# Should angular velocities always be referred to frames?

• etotheipi
In summary, the conversation discusses the definition of angular velocity between frames and its property of addition, as well as the terminology of referring to the angular velocity of a particle with respect to a frame. It is suggested that it is simpler to analyze rigid body dynamics in an inertial frame and introduce a non-inertial body-fixed frame, which is unambiguous for determining the angular velocity of a rigid body. The conversation also mentions the application of a spinning top in a rotating frame and its connection to the theory of gyrocompass.
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

Lnewqban
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
vanhees71 said:
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!

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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

berkeman and 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

vanhees71

## 1. What is the definition of angular velocity?

Angular velocity is a measure of the rate of change of angular displacement over time. It is typically represented by the symbol ω and is measured in radians per second.

## 2. Why is it important to refer to frames when discussing angular velocities?

Frames provide a reference point for measuring angular velocities. Without a defined frame of reference, it is impossible to accurately determine the direction and magnitude of an object's angular velocity.

## 3. Can angular velocities be referred to multiple frames?

Yes, angular velocities can be referred to multiple frames as long as the frames are clearly defined and the calculations are consistent. However, it is generally recommended to refer to a single frame to avoid confusion.

## 4. What happens if angular velocities are not referred to frames?

If angular velocities are not referred to frames, it can lead to incorrect calculations and confusion. Without a defined frame of reference, it is impossible to accurately determine the direction and magnitude of an object's angular velocity.

## 5. Are there any exceptions to referring to frames when discussing angular velocities?

In some cases, it may not be necessary to refer to frames when discussing angular velocities. For example, if the object's motion is purely rotational and there is no translation, the angular velocity can be described using a fixed frame of reference. However, it is still important to clearly define the frame of reference to avoid confusion.

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