Should angular velocities always be referred to frames?

[etotheipi]

This is a semantic question, without any implications really, but I wondered if someone could check if I understand this correctly? The angular velocity $\vec{\Omega}$ of a frame $\mathcal{F}$ with respect to another frame $\mathcal{F}'$ is defined such that, for any vector $\vec{a}$,$$\left(\frac{d\vec{a}}{dt} \right)_{\mathcal{F}'} = \left(\frac{d\vec{a}}{dt} \right)_{\mathcal{F}} + \vec{\Omega} \times \vec{a}$$The property of addition of angular velocities between frames follows quite naturally,$$\left(\frac{d\vec{a}}{dt} \right)_{\mathcal{F}''} = \left [\left(\frac{d\vec{a}}{dt} \right)_{\mathcal{F}} + \vec{\Omega}_1 \times \vec{a} \right] + \vec{\Omega}_2 \times \vec{a} = \left(\frac{d\vec{a}}{dt} \right)_{\mathcal{F}} + (\vec{\Omega}_1 + \vec{\Omega}_2) \times \vec{a}$$Sometimes, we might say a particle or a rigid body has an angular velocity of $\vec{\omega}$ with respect to some coordinate system. Whilst this seems passable if there are only two frames involved, it doesn't seem like a good terminology otherwise. For instance, whilst it makes perfect sense to say that the angular velocity of frame $\mathcal{F}$ w.r.t. $\mathcal{F}''$ is $(\vec{\Omega}_1 + \vec{\Omega}_2)$, it doesn't seem to make sense to say a particle has an angular velocity of $(\vec{\Omega}_1 + \vec{\Omega}_2)$ w.r.t. $\mathcal{F}$. Instead, to get anything meaningful for the particle, you need to explicitly perform the change of coordinates $\vec{r}'' = \vec{R} + \vec{r}$ and differentiate (making use of the second equation), e.g. casting it in terms of $\vec{v}'' = \vec{V} + (\vec{\Omega}_1 + \vec{\Omega}_2) \times \vec{r}$.

I wondered if you guys would agree that it's better to talk about the angular velocity of the frame in which the particle is at rest (or the body fixed frame of a rigid body), rather than the angular velocity of the particle itself? Thanks

 

[vanhees71]

I think it simplifies things a lot to treat the rigid-body dynamics as usual, i.e., in an inertial "space-fixed frame" and introducing the "body-fixed frame" which is non-inertial (in the general case the body-fixed origin can be accelerated and the body, i.e., the body-fixed Cartesian basis of the body-fixed frame is rotating against the space-fixed inertial frame).

It can of course be interesting to consider the equation of motion of a rigid body/gyroscope as observed in a non-inertial (particularly a rotating) frame.

 

[etotheipi]

I think it simplifies things a lot to treat the rigid-body dynamics as usual, i.e., in an inertial "space-fixed frame" and introducing the "body-fixed frame" which is non-inertial (in the general case the body-fixed origin can be accelerated and the body, i.e., the body-fixed Cartesian basis of the body-fixed frame is rotating against the space-fixed inertial frame).

Yes that is also my preferred way of analysing the motion. For rigid bodies it is very simple conceptually, since the body fixed frame is easily realized and it is completely unambiguous to talk about the angular velocity of a rigid body.

I don't think I have tried much extended body dynamics in a rotating frame, though, so maybe I will try and find some problems. Thanks!

 

[vanhees71]

An interesting application of a spinning top in a rotating frame is the theory of the gyrocompass. Interestingly Einstein was involved as an expert in patent issues about the subject during WW1.

https://en.wikipedia.org/wiki/Gyrocompass

 

[etotheipi]

That's very cool! I would like to try and do the derivation but I fear I will make an algebraic mistake somewhere . In any case it looks like a nice exercise for the Lagrangian dynamics