible.  Thus the rotation of a rigid body over a fixed axis is referred to as [[rotational motion]].  
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Example==
Registered users will be able to choose between the following three rendering modes:  
In the example illustrated to the right, a particle on object P at a fixed distance ''r'' from the origin, ''O'', rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (''r'', ''θ'').  In this particular example, the value of ''θ'' is changing, while the value of the radius remains the same.  (In rectangular coordinates (''x'', ''y'') both ''x'' and ''y'' vary with time). As the particle moves along the circle, it travels an [[Arc (geometry)|arc length]] ''s'', which becomes related to the angular position through the relationship:


:<math>
'''MathML'''
s=r\theta \,</math>
:<math forcemathmode="mathml">E=mc^2</math>


==Measurements of angular displacement==
<!--'''PNG''' (currently default in production)
Angular displacement may be measured in [[radian]]s or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance ''r'' from the centre.
:<math forcemathmode="png">E=mc^2</math>


:<math>\theta=\frac sr</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


For example if an object rotates 360 degrees around a circle of radius ''r'', the angular displacement is given by the distance traveled around the circumference - which is 2π''r''
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
divided by the radius: <math>\theta= \frac{2\pi r}r</math> which easily simplifies to <math>\theta=2\pi</math>. Therefore 1 revolution is <math>2\pi</math> radians.


<!-- Image with unknown copyright status removed: [[Image:angulardisplacement2.jpg|250px|left|thumb|A particle that is rotating from point P to point Q along the acr of the circle.  In the time that elapses, the change in time is equal to the final time minus the original time, and the radius travels an angle theta, or the original angle subtracted from the final angle.]] -->
==Demos==


When object travels from point P to point Q, as it does in the illustration to the left, over <math>\delta t</math> the radius of the circle goes around a change in angle. <math>\Delta \theta = \Delta \theta_2 - \Delta \theta_1 </math>  which equals the '''Angular Displacement'''.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


== Three dimensions ==
In three dimensions, angular displacement is an entity with a direction and a magnitude.  The direction specifies the axis of rotation, which always exists by virtue of the [[Euler's rotation theorem]]; the magnitude specifies the rotation in [[radian]]s about that axis (using the [[right-hand rule]] to determine direction).


Despite having direction and magnitude, angular displacement is not a [[vector (geometry)|vector]] because it does not obey the [[commutative law]] for addition.<ref>{{cite book|last1=Kleppner|first1=Daniel|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|publisher=McGraw-Hill|year=1973|pages=288–89}}</ref>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


=== Matrix notation ===
==Test pages ==
Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being <math>A_0</math> and <math>A_f</math> two matrices, the angular displacement matrix between them can be obtained as <math>dA =  A_f . A_0^{-1}</math>


== References ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
<references />
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==See also==
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Second moment of area]]
*[[Url2Image|Url2Image (private Wikis only)]]
*[[Linear elasticity]]
==Bug reporting==
*[[Rotation_matrix#Infinitesimal_rotations|Infinitesimal rotation]]
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
*[[Angular distance]]
*[[Angular velocity]]
 
{{Classical mechanics derived SI units}}
 
[[Category:Angle]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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