Understanding the Body Frame of a Spinning Symmetric Top

[darkSun]

Hello, I have a question about a spinning symmetric top:

When the equations of motion are solved, they are solved in two frames--the space frame and the body frame. I understand the space frame, but in the body frame you are looking at the top from a frame that is rotating with it, right? So you don't see any rotation, so do the values of w_i mean anything?
I think my thinking is wrong here (but I'm not sure why), that although we are in the body frame, we still see w. In fact I know what the solved w looks like, but I just can't wrap my head around it. If there are any good animations for this kind of thing, I'd appreciate being shown them.

 

[Riyuki]

I'm not exactly sure what you're representing by the variable w_i. Angular velocity?

I think the best I can do is to say that rotational motion is always relevant because:

1) A centripetal acceleration/force is required to keep an object moving in this fashion, in contrast to straight-line motion. In the case of the spinning top the centripetal force is provided by the molecular bonds of its particles. Similarly gravity "maintains" the orbits involved in solar systems and galaxies.

If the centripetal force were suddenly terminated, a given rotating mass would fly out along a path tangential to its original orbit (though of course if there were angular acceleration near the time of release you'd have to account for that). Which is of course unlikely to happen in the case of the top (I guess you could explode it somehow, though that would introduce a lot of other variables so that the effect of the original rotation wouldn't be very obvious) but you can imagine e.g. twirling a mass on a string in a circle above your head and suddenly cutting or otherwise releasing the string.
I'm not saying that the only reason this matters is because "there's a chance that the centripetal force might be interrupted"; I'm just saying it in illustration of inertia, which is relevant to the overall mechanics of the system regardless of whether or not the force might be interrupted.

and

2) Although there are many cases where the rotational motion of a given system can be fairly easily neglected because everything of interest is rotating together along nearly the same path the simplification doesn't change the fact that the system is still moving appreciably relative to more distant objects.
For example: microscopic phenomena on/inside the spinning top, most human-scale motion on the surface of a revolving (and possibly rotating) planet, or the motion of objects in a star system which is rotating around a galaxy's center.

And even within the system the simplification is (I think) only an approximation, which depending on the size of the system and the degree of precision needed for a given purpose might not be sufficiently accurate. Though I can't think offhand of an example of such a precision issue due to neglected rotational motion, and it might be a very rare issue in practice.

 

[Cleonis]

Hello, I have a question about a spinning symmetric top:

When the equations of motion are solved, they are solved in two frames--the space frame and the body frame. I understand the space frame, but in the body frame you are looking at the top from a frame that is rotating with it, right? So you don't see any rotation, so do the values of w_i mean anything?

I think this is best addressed by making a comparison with formulating equations of motion for a rotating coordinate system. For example, a coordinate system that is co-rotating with the Earth.

In an equation of motion for the Earth-corotating system there is a centrifugal term and a coriolis term. The centrifugal and coriolis term are proportional to w_i (assuming you are using the letter 'w' to represent the Earth angular velocity.)

In other words, an equation of motion for the co-rotating coordinate system references the inertial coordinate system.

One can formulate an equation of motion for motion wrt to the inertial coordinate system, or an equation of motion for motion wrt to a rotating coordinate system; both will feature the angular velocity wrt to the inertial coordinate system.

 

[darkSun]

Thanks for your reply, Riyuki.
Yes, I do mean angular velocity by w_i, maybe I should have written omega_i, I meant the different components of angular velocity.

My question was a little more specific, although it was probably poorly worded. In mechanics texts, when the motion of the symmetric spinning top is analyzed (in chapters on Rigid body motion a la Goldstein for example), the equations of motion are derived in both the space frame and a frame which has its axes fixed in the body. My question is, if you are observing the top from a frame fixed in itself, how can you observe any angular velocity?

I know the Euler equations apply in this case, and that they do of course lead to non-zero angular velocities. But if you're in a frame rotating with the body, what angular velocity do you see?

 

[diazona]

You can observe the angular velocity of the rest of the universe with respect to your (rotating) reference frame.

 

[darkSun]

Thanks diazona, I guess it was a simple question after all. But that really helped me.

 

[Cleonis]

But if you're in a frame rotating with the body, what angular velocity do you see?

Giving it another try:
It's not about thinking in terms of angular velocity that you see. It's about the consequences of inertia.

If you represent motion as motion relative to an inertial coordinate system then inertial motion is along straight lines in that coordinate system.
If you represent the physics taking place in terms of axes fixed to the spinning top, then just as well it's about finding the consequences of inertia. Same physics, just different representation.

 

[diazona]

Thanks diazona, I guess it was a simple question after all. But that really helped me.

Cool, glad it helped.

But Cleonis does have a good point, that even if you were inside a closed windowless spinning top so that you couldn't see the universe outside, you'd still be able to tell that you were spinning by measuring the deviation from the law of inertia. You could apply some known force to an object and measure its acceleration, and compute the quantity $F - ma$ which tells you what fictitious force(s) you need to add to the equation to make it work in your coordinates. If the fictitious forces you find match the descriptions of the centrifugal force and Coriolis force, you can work backwards from those to identify the angular velocity that you presumably would see if there were windows in the top.

(Technically this technique doesn't really tell you that you're spinning per se - you could be at rest in some weird distorted spacetime, but that's kind of another topic)

 

[darkSun]

Ok cool, that's what I suspected. But they kept talking about solving for the angular velocity along this and that axis, I wasn't sure. But this clears it up for me.

So in the Euler equations, it's like -w x L is a pseudotorque, a consequence of looking at the top from its own spinning frame.