In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, angular velocity or rotational velocity ( or ), also known as angular frequency vector,
[(UP1)] is a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
representation of how fast the
angular position
In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies.
More specifically, it refers to the imagin ...
or orientation of an object changes with time (i.e. how quickly an object
rotate
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s or revolves relative to a point or axis). The magnitude of the pseudovector represents the ''
angular speed
Angular may refer to:
Anatomy
* Angular artery, the terminal part of the facial artery
* Angular bone, a large bone in the lower jaw of amphibians and reptiles
* Angular incisure, a small anatomical notch on the stomach
* Angular gyrus, a regio ...
'', the rate at which the object rotates or revolves, and its direction is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
to the instantaneous
plane of rotation
In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as t ...
or angular displacement. The orientation of angular velocity is conventionally specified by the
right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
.
[(EM1)]
There are two types of angular velocity.
* Orbital angular velocity refers to how fast a point object
revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
.
* Spin angular velocity refers to how fast a rigid body rotates with respect to its
center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity.
In general, angular velocity has
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of angle per unit time (angle replacing
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from linear
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
with time in common). The
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
of angular velocity is
radians per second
The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omega). ...
, with the
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
being a
dimensionless quantity
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, thus the SI units of angular velocity may be listed as s
−1. Angular velocity is usually represented by the symbol
omega
Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μÎγα, Modern Greek ωμÎγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/isopsephy (gematria), it has a value of 800. The wo ...
(, sometimes ). By convention, positive angular velocity indicates counter-
clockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
rotation, while negative is clockwise.
For example, a
geostationary
A geostationary orbit, also referred to as a geosynchronous equatorial orbit''Geostationary orbit'' and ''Geosynchronous (equatorial) orbit'' are used somewhat interchangeably in sources. (GEO), is a circular geosynchronous orbit in altitude ...
satellite completes one orbit per day above the
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
, or 360 degrees per 24 hours, and has angular velocity ''ω'' = (360°)/(24 h) = 15°/h, or (2π rad)/(24 h) ≈ 0.26 rad/h. If angle is measured in radians, the linear velocity is the radius times the angular velocity,
. With orbital radius 42,000 km from the earth's center, the satellite's speed through space is thus ''v'' = 42,000 km × 0.26/h ≈ 11,000 km/h. The angular velocity is positive since the satellite travels eastward with the Earth's rotation (counter-clockwise from above the north pole.)
Orbital angular velocity of a point particle
Particle in two dimensions
In the simplest case of circular motion at radius
, with position given by the angular displacement
from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time:
. If
is measured in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s, the arc-length from the positive x-axis around the circle to the particle is
, and the linear velocity is
, so that
.
In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector
from the origin
to a particle
, with its
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
. (All variables are functions of time
.) The particle has linear velocity splitting as
, with the radial component
parallel to the radius, and the cross-radial (or tangential) component
perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity.
The angular velocity ''ω'' is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:
:
Here the cross-radial speed
is the signed magnitude of
, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity
gives magnitude
(linear speed) and angle
relative to the radius vector; in these terms,
, so that
:
These formulas may be derived doing
, being
a function of the distance to the origin with respect to time, and
a function of the angle between the vector and the x axis. Then Which is equal to (See
Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
in cylindrical coordinates). Knowing we conclude that the radial component of the velocity is given by because
is a radial unit vector; and the perpendicular component is given by
because
is a perpendicular unit vector.
In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a
pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...
, a numerical quantity which changes sign under a
parity inversion
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
, such as inverting one axis or switching the two axes.
Particle in three dimensions
In
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
, we again have the position vector r of a moving particle. Here, orbital angular velocity is a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
whose magnitude is the rate at which r sweeps out angle, and whose direction is perpendicular to the instantaneous plane in which r sweeps out angle (i.e. the plane spanned by r and v). However, as there are ''two'' directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, the
right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
is used.
Let the pseudovector
be the unit vector perpendicular to the plane spanned by r and v, so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top of
). Taking polar coordinates
in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:
:
where ''θ'' is the angle between r and v. In terms of the cross product, this is:
:
From the above equation, one can recover the tangential velocity as:
:
Spin angular velocity of a rigid body or reference frame
Given a rotating frame of three unit coordinate vectors, all the three must have the same angular speed at each instant. In such a frame, each vector may be considered as a moving particle with constant scalar radius.
The rotating frame appears in the context of
rigid bodies
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
, and special tools have been developed for it: the spin angular velocity may be described as a vector or equivalently as a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
.
Consistent with the general definition, the spin angular velocity of a frame is defined as the orbital angular velocity of any of the three vectors (same for all) with respect to its own center of rotation. The addition of angular velocity vectors for frames is also defined by the usual vector addition (composition of linear movements), and can be useful to decompose the rotation as in a
gimbal
A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...
. All components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices). As in the general case, addition is commutative:
.
By
Euler's rotation theorem
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed p ...
, any rotating frame possesses an
instantaneous axis of rotation
The instant center of rotation (also, instantaneous velocity center, instantaneous center, or instant center) is the point fixed to a body undergoing planar movement that has zero velocity at a particular instant of time. At this instant, the vel ...
, which is the direction of the angular velocity vector, and the magnitude of the angular velocity is consistent with the two-dimensional case.
If we choose a reference point
fixed in the rigid body, the velocity
of any point in the body is given by
:
Components from the basis vectors of a body-fixed frame
Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors
fixed to the body and with their common origin at O. The spin angular velocity vector of both frame and body about O is then
:
where
is the time rate of change of the frame vector
due to the rotation.
Note that this formula is incompatible with the expression for ''orbital'' angular velocity
:
as that formula defines angular velocity for a ''single point'' about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body a ''single''
has to account for the motion of ''all'' particles in the body.
Components from Euler angles
The components of the spin angular velocity pseudovector were first calculated by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
using his
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189†...
and the use of an intermediate frame:
* One axis of the reference frame (the precession axis)
* The line of nodes of the moving frame with respect to the reference frame (nutation axis)
* One axis of the moving frame (the intrinsic rotation axis)
Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous
Euler rotations In physics and engineering, Davenport chained rotations are three chained intrinsic rotations about body-fixed specific axes. Euler's rotation theorem, Euler rotations and Tait–Bryan rotations are particular cases of the Davenport general rotation ...
). Therefore:
K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics
/ref>
:
This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:
:
where are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles.
Tensor
The angular velocity vector defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) ''W'' = ''W''(''t'') defined by:
:
This is an infinitesimal rotation matrix. The linear mapping ''W'' acts as :
:
Calculation from the orientation matrix
A vector undergoing uniform circular motion around a fixed axis satisfies:
:
Given the orientation matrix ''A''(''t'') of a frame, whose columns are the moving orthonormal coordinate vectors , we can obtain its angular velocity tensor ''W''(''t'') as follows. Angular velocity must be the same for the three vectors , so arranging the three vector equations into columns of a matrix, we have:
:
(This holds even if ''A''(''t'') does not rotate uniformly.) Therefore the angular velocity tensor is:
:
since the inverse of the orthogonal matrix is its transpose .
Properties
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 ''W'' will have independent components, which is the dimension of the Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of rotations
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of an ''n''-dimensional inner product space.[Rotations and Angular Momentum](_blank)
on the Classical Mechanics page o
especially Questions 1 and 2.
Duality with respect to the velocity vector
In three dimensions, angular velocity can be represented by a pseudovector because second rank tensors are dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
to pseudovectors in three dimensions. Since the angular velocity tensor ''W'' = ''W''(''t'') is a skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
:
:
its Hodge dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the al ...
is a vector, which is precisely the previous angular velocity vector