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[[Berkas:Angular velocity.svg|thumbjmpl|250px|Kecepatan sudut menjelaskan laju dan orientasi [[rotasi]] suatu benda pada sumbunya. Arah vektor kecepatan sudut berimpit dengan sumbu rotasi; pada gambar ini (rotasi berlawanan arah jarum jam) vektor naik.]]
Di dalam [[fisika]], '''kecepatan sudut''' adalah besaran [[vektor (spasial)|vektor]] (lebih tepatnya, [[vektor semu]]) yang menyatakan [[frekuensi sudut]] suatu benda dan sumbu putarnya. Satuan [[Sistem Satuan Internasional|SI]] untuk kecepatan sudut adalah [[radian per detik]], meskipun dapat diukur pula menurut derajat per detik, rotasi per detik, derajat per jam, dan lain-lain. Ketika diukur dalam putaran per waktu (misalnya [[rotasi per menit]]), kecepatan sudut sering dikatakan sebagai kecepatan rotasi dan besaran skalarnya adalah [[laju rotasi]]. Kecepatan sudut biasanya dinyatakan oleh simbol [[omega]] ('''Ω''' atau '''ω'''). Arah vektor kecepatan sudut adalah tegak lurus dengan bidang rotasi, dalam arah yang biasa disebut [[kaidah tangan kanan]].<ref name= EM1>{{cite book
|last = Hibbeler
|first = Russell C.
|authorlink =
|coauthors =
|title = Engineering Mechanics
|publisher = Pearson Prentice Hall
|date = 2009
|location = Upper Saddle River, New Jersey
|pages = 314, 153
|url =http://books.google.com/books?id=tOFRjXB-XvMC&pg=PA314&dq=angular+velocity&rview=1
|doi =
|id =
|isbn = 9780136077916}}(EM1)</ref>
== Kecepatan sudut suatu partikel ==
=== Dimensi dua ===
[[Berkas:AngularVelocity01.png|rightka|256 px|thumbjmpl|Kecepatan sudut suatu partikel pada P relatif terhadap titik asal O ditentukan oleh [[komponen tangensial dan normal]] vektor kecepatan '''v'''.]]
Kecepatan sudut suatu partikel di dalam bidang dua dimensi adalah yang paling mudah dipahami. Seperti yang ditunjukkan pada gambar di kanan (biasanya menyatakan ukuran sudut ''φ'' dan ''θ'' di dalam [[radian]]), jika garis dilukiskan dari titik asal (O) ke partikel yang dimaksud (P), maka vektor kecepatan ('''v''') partikel akan memiliki komponen sepanjang jari-jari ([[komponen jari-jari]], '''v'''<sub>∥</sub>) dan komponen yang tegak lurus dengan jari-jari ([[komponen silang jari-jari]], '''v'''<sub><math>_\perp</math></sub>). TetapiNamun, harus diingat bahwa vektor kecepatan dapat juga diuraikan menjadi [[komponen tangensial dan normal]].
Gerak radial (gerak memancar) tidak menghasilkan perubahan jarak partikel terhadap titik asal; sehingga untuk menentukan kecepatan sudut, komponen sejajar (radial) dapat diabaikan. Oleh karena itu, rotasi sepenuhnya dihasilkan oleh gerak tangensial (seperti yang terjadi pada partikel yang bergerak pada lingkaran), dan kecepatan sudut sepenuhnya ditentukan oleh komponen tegak lurus (tangensial).
:<math>\boldsymbol\omega=\frac{\mathbf{r}\times\mathbf{v}}{|\mathrm{\mathbf{r}}|^2}</math>
== Kerangka berputar ==
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[[Euler's rotation theorem]] states that, in an instant, for any dt there always exists a momentary axis of rotation. Therefore, any transversal section of the body by a plane perpendicular to this axis has to behave as a two dimensional rotation. The angular speed vector will be defined over the rotation axis ([[eigenvector]] of the [[linear map]]), and such as its value is the derivative of the angle rotated with respect to time.
:<math>\omega = \frac{d\theta}{dt}</math>
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=== Dimensi yang lebih tinggi ===
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In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank [[skew-symmetric]] tensor. This tensor will have n(n-1)/2 independent components and this number is the dimension of the [[Lie algebra]] of the [[Lie group]] of [[rotations]] of an ''n''-dimensional inner product space.<ref name="baez"> [http://math.ucr.edu/home/baez/classical/galilei2.pdf Rotations and Angular Momentum] on the Classical Mechanics page of [http://math.ucr.edu/home/baez/README.html the website of John Baez], especially Questions 1 and 2.</ref> It turns out that in three dimensional space angular velocity can be represented by vector because number of independent components is equal to number of dimensions of space.
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== Kecepatan sudut suatu benda tegar ==
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[[Image:AngularVelocity02.png|right|256 px|thumb|Position of point P located in the rigid body (shown in blue). '''R'''<sub>i</sub> is the position with respect to the lab frame, centered at ''O'' and '''r'''<sub>i</sub> is the position with respect to the rigid body frame, centered at ''O' '' . The origin of the rigid body frame is at vector position '''R''' from the lab frame.]]
In order to deal with the motion of a [[rigid body]], it is best to consider a coordinate system that is fixed with respect to the rigid body, and to study the coordinate transformations between this coordinate and the fixed "laboratory" system. As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is '''R'''. A particle (''i'') in the rigid body is located at point P and the vector position of this particle is '''R'''<sub>i</sub> in the lab frame, and at position '''r'''<sub>i</sub> in the body frame. It is seen that the position of the particle can be written:
:<math>\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i</math>
The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector <math>\mathbf{r}_i</math> is unchanging. By [[Euler's rotation theorem]], we may replace the vector <math>\mathbf{r}_i</math> with <math>\mathcal{R}\mathbf{r}_{io}</math> where <math>\mathcal{R}</math> is a [[rotation matrix]] and <math>\mathbf{r}_{io}</math> is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix <math>\mathcal{R}</math> which is changing in time and not the reference vector <math>\mathbf{r}_{io}</math>, as the rigid body rotates about point O'. The position of the particle is now written as:
:<math>\mathbf{R}_i=\mathbf{R}+\mathcal{R}\mathbf{r}_{io}</math>
Taking the time derivative yields the velocity of the particle:
:<math>\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathbf{r}_{io}</math>
where '''V'''<sub>i</sub> is the velocity of the particle (in the lab frame) and '''V''' is the velocity of O' (the origin of the rigid body frame). Since <math>\mathcal{R}</math> is a rotation matrix its inverse is its transpose. So we substitute <math>\mathcal{I}=\mathcal{R}^T\mathcal{R}</math>:
:<math>\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{I}\mathbf{r}_{io}</math>
:<math>\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}^T\mathcal{R}\mathbf{r}_{io}</math>
:<math>\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}^T\mathbf{r}_{i}</math>
Continue by taking the time derivative of <math>\mathcal{R}\mathcal{R}^T</math>:
:<math>\mathcal{I}=\mathcal{R}\mathcal{R}^T</math>
:<math>0=\frac{d\mathcal{R}}{dt}\mathcal{R}^T+\mathcal{R}\frac{d\mathcal{R}^T}{dt}</math>
Applying the formula ('''AB''')<sup>T</sup> = '''B'''<sup>T</sup>'''A'''<sup>T</sup>:
:<math>0=\frac{d\mathcal{R}}{dt}\mathcal{R}^T+\left(\frac{d\mathcal{R}}{dt}\mathcal{R}^T\right)^T</math>
<math>\frac{d\mathcal{R}}{dt}\mathcal{R}^T</math> is the negative of its transpose. Therefore it is a [[Skew-symmetric matrix|skew symmetric 3x3 matrix]]. We can therefore take its dual to get a 3 dimensional vector. <math>\frac{d\mathcal{R}}{dt}\mathcal{R}^T</math> is called the [[angular velocity tensor]]. If we take the dual of this tensor, matrix multiplication is replaced by the cross product. Its dual is called the angular velocity pseudovector, ω.
:<math>\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]</math>
Substituting ω into the above velocity expression:
:<math>\mathbf{V}_i=\mathbf{V}+\boldsymbol\omega\times\mathbf{r}_i.</math>
It can be seen that the velocity of a point in a rigid body can be divided into two terms - the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.
It is an '''important point''' that the spin angular velocity of every particle in the rigid body is the same, and that the spin angular velocity is independent of the choice of the origin of the rigid body system or of the lab system. In other words, it is a physically real quantity which is a property of the rigid body, independent of one's choice of coordinate system. The angular velocity of the reference point about the origin of the lab frame will, however, depend on these choices of coordinate system. It is often convenient to choose the [[center of mass]] of the rigid body as the origin of the rigid body system, since a considerable mathematical simplification occurs in the expression for the [[angular momentum]] of the rigid body.
If the reference point is the "Instantaneous axis of rotation" the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a pure rolling spherical rigid body.
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== Lihat pula ==
== Pranala luar ==
[http://books.google.com/books?id=QBc5AAAAMAAJ&pg=PA88&dq=angular+velocity+of+a+particle&lr=&rview=1 Buku kuliah fisika] oleh Arthur Lalanne Kimball (''Angular Velocity of a [[Particle|<sup>particle</sup>]]'')
[[Kategori:Persamaan mekanika klasik]]
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