7.1: Linear and Angular Velocity

We related the linear and angular velocities of a rotating object in two dimensions in Section 5.1. There, we also already stated the relation between the linear velocity vector and rotation vector in three dimensions (Equation 5.1.5):

\[\boldsymbol{v}=\boldsymbol{\omega} \times \boldsymbol{r}\]

It is not hard to see that this expression indeed simplifies to the scalar relationship \(v = \omega r\) for rotations in a plane, with the right sign for the linear velocity. That’s hardly a proof though, so let’s put this on some more solid footing. Suppose \(\boldsymbol{r}\) makes an angle \(\phi\) with \(\boldsymbol{\omega}\). Suppose also that it changes by \(\mathrm{d}\boldsymbol{r}\) in a time interval \(\mathrm{d}t\), then if we have pure rotation, \(\mathrm{d}\boldsymbol{r}\) is perpendicular to both \(\boldsymbol{r}\) and \(\boldsymbol{\omega}\), and its magnitude is given by \(|d \boldsymbol{r}| = \omega r \sin \phi \mathrm{d} t=|\boldsymbol{\omega} \times \boldsymbol{r}| \mathrm{d} t\), where \(\omega\) and \(r\) are the lengths of their respective vectors. Finally, as seen from the top (i.e., looking down the vector \(\boldsymbol{\omega}\)), the rotation should be counter-clockwise (by definition of the direction of \(\boldsymbol{\omega}\)), which corresponds with the direction of \(\boldsymbol{\omega} \times \boldsymbol{r}\). We thus find that both the magnitude and direction of \(\boldsymbol{v}=\mathrm{d} \boldsymbol{r} / \mathrm{d} t\) indeed equal \(\boldsymbol{\omega} \times \boldsymbol{r}\), and Equation 5.1.5 holds.