In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. 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. In general, angular velocity is measured in angle per unit time, e.g. radians per second. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec. Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.
In three dimensions, angular velocity is a pseudovector, with its magnitude measuring the rate at which an object rotates or revolves, and its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule.
Orbital angular velocity of a point particle
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:
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, 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.
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:
Notice that this also defines the subtraction as the addition of a negative vector.
Spin angular velocity vector for a 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.
By Euler's rotation theorem, any rotating frame possesses an instantaneous axis of rotation, which is the direction of the angular velocity vector, and the magnitude of the angular velocity is consistent with the two-dimensional case.
- 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:
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:
Angular velocity tensor
(This holds even if A(t) does not rotate uniformly.) Therefore the angular velocity tensor is:
Properties of angular velocity tensors
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 three dimensions, angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions. Since the angular velocity tensor W = W(t) is a skew-symmetric matrix:
If we know an initial frame A(0) and we are given a constant angular velocity tensor W, we can obtain A(t) for any given t. Recall the matrix differential equation:
This equation can be integrated to give:
which shows a connection with the Lie group of rotations.
Thus, W is the negative of its transpose, which implies it is skew symmetric.
Because W is the derivative of an orthogonal transformation, the bilinear form
Taking the sharp L♯ of L we get
Since the spin angular velocity tensor of a rigid body (in its rest frame) is a linear transformation that maps positions to velocities (within the rigid body), it can be regarded as a constant vector field. In particular, the spin angular velocity is a Killing vector field belonging to an element of the Lie algebra SO(3) of the 3-dimensional rotation group SO(3).
Also, it can be shown that the spin angular velocity vector field is exactly half of the curl of the linear velocity vector field v(r) of the rigid body. In symbols,
Rigid body considerations
The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Here is not assumed that the rigid body rotates around the origin. Instead, it can be supposed rotating around an arbitrary point that is moving with a linear velocity V(t) in each instant.
To obtain the equations, it is convenient to imagine a rigid body attached to the frames and consider a coordinate system that is fixed with respect to the rigid body. Then we will 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 Ri in the lab frame, and at position ri in the body frame. It is seen that the position of the particle can be written:
Taking the time derivative yields the velocity of the particle:
Substituting ω for W into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:
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 orbital angular velocity of the particle with respect to the reference point. This angular velocity is what physicists call the "spin angular velocity" of the rigid body, as opposed to the orbital angular velocity of the reference point O′ about the origin O.
We have supposed that the rigid body rotates around an arbitrary point. We should prove that the spin angular velocity previously defined is independent of the choice of origin, which means that the spin angular velocity is an intrinsic property of the spinning rigid body. (Note the marked contrast of this with the orbital angular velocity of a point particle, which certainly does depend on the choice of origin.)
The above two yields that
If the reference point is the instantaneous axis of rotation the expression of the velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of the instantaneous axis of rotation is zero. An example of the instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a purely rolling spherical (or, more generally, convex) rigid body.