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?
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.
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