| Angular velocity | |
| Symbols: | |
| Unit: | rads−1 |
| Baseunits: | s−1 |
| Dimension: | wikidata |
| Extensive: | yes |
| Intensive: | yes (for rigid body only) |
| Conserved: | no |
| Transformsas: | pseudovector |
In kinematics, angular velocity (symbol or, the lowercase Greek letter omega), also known as the angular frequency vector,[1] is a pseudovector that is equal to the derivative, with respect to time, of angular position with respect to a fixed half-line (angular position also being a pseudovector, which is angular equivalent of position vector). [2]
The magnitude of the pseudovector,
\omega=\|\boldsymbol{\omega}\|
\hat\boldsymbol{\omega}=\boldsymbol{\omega}/\omega
There are two types of angular velocity:
Angular velocity has dimension of angle per unit time; this is analogous to linear velocity, with angle replacing distance, with time in common. The SI unit of angular velocity is radians per second,[3] although degrees per second (°/s) is also common. The radian is a dimensionless quantity, thus the SI units of angular velocity are dimensionally equivalent to reciprocal seconds, s−1, although rad/s is preferable to avoid confusion with rotational velocity in units of hertz (also equivalent to s−1).[4]
The sense of angular velocity is conventionally specified by the right-hand rule, implying clockwise rotations (as viewed on the plane of rotation); negation (multiplication by −1) leaves the magnitude unchanged but flips the axis in the opposite direction.[5]
For example, a geostationary satellite completes one orbit per sidereal day above the equator (approximately 360 degrees per 24 hours) has angular velocity magnitude (angular speed) ω = 360°/24 h = 15°/h (or 2π rad/24 h ≈ 0.26 rad/h) and angular velocity direction (a unit vector) parallel to Earth's rotation axis (in the geocentric coordinate system). If angle is measured in radians, the linear velocity is the radius times the angular velocity, . With orbital radius from the Earth's center, the satellite's tangential speed through space is thus v = × 0.26/h ≈ . The angular velocity is positive since the satellite travels prograde with the Earth's rotation (the same direction as the rotation of Earth).
In the simplest case of circular motion at radius, with position given by the angular displacement
\phi(t)
\phi
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
r
O
v\|
v\perp
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
v\perp
v
v
\theta
These formulas may be derived doing, being
r
\varphi
Knowing, we conclude that the radial component of the velocity is given by, because
\hat{r}
| r\varphi |
\hat{\varphi}
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.
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 (in radians per unit of time), 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
u
(r,\phi)
\boldsymbol\omega=\omegau=
| d\phi | |
| dt |
|
| \boldsymbol\omega= | r x v |
| r2 |
.
From the above equation, one can recover the tangential velocity as:
v\perp=\boldsymbol{\omega} x r
Given a rotating frame of three linearly independent unit coordinate vectors, at each instant in time, there always exists a common axis (called the axis of rotation) around which all three vectors rotate with the same angular speed and in the same angular direction (clockwise or counterclockwise). In such a frame, each vector may be considered as a moving particle with constant scalar radius. A collection of such particles is called a rigid body.
Euler's rotation theorem says that in a rotating frame, the axis of rotation one obtains from one choice of three linearly independent unit vectors is the same as that for any other choice; that is, there is one single instantaneous axis of rotation to the frame, around which all points rotate at the same angular speed and in the same angular direction (clockwise or counterclockwise). The spin angular velocity of a frame or rigid body is defined to be the pseudovector whose magnitude is this common angular speed, and whose direction is along the common axis of rotation in accordance with the right-hand rule (that is, for counterclockise rotation, it points "upward" along the axis, while for clockwise rotation, it points "downward").
In larger than 3 spatial dimensions, the interpretation of spin angular velocity as a pseudovector is not valid; however, it may be characterized by a more general type of object known as an antisymmetric rank-2 tensor.
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: .
If we choose a reference point
{\boldsymbolr0}
| \boldsymbolr |
| \boldsymbolr |
=
| \boldsymbolr0 |
+{\boldsymbol\omega} x ({\boldsymbolr}-{\boldsymbolr0})
Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors
e1,e2,e3
\boldsymbol\omega=\left(
| e |
1 ⋅ e2\right)e3+\left(
| e |
2 ⋅ e3\right)e1+\left(
| e |
3 ⋅ e1\right)e2,
| e |
i=
| dei | |
| dt |
This formula is incompatible with the expression for orbital angular velocity
| \boldsymbol\omega = | \boldsymbol{r |
| x \boldsymbol{v}}{r |
2},
\boldsymbol\omega
The components of the spin angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and the use of an intermediate frame:
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:[7]
\boldsymbol\omega=
| \alphau |
|
|
u3
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:
\boldsymbol\omega= (
| \alpha |
\sin\beta\sin\gamma+
| \beta\cos\gamma) |
\hat
|
\cos\gamma-
| \beta\sin\gamma) | \hatj+ ( |
| \alpha |
\cos\beta+
| \gamma) |
\hatk
where
\hati,\hatj,\hatk
