angular velocity

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We may imagine that a particle is rigidly connected to some central, fixed point $P\,$ , and that $\mathbf{r}\,$ be measured from that point. Define the angular velocity of the particle relative to $P\,$ by

$\boldsymbol{\omega} = \hat{\mathbf{r}} \times \frac{\mathbf{v}}{r} = \frac{\mathbf{r}\times\mathbf{v}}{r^{2}}\,$ .

One sees that its magnitude coincides with the earlier notion of angular frequency, $\omega = \frac{v_\perp}{r}\,$ , and is perpendicular to the plane of motion spanned by $\mathbf{v}\,$ and $\mathbf{r}\,$ . Furthermore, it is not a vector, but rather a pseudovector.

Now, using the BAC-CAB identity:

$\boldsymbol{\omega} \times \mathbf{r} = \frac{\mathbf{r}\times\mathbf{v}}{r^{2}} \times \mathbf{r} = \mathbf{v} - (\hat{\mathbf{r}}\cdot\mathbf{v})\hat{\mathbf{r}} \,$ .

Since $r\,$ is fixed, $\hat{\mathbf{r}}\cdot\mathbf{v}= 0\,$ , and so

$\mathbf{v} = \boldsymbol{\omega}\times\mathbf{r}\,$ .

Thus $\boldsymbol{\omega}\,$ is to be identified with the generator of infinitesimal rotations.

We may take $P\,$ to be the center of mass $\mathbf{R}\,$ and assume it is in uniform motion with velocity $\mathbf{V}\,$ . Then the particle has position $\mathbf{R} = \mathbf{R}_{com} + \mathbf{r}\,$ and velocity $\mathbf{V} = \mathbf{V}_{com} + \mathbf{v}\,$ , i.e.,

$\mathbf{V} = \mathbf{V}_{com} + \boldsymbol{\omega}\times\mathbf{r}\,$ .

Suppose we pick a different point $P_{new}\,$ which is rigidly connected to $P\,$ by an instantaneous vector $\mathbf{b}\,$ , from which to measure the position of the particle. This new reference point has position $\mathbf{R}_{new} \,$ and velocity $\mathbf{V}_{new} \,$ , and the new coordinate system has angular velocity $\boldsymbol{\omega}_{new} \,$ . Then $\mathbf{r}_{new} = \mathbf{r} - \mathbf{b}\,$ , so that

$\mathbf{V} = \mathbf{V}_{com} + \boldsymbol{\omega}\times ( \mathbf{r}_{new} + \mathbf{a} )\,$ .

Repeating our earlier analysis in the new coordinate system, we would require that

$\mathbf{V} = \mathbf{V}_{new} + \boldsymbol{\omega}_{new} \times \mathbf{r}_{new} \,$ ,

which is only possible if

$\boldsymbol{\omega}_{new} = \boldsymbol{\omega}\,$ , and

$\mathbf{V}_{new} = \mathbf{V}_{com} + \boldsymbol{\omega}\times \mathbf{b}\,$ .

Thus the angular velocity is independent of the chosen reference point.

Relation to Euler angles

Euler angles - The xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes, labeled N, is shown in green.

The angular velocity of a rotating body, expressed in a body-fixed basis, can be written in terms of Euler angles (ZXZ convention) as follows^[1]^[2]:

$\boldsymbol{\omega}' = \begin{pmatrix} \omega_{x'}\\ \omega_{y'}\\ \omega_{z'} \end{pmatrix}= \begin{pmatrix} \sin\psi \sin\theta \dot{\phi} + \cos\psi \dot\theta\\ \cos\psi \sin\theta \dot{\phi} - \sin\psi \dot\theta\\ \cos\theta \dot{\phi} + \dot\psi \end{pmatrix}\,$

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angular velocity

From Physics wiki

Relation to Euler angles

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