# Component of angular velocity along an axis inclined at $\theta$

If an arbitrary rigid body rotates with angular velocity $\omega_0$ about some axis, can it be said that the body will rotate with an angular velocity $\omega_0 \cos(\theta)$ about an axis which is at an angle of $\theta$ with this axis. If yes, what is the physical significance of such an equation?

• Simultaneous rotation around more than one axis at the same time is impossible (see Euler's Rotation Theorem). – David H Jun 25 '13 at 18:20
• @DavidH you ever seen an olympic gymnast do that vault and spin thing? Sure looks like they have some multi-axis rotation going on. :-P – Jim Jun 25 '13 at 19:56

## 1 Answer

You can decompose an angular velocity into components e.g. express it as:

$$\vec{\omega_0} = \omega_x \vec{i} + \omega_y \vec{j} + \omega_z \vec{k}$$

and this is is essentially what you're doing by calculating the component along some axis at an angle $\theta$, though obviously you will need the value in two other directions as well. However there is no physical significance to the values of the components. They are just the representation of the angular velocity vector in whatever coordinate system you have chosen and will be different in different coordinate systems.

• I think this question should be approached from spherical coordinates where $\omega_0$ is the $\hat\theta$ component of the velocity, then the angle $\theta$ could be a projection to some new angle along the $\hat\phi$ direction – Jim Jun 25 '13 at 19:50
• You can choose spherical coordinates or indeed any other coordinate system. $\vec{\omega_0}$ doesn't care! – John Rennie Jun 25 '13 at 19:54
• I know that, I meant it could help you explain it to the people that don't. But it's your answer. +1 anyway – Jim Jun 25 '13 at 19:58