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Well, the key difference here is that one is a vector quantity while the other is a scalar. If your angle is measured in radians then angular frequency $\omega$ is given by $$\omega = 2 \pi f \space \mbox{(rad)} s^{-1}$$ while angular velocity is $$\vec{\Omega} = \frac{d \vec{v}}{dt} \mbox{m} \space s^{-1}$$ What you have above is the magnitude of ...

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ja72's answer is probably right, but I was confused by his notation, so I will give my own answer. Suppose we have two object rotating with angular velocity $\vec{\omega}_1$ and $\vec{\omega}_2$. Then the velocity of a point $\vec{r}$ of object $1$ in the lab frame is $\vec{v}_{1,lab}=\vec{\omega}_1 \times \vec{r}$. Similarly, the velocity of a point ...

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Lets pick a coordinate system where the second body rotates about the local $Z$ axis. The rotational kinematics if the second body are defined as $$E_2 = E_1 {\rm Rot}(\hat{z}, \theta)$$ where $E_i$ are the 3×3 rotation matrices, and $\hat{z}=(0,0,1)$ . If the angular velocity of the first body is $\hat{\omega}_1$ then differentiating the above ...

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