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| {{lead too long|date=September 2013}}
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| {{Classical mechanics|cTopic=Fundamental concepts}}
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| [[Image:angulardisplacement1.jpg|300px|right|thumb|Rotation of a rigid object ''P'' about a fixed object about a fixed axis ''O''.]]
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| '''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]].
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| 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]].
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| 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:
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| :<math>
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| s=r\theta \,</math>
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| ==Measurements of angular displacement==
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| 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.
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| :<math>\theta=\frac sr</math>
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| 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''
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| 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.
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| <!-- 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.]] -->
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| 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'''.
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| == Three dimensions ==
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| 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).
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| 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>
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| === Matrix notation ===
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| 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>
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| == References ==
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| <references />
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| ==See also==
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| *[[Second moment of area]]
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| *[[Linear elasticity]]
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| *[[Rotation_matrix#Infinitesimal_rotations|Infinitesimal rotation]]
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| *[[Angular distance]]
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| [[Category:Angle]]
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Nothing to write about myself I think.
Great to be a member of this site.
I just wish I'm useful in some way .
Feel free to surf to my weblog system mechanic