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I intend to determine the angular velocity of a referential system, $S$, translating and rotating in relation to an inertial referential, $S_o$.

I need to know this concept to determine, for example, the velocity of a fixed particle on $S$, measured on the inertial referential:

$$\vec{v}^{S_o}=\vec{v}_O^{S_o} +\vec{v}^S+\left[\vec{w}_S^{S_o}\times\vec{r}^S\right]\tag1$$

where:

  • $\vec{v}_O^{S_o}$ is the velocity of the origin of $S$ in relation to $S_o$;

  • $\vec{v}^S$ is the velocity of the particle on the referential $S$;

  • $\vec{w}_S^{S_o}$ is the angular velocity of the referential $S$ measured on the referential $S_o$;

  • $\vec{r}^s$ is the position of the particle on the referential $S$.

I learnt expression $(1)$ on my Satellite Engineering course, but I didn't totally understand the meaning of $\vec{w}_S^{S_o}$. Is $\vec{w}_S^{S_o}$ just the frequency of rotation of $S$? And is the angular frequency of translation (let it be $\vec{{\Omega}}$) introduced on $\vec{v}_O^{S^o}$ expression by $$\vec{v}_O^{S^o}=\vec{v}_{(O,n)}^{S^o}+\vec{v}_{(O,t)}^{S^o}=\vec{v}_{(O,n)}^{S^o}+\left[\vec{\Omega}\times \vec{r_O}^{S^o}\right]?$$

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  • $\begingroup$ Related (if not duplicate) : Velocity in a turning reference frame. $\endgroup$
    – Frobenius
    Jun 10, 2018 at 11:55
  • $\begingroup$ Is $\vec{v}^S$ and $\vec{v}^{S_o}$ on the same basis vectors? There seems to be a rotation transformation missing between the two. $\endgroup$
    – John Alexiou
    Sep 22, 2022 at 4:39

2 Answers 2

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There's no such thing as an "angular velocity of translation," translation is different from rotation. Think of a frisbee flying through the air-- that motion is a combination of the translation of the center of mass (which is just a smooth gentle arc that is instantaneously just a single velocity, i.e., a single arrow) and the rotation around the center of mass (which involves a different velocity at every point on the frisbee), those are the two corrections to the velocity when you go from the frame of the ground to the frame of an ant on the frisbee. So omega is indeed the angular velocity of one frame relative to the other, but there's no capital-Omega in going between frames.

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Yes $\vec{w}_{S_{0}}^{S}$ just the frequency of rotation of $S$ because "the angular velocity of rotation, at any instant, of a system of co-ordinates fixed in the body is independent of the particular system chosen".(demonstration)

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