Angular velocity

Not to be confused with angular frequency.

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 up.

In physics, the angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed of an object and the axis about which the object 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, revolutions 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.^[1]

1 The angular velocity of a particle
- 1.1 Two dimensions
- 1.2 Three dimensions
2 For a moving frame
3 Angular velocity tensor
- 3.1 Coordinate-free description
4 Viewing as a vector field
5 Angular velocity of a rigid body
6 See also
7 References
8 External links

The angular velocity of a particle

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 (typically expressing the angular measures φ and θ in radians), if we draw a line from the origin (O) to the particle (P), then the velocity vector (v) of the particle will have a component along the radius (radial component, v_∥) and a component perpendicular to the radius (cross-radial component, v). However, it must be remembered that the velocity vector can be also decomposed into tangential and normal components.

A radial motion produces no change in the distance 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 cross-radial velocity by:^[1]

$\mathrm{v}_\perp=r\,\frac{d\phi}{dt}$

Utilizing θ, the angle between vectors v_∥ and v, or equivalently as the angle between vectors r and v, gives:

$\mathrm{v}_\perp=|\mathrm{\mathbf{v}}|\,\sin(\theta).$

Combining the above two equations and defining the angular velocity as ω=dΦ/dt 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.

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

Three dimensions

Classical mechanics

$\mathbf{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \mathbf{v})$
Newton's Second Law

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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.

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 normal 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:

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

which, by the definition of the cross product, can be written:

$\boldsymbol\omega=\frac{\mathbf{r}\times\mathbf{v}}{|\mathrm{\mathbf{r}}|^2}$

For a moving frame

Euler's rotation theorem states that, in an instant, for any dt there always exists a momentary axis of rotation. Therefore, any transversal section of the body by a plane perpendicular to this axis has to behave as a two dimensional rotation. The angular speed vector will be defined over the rotation axis (eigenvector of the linear map), and as such its value is the derivative of the angle rotated with respect to time.

$\omega = \frac{d\theta}{dt}$

Angular velocity tensor

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.^[2]

In three dimensions angular velocity can be represented by a vector because second rank tensors are dual to vectors in three dimensions. The tensor can be defined as a matrix T(t) such that:

$\boldsymbol\omega(t) \times \mathbf{r}(t) = T(t) \mathbf{r}(t)$

with:

$T(t) = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0 & -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix}$

So the cross product

$\boldsymbol\omega(t) \times \mathbf{r}(t)$

can be expressed as a matrix multiplication.

Coordinate-free description

At any instant, $t$ , the angular velocity tensor is a linear map between the position vectors $\mathbf{r}(t)$ and their velocity vectors $\mathbf{v}(t)$ of a rigid body rotating around the origin:

$\mathbf{v} = T\mathbf{r}$

where we omitted the $t$ parameter, and regard $\mathbf{v}$ and $\mathbf{r}$ as elements of the same 3-dimensional Euclidean vector space $V$ .

The relation between this linear map and the angular velocity pseudovector $\omega$ is the following.

Because of T is the derivative of an orthogonal transformation, the

$B(\mathbf{r},\mathbf{s}) = (T\mathbf{r}) \cdot \mathbf{s}$

bilinear form is skew-symmetric. (Here $\cdot$ stands for the scalar product). So we can apply the fact of exterior algebra that there is a unique linear form $L$ on $\Lambda^2 V$ that

$L(\mathbf{r}\wedge \mathbf{s}) = B(\mathbf{r},\mathbf{s})$ ,

where $\mathbf{r}\wedge \mathbf{s} \in \Lambda^2 V$ is the wedge product of $\mathbf{r}$ and $\mathbf{s}$ .

Taking the dual vector L* of L we get

$(T\mathbf{r})\cdot \mathbf{s} = L^* \cdot (\mathbf{r}\wedge \mathbf{s})$

Introducing $\omega�:= *L^*$ , as the Hodge dual of L* , and apply further Hodge dual identities we arrive at

$(T\mathbf{r}) \cdot \mathbf{s} = * ( *L^* \wedge \mathbf{r} \wedge \mathbf{s}) = * (\omega \wedge \mathbf{r} \wedge \mathbf{s}) = *(\omega \wedge \mathbf{r}) \cdot \mathbf{s} = (\omega \times \mathbf{r}) \cdot \mathbf{s}$

where

$\omega \times \mathbf{r}�:= *(\omega \wedge \mathbf{r})$

by definition.

Because $\mathbf{s}$ is an arbitrary vector, from nondegeneracy of scalar product follows

$T\mathbf{r} = \omega \times \mathbf{r}$

Viewing as a vector field

For angular velocity tensor maps velocities to positions, it is a vector field. In particular, this vector field is a Killing vector field belonging to an element of the Lie algebra so(3) of the 3-dimensional rotation group, SO(3).

Angular velocity of a rigid body

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:

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

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:

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

Taking the time derivative yields the velocity of the particle:

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathbf{r}_{io}$

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). Since $\mathcal{R}$ is a rotation matrix its inverse is its transpose. So we substitute $\mathcal{I}=\mathcal{R}^T\mathcal{R}$ :

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{I}\mathbf{r}_{io}$

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}^T\mathcal{R}\mathbf{r}_{io}$

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}^T\mathbf{r}_{i}$

Continue by taking the time derivative of $\mathcal{R}\mathcal{R}^T$ :

$\mathcal{I}=\mathcal{R}\mathcal{R}^T$

$0=\frac{d\mathcal{R}}{dt}\mathcal{R}^T+\mathcal{R}\frac{d\mathcal{R}^T}{dt}$

Applying the formula (AB)^T = B^TA^T:

$0=\frac{d\mathcal{R}}{dt}\mathcal{R}^T+\left(\frac{d\mathcal{R}}{dt}\mathcal{R}^T\right)^T$

$\frac{d\mathcal{R}}{dt}\mathcal{R}^T$ is the negative of its transpose. Therefore it is a skew symmetric 3x3 matrix. We can therefore take its dual to get a 3 dimensional vector. $\frac{d\mathcal{R}}{dt}\mathcal{R}^T$ is called the angular velocity tensor. If we take the dual of this tensor, matrix multiplication is replaced by the cross product. Its dual is called the angular velocity pseudovector, ω.

$\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]$

Substituting ω into the above velocity expression:

$\mathbf{V}_i=\mathbf{V}+\boldsymbol\omega\times\mathbf{r}_i.$

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.

If the reference point is the instantaneous axis of rotation the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a pure rolling spherical rigid body.

References

↑ ^1.0 ^1.1 Hibbeler, Russell C. (2009). Engineering Mechanics. Upper Saddle River, New Jersey: Pearson Prentice Hall. pp. 314, 153. ISBN 9780136077916. http://books.google.com/books?id=tOFRjXB-XvMC&pg=PA314&dq=angular+velocity&rview=1. (EM1)
↑ 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.

External links

A college text-book of physics By Arthur Lalanne Kimball (Angular Velocity of a particle)