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 rel ...
, angular velocity or rotational velocity ( or ), also known as angular frequency vector,
[(UP1)] is a
pseudovector 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 region ...
'', the rate at which the object rotates or revolves, and its direction is
normal to the instantaneous
plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the
right-hand rule.
[(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), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
.
* 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 mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of angle per unit time (angle replacing
distance 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 i ...
with time in common). The
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
of angular velocity is
radians per second, with the
radian being a
dimensionless quantity, 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 μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. Th ...
(, sometimes ). By convention, positive angular velocity indicates counter-
clockwise rotation, while negative is clockwise.
For example, a
geostationary 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 al ...
, 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
radians, 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 t ...
. (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 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, a numerical quantity which changes sign under a
parity inversion, such as inverting one axis or switching the two axes.
Particle in three dimensions
In
three-dimensional space, we again have the position vector r of a moving particle. Here, orbital angular velocity is a
pseudovector 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 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, 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 tens ...
.
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. 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, 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 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). 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 In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ' ...
.
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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
of the Lie group 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:
:
its Hodge dual is a vector, which is precisely the previous angular velocity vector