Degrees of freedom of thin rods

[hkyriazi]

TL;DR Summary

How does one determine the degrees of freedom of a "gas" of thin rods, so as to know the rods' equilibrium energy distribution between translation and the two forms of rotation?

(Note: I had this question posted at the intermediate level of difficulty for 11 days, but got only one, cryptic (to me) response that was rather quickly removed. So, I figured perhaps it's actually an advanced question, requiring more than a cursory understanding.)

Assuming they've had an infinite amount of time to interact (and thus have achieved an equilibrium regarding energy partition), I'm trying to decide if a "gas" of thin rods will have 3 parts of its kinetic energy in translation, 2 in "twirling" or precessional motion, and 1 part in rotation about the main axis ("rolling"). (Imagine that they're in outer space, free of gravity and all other action-at-a-distance forces, subject only to the force of impact and recoil. The rods are assumed to have surface friction--otherwise the rotation about the main rod axis could never change. In coming to this very tentative conclusion, I'm thinking that there are two directional axes of collision which can set a rod, with main axis Z, twirling (like a baton): the X and Y. But to get it rolling along its axis, collisions have to come in along those same two axes, but aimed not right at the axis, but tangentially to some degree. Intuitively this amount of energy seems necessarily less than that in twirling, but the fact that collisions can still come in along both the X and Y axes is throwing me off.

What's the correct thinking here? What's the proper way to approach the problem?

(Note #2: After receiving a cryptic-to-me reply about "thermodynamic conjugate variables," I added the following, which may or may not be clarifying: What are these conjugate variables? Is this similar to the idea that the degrees of freedom is equal to the number of independent variables needed to specify the object's motion? I'm thinking 3 translational velocity coordinates (for the rigid rod's center of mass), plus the angular momentum vector, precession angle, and rate of precession. But aren't 3 variables needed to define just the angular momentum vector (either 2 for direction in polar coordinates plus another for magnitude, or 3 points in Cartesian space)? I suppose the angular velocity of precession could be computed from the combination of angular momentum and precession angle (or vice-versa), which would also give one the ratio of Z-axis spin vs. XY-axis twirling energy. But that's still 7 independent variables.)

 

[BvU]

Hi,

You call them 'thin rods' for a reason. My interpretation is that there is no rotation degree of freedom around the length axis (call that the $z$-axis ) and the length of the rods is fixed. That leaves two axes of rotation and three velocities. Like with the velocities, the degree of freedom is in the direction, not in the magnitude.

 

[hkyriazi]

You call them 'thin rods' for a reason. My interpretation is that there is no rotation degree of freedom around the length axis (call that the $z$-axis ) and the length of the rods is fixed. That leaves two axes of rotation and three velocities. Like with the velocities, the degree of freedom is in the direction, not in the magnitude.

Thanks, BvU. Apologies for the multiplicity of questions, below.

If that were the case, then we'd have 3/5ths of the energy in translation, and 2/5ths in twirling, correct?

I specified "thin rods" mostly to minimize end collisions, not actually to eliminate "rolling" along their z-axis. But now that you mention it, I'm wondering: if there truly were no end collisions, would that make the energy distribution 50-50? (But they'd still be able to translate and twirl, such that sometimes their z-axis would line up with their direction of motion, and there nevertheless would be motion, momentarily, along the z-axis. See my confusion? Not sure what to pay attention to--their actual motion, or how many different ways they can be hit.)

But I'd like there to be friction in collisions, so that there *would* be rolling.

 

[BvU]

See my confusion?

Not quite. Yes, the $z$-axis of the stick can momentarily align with the velocity vector; so what ?

I have no idea what 'end collisions' are.

Thin rods to me means no moment of inertia around the $z$-axis. No rolling.

Seems to me you think of massive tubes with a nonzero diameter ?

 

[hkyriazi]

Not quite. Yes, the $z$-axis of the stick can momentarily align with the velocity vector; so what ?

I'm not understanding whether the phrase "degrees of freedom" relates solely to an object's possible motions, solely to the number of ways it can be set in motion, or somehow to both. I mentioned this possible, momentary z-axis motion, thinking that perhaps that indicated a "degree of freedom" along the z-axis.

I have no idea what 'end collisions' are.

I meant thin rods, but not infinitesimally thin, that thus have actual, circular ends. A hit on a rod's end could propel it along its z-axis without much twirling involved. Again, not sure how this affects its "degrees of freedom."

Thin rods to me means no moment of inertia around the $z$-axis. No rolling.
Seems to me you think of massive tubes with a nonzero diameter ?

Yes--massive rods (not tubes, but that really doesn't matter here), with non-zero diameter, with surface friction during impacts, and thus capable of rolling.

 

[BvU]

How does one determine the degrees of freedom of a "gas" of thin rods, so as to know the rods' equilibrium energy distribution between translation and the two forms of rotation?

Where is this going ?

The conventional model of an ideal gas starts out with point masses (funy enough, with collisions) and is then expanded perturbation-wise in the direction of real gas phenomena. I'm not sure how your exploration fits in with the limitations of the model ?

 

[hkyriazi]

Where is this going ?

The conventional model of an ideal gas starts out with point masses (funy enough, with collisions) and is then expanded perturbation-wise in the direction of real gas phenomena. I'm not sure how your exploration fits in with the limitations of the model ?

I'm trying to understand the thinking behind the notion of degrees of freedom. I'm fairly certain those dealing with diatomic gases, for example, don't consider the possibility of friction, and thus rotation along the z-axis. Do I need to go back and read the original works of Clausius and other pioneers of the kinetic theory of gases to learn the reasoning involved?

 

[hkyriazi]

Where is this going ?

Where does it need to go, other than achieving a grasp of the fundamentals? In any case, I'm now thinking the equilibrium energy distribution should be half in translation, 1/3rd in twirling, and 1/6th in rolling, based on 6 degrees of freedom in total (3 translation and 3 rotation).