NationMaster - Encyclopedia: Angular velocity

Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. The direction of the angular velocity vector will be along the axis of rotation; in this case (counter-clockwise rotation) the vector points toward the viewer.

In physics, the angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed at which an object is rotating along with the direction in which it is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, degrees per hour, etc. When measured in cycles or rotations per unit time (e.g. revolutions per minute), it is often called the rotational velocity and its magnitude the rotational speed. Angular velocity is usually represented by the symbol omega (Ω or ω). The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the right hand rule. angular velocity File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... angular velocity File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A sphere rotating around its axis. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... Look up vector in Wiktionary, the free dictionary. ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency Ï‰ (also called angular speed) is a scalar measure of rotation rate. ... Revolutions per minute (abbreviated rpm, RPM, r/min, or min-1) is a unit of frequency, commonly used to measure rotational speed, in particular in the case of rotation around a fixed axis. ... Rotational speed (sometimes called speed of revolution) indicates for example how fast the motor is running. ... Look up Î©, Ï‰ in Wiktionary, the free dictionary. ... The right hand rule is also an algorithm used to solve Mazes In mathematics and physics, the right-hand rule is a convention for determining relative directions of certain vectors. ...

$I n s e r t f o r m u l a h e r e$ == The angular velocity of a particle ==

1 Two dimensions
2 Higher dimensions
3 Angular velocity of a rigid body
4 See also
5 References

[edit] Two dimensions

The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector V .

The angular velocity of a particle in a 2-dimensional plane is the easiest to understand. As shown in the figure on the right, if we draw a line from the origin (O) to the particle (P), then the velocity vector ( $mathbf{v}$ ) of the particle will have a component along the radius ( $mathrm{v}_parallel,$ - the radial component) and a component perpendicular to the radius ( $mathrm{v}_perp$ - the tangential component). Image File history File links Size of this preview: 600 Ã— 599 pixel Image in higher resolution (1790 Ã— 1788 pixel, file size: 70 KB, MIME type: image/png) // File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Angular velocity ... Image File history File links Size of this preview: 600 Ã— 599 pixel Image in higher resolution (1790 Ã— 1788 pixel, file size: 70 KB, MIME type: image/png) // File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Angular velocity ... Illustration of tangential and normal components of a vector to a surface. ... Illustration of tangential and normal components of a vector to a surface. ...

A radial motion produces no rotation of the particle (relative to the origin), so for purposes of finding the angular velocity the parallel (radial) component can be ignored. Therefore, the rotation is completely produced by the tangential motion (like that of a particle moving along a circumference), and the angular velocity is completely determined by the perpendicular (tangential) component.

It can be seen that the rate of change of the angular position of the particle is related to the tangential velocity by:

$mathrm{v}_perp=r,frac{dphi}{dt}$

Defining ω=dφ/dt as the angular velocity, and realizing that $mathrm{v}_perp$ is equal to $|mathrm{mathbf{v}}|,sin(theta)$ where θ is the angle between vectors r and v yields:

$omega=frac{|mathrm{mathbf{v}}|sin(theta)}{|mathrm{mathbf{r}}|}$

In two dimensions the angular velocity is a single number which has no direction. A single number which has no direction is either a scalar or a pseudoscalar, the difference being that a scalar does not change its sign when the x and y axes are exchanged (or inverted), while a pseudoscalar does. The angle as well as the angular velocity is a pseudoscalar. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis. If the axes are inverted, but the sense of a rotation does not, then the sign of the angle of rotation, and therefore the angular velocity as well, will change. A scalar may be: Look up scalar in Wiktionary, the free dictionary. ... In mathematics, a pseudoscalar in a geometric algebra is the highest-grade basis element of the algebra. ...

It is important to note that the pseudoscalar angular velocity of a particle depends upon the choice of the origin and upon the orientation of the coordinate axes.

=== Three dimensions === :)

In three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a pseudovector. It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The right hand rule indicates the positive direction of the angular velocity pseudovector, namely: Look up vector in Wiktionary, the free dictionary. ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... The right hand rule is also an algorithm used to solve Mazes In mathematics and physics, the right-hand rule is a convention for determining relative directions of certain vectors. ...

If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb.

Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line perpendicular to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Define a unit vector $hat{n}$ which points in the direction of the angular velocity pseudovector. The angular velocity may be written in a manner similar to that for two dimensions:

$boldsymbolomega=frac{|mathrm{mathbf{v}}|sin(theta)}{|mathrm{mathbf{r}}|},hat{n}$

which, by the definition of the cross product, can be written: For the crossed product in algebra and functional analysis, see crossed product. ...

$boldsymbolomega=frac{mathbf{r}timesmathbf{v}}{|mathrm{mathbf{r}}|^2}$

[edit] Higher dimensions

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. ^[1] In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... This article is about rotation as a movement of a physical body. ...

[edit] Angular velocity of a rigid body

Main article: Rigid body dynamics

Position of point P located in the rigid body (shown in blue). R_i is the position with respect to the lab frame, centered at O and r_i 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_i in the lab frame, and at position r_i in the body frame. It is seen that the position of the particle can be written: Rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate, which introduces other considerations. ... Image File history File links Size of this preview: 800 Ã— 490 pixel Image in higher resolution (1280 Ã— 784 pixel, file size: 120 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Angular velocity ... Image File history File links Size of this preview: 800 Ã— 490 pixel Image in higher resolution (1280 Ã— 784 pixel, file size: 120 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Angular velocity ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

$mathbf{R}_i=mathbf{R}+mathbf{r}_i$

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 $mathbf{r}_i$ is unchanging. By Euler's rotation theorem, we may replace the vector $mathbf{r}_i$ with $mathcal{R}mathbf{r}_{io}$ where $mathcal{R}$ is a rotation matrix and $mathbf{r}_{io}$ 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 $mathcal{R}$ which is changing in time and not the reference vector $mathbf{r}_{io}$ , as the rigid body rotates about point O'. The position of the particle is now written as: In mathematics, Eulers rotation theorem states that any rotation has an axis. ... A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...

$mathbf{R}_i=mathbf{v}+mathcal{R}mathbf{r}_{io}$

Taking the time derivative yields the velocity of the particle:

$mathbf{V}_i=mathbf{x}+frac{dmathcal{R}}{dt}mathbf{r}_{io}$

where V_i is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). The velocity of the particle is given by:

$mathbf{V}_i=mathbf{V}+boldsymbolOmegamathbf{r}_i$

Where Ω is the angular velocity tensor. If we take the dual of the angular velocity tensor, we get the angular velocity pseudovector In physics, the angular velocity tensor is defined as It has the important property that when the cross product is written with the matrix multiplication (A is a orientation matrix), this matrix is a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of...

$boldsymbolomega=[omega_x,omega_y,omega_z]$

and the matrix multiplication is replaced by the cross product, yielding:

$mathbf{V}_i=mathbf{V}+boldsymbolomegatimesmathbf{r}_i$

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. In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... This gyroscope remains upright while spinning due to its angular momentum. ...

[edit] See also

It has been suggested that this article or section be merged into Angular velocity. ... This gyroscope remains upright while spinning due to its angular momentum. ... Areal velocity is the rate at which area is swept by the position vector of a point which moves along a curve. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

[edit] References

^ Rotations and Angular Momentum on the Classical Mechanics page of the website of John Baez, especially Questions 1 and 2.

Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7.

Landau, L.D.; Lifshitz, E.M. (1997). Mechanics. Butterworth-Heinemann. ISBN 0-750-62896-0.

Categories: Physical quantity | Rotational symmetry | Angle Evgeny Mikhailovich Lifshitz (Russian: ; February 21, 1915 â€“ October 29, 1985) was a notable Soviet physicist. ...

Results from FactBites:

Angular velocity - Wikipedia, the free encyclopedia (860 words)
	The magnitude of the angular velocity is the angular speed (or angular frequency) and is denoted by ω.
	With constant angular acceleration, the angular velocity conforms to the rotational equations of motion, equivalent to the standard linear equations of motion under constant linear acceleration.
	In the case of pure circular motion, the angular velocity is equal to linear velocity divided by the radius.

Angular frequency - Wikipedia, the free encyclopedia (193 words)
	Angular frequency is the magnitude of the vector quantity angular velocity.
	is sometimes used as a synonym for the vector quantity angular velocity.
	Angular frequency is therefore a simple multiple of ordinary frequency.

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Encyclopedia > Angular velocity

Contents

[edit] Two dimensions

[edit] Higher dimensions

[edit] Angular velocity of a rigid body

[edit] See also

[edit] References