A Curious Twist -- A Physics Question

Factual Questions
1

If you hold in your hand, at arm’s length, a rectangular solid of consistent density (e.g. a short section of a 2 X 6, a pack of playing cards, a box of facial tissues) and toss it up into the air with sufficient force so that it flips it will behave differently depending on which of three imaginary axes you attempt to make it pivot around.

The three axes would correspond to the three primary dimensions of the object (length, width, and thickness). (Of course there are many imaginary axes, but we are only considering the three “obvious” axes of symmetry, the ones that pass through the center of the solid; correspond with the dimensions of length, width, and thickness; and are parallel with the surfaces of the object.)

Toss the object three times, once with each of the three axes perpendicular with your arm and parallel with the ground.

(Of course, this is not, strictly speaking, a “scientifically controlled” experiment, since your fingers and thumb might put more force on one side of the solid or the other. Nevertheless, you will immediately see that the solid behaves
quite differently when tossed with the intent of rotating it around the intermediate axis.)

These are the three possibilities:

Axis of rotation = the longest axis….The object will “spin,” almost as if on a material axle.

Axis of rotation = the shortest axis…. The object will “spin,” almost as if on a material axle.

Axis of rotation = the intermediate axis….The object will “twist” rather than spin around this axis.

To me this is curious and I have never heard a clear explanation for WHY the object twists when you attempt to rotate it around the intermediate axis. I would be obliged if someone can explain it.

2

I’m at work right now, but I have notes buried somewhere at home from a Differential Equations “guest” lecture back in school that explained this exact problem. Alar Toomre was the “guest lecturer” (he’s another math professor at MIT and was also my TA at the time I took 18.03 from Giancarlo Rota – that’s a shout-out for all my MIT peeps) once and he gave the famous “Yellow Box” lecture and explained the physics behind all of this… Almost wholly unrelated to differential equations, but fascinating stuff nonetheless. I’ll see if I can dig up my notes when I get home tonight.

3

The short answer is that while all three axes are equilbrium states of rotational kinetic energy, the rotation around the secondary axis is an unstable equilbrium. That is, if you spin the box so that it rotates exactly around the axis, it will continue to rotate around that axis until disturbed, regardless of which axis of symmetry you choose. However, if your rotation is not exactly perfect, there will be twisting forces applied by the out-of-alignment rotation. In the case of the first and third axes, these forces tend to return the axis of rotation toward the equilibrium-the box may ‘wobble’ a bit, but the axis will continue to point in the same general direction. For rotation around the second axis, though, these forces point away from those needed to return the rotation to that axis, so a slight wobble is quickly magnified to complete chaos.

Yeah, that was the short answer. :slight_smile: For the long answer, there’s really nothing you can do but the calculus-find the moment of inertia as a function of the rotational axis in the vicinity of those three axes of symmetry, and see whether the value of the moment when the axes coincide is a local maximum or a local minimum …

4

It’s been a long time since I studied mechanics, and I don’t remember the mathematics of it, but it’s not just symmetrical solids which have three orthogonal axes of rotation like that: it’s any solid object, no matter how irregular. I once saw the proof, in a course on applied mathematics, so I know it exists, but I have no idea how it goes.

5

Clear in the context of a random person on the street, or clear in the context of a grduate level physics class in mechanics? :slight_smile:

I understood in once in the second context, about fifteen years ago. Briefly, the spinning object has to satisfy conservation of energy and conservation of angular momentum. If you set up coordinates aligned with the body’s three axes, and scaled so that the energy as a function of the body’s angle is a sphere, then the corresponding surface of constant angular momentum is an ellipsoid with three unequal axes.

Spinning the body around one of the stable axes corresponds to the case where the constant-energy surface intersects the ellipsoid at the longest axis. If your energy sphere is a little smaller, it intersects the momentum ellipsod in a small circle near the axis.

Spinning the body around the other stable axis corresponds to the case where the constant-energy surface intersects the ellipsoid at the shortest axis. If your energy sphere is a little bigger, it intersects the momentum ellipsod in a small circle near that axis.

Spinning the body around the unstable axes corresponds to the case where the constant-energy surface intersects the ellipsoid at the longest axis. If your energy sphere is a little smaller, it intersects the momentum ellipsod not just at the middle axis, but in a shape corresponding to two perpendicular Great Circles intersecting at the middle axis.

If your initial rotation axis is not exactly along one of the ellipsoid’s axes, the point your body is at moves around the curve of the intersection. For the two stable cases, that’s just going around the small circle, but for the unstable case, it covers a large part of the body.

That might be about as clear as your going to get.

6

We definitely did this in a lecture, and I remember that the proof was convincing at the time, but no-one produced a way you could see why, it was a matter of “here’s what it is, and here’s a matrix that’s representative of it, and Lo! it has three axes, two stable and one unstable…” But then I became a pure mathematician where I had a good reason for not understanding stuff :slight_smile:

7

Should read:

Spinning the body around the unstable axes corresponds to the case where the constant-energy surface intersects the ellipsoid at the middle axis. But it intersects the momentum ellipsod not just at the middle axis, but in a shape corresponding to two perpendicular Great Circles intersecting at the middle axis.

Sorry about that …

8

You’re talking about the principal axes of a rigid body. These are the axes about which the “products of inertia” (the off-diagonal terms of the inertia matrix) vanish. Hence, a body rotating with its angular velocity vector along a principal axis will also have its angular momentum vector aligned along that axis, so no external moments are needed to maintain the motion, which is another way of saying it is an equilibrium condition. The equilibrium may be shown to be stable if the spin is about the axis with the largest or smallest moment of inertia, but unstable if the spin is about the axis with the intermediate moment of inertia. On preview I see that this adds nothing to the discussion, but I believe there is a proof in Kleppner and Kolenkow’s mechanics textbook.

Incidentally, this is why a football thrown with a spiral will tend to keep spiraling, whereas “wounded-duck” pass will wobble erratically about all axes. It is also the basis for “spin stabilization” of projectiles and spacecraft.

9

Here is a good explanation of why the principal axes of an object cause nice neat spinning and other axes cause wobbling.

When you are spinning an object, all parts of that object are feeling the pull of centripetal force away from the axis of rotation. If you pick a principal axis to spin it about you will have all those forces balance nicely out. If the object is symmetrically constructed it is easy to see where these axes lie, but it can be more difficult with odder shapes. When you spin the object around and axis other than a principal axis it will wobble because these forces don’t balance out anymore.

Experiment: Spin on one foot (or preferably a roller or ice skate) with both arms stuck out. You are close to symmetric about the spin axis so you don’t have much wobble. Now pull one arm in to your body. You will wobble (and probably fall down if you are going fast enough). You will feel an extra pull in the direction of the extended arm.

Depending upon the shape of the object that extra pull from spinning about a non-principal axis can result in some very odd motion, but it all comes from imbalance in the centripetal forces (as well as secondary effects from coriolis forces once you have started to wobble).

10

I realize I just explained why non-principal axes result in wobble, not why spinning about one of the principal axes is more likely to degrade to wobble than another.

ZenBeam has the concept right but I may be able to explain a little better.

Back to the spinning skater. If you have your arms out and tilt one a little up it will tend to be pulled back out, making you stable. If you have your arms straight up and tilt one a little out, it will tend to be pulled out, wobbling you all over the place, unstable. Yes, this is not exactly the same thing, but I wanted to keep the same analogy and it is easy to follow.

11

This is known to some as the “Tennis racket theorem”. For a tennis racket, the unstable axis is when you hold racket by the handle and try to flip it to rotate about an axis parallel to its face. It’s surprisingly hard to do.

As a grad student, I once tried to impress my undergrad friends by betting them they couldn’t get it to flip over perfectly without twisting. One of them grabbed the tennis racket and executed a perfect flip. “C’mon, I’m a juggler”, he said.

:frowning:

(I would have paid up but he wouldn’t accept the money.)

12

Ladies & Gentlemen,

This has become a fascinating discussion for me, although I never had a single college level course in mathematics! So, as you might expect, even though I posted the question (and you all have been so kind as to offer learned explanations) I basically still have no idea why the object twists as it does. It’s not your fault, of course! You’re doing great.

I wrote, “I have never heard a clear explanation.” ZenBean responded: “Clear in the context of a random person on the street, or clear in the context of a graduate level physics class in mechanics?”

I guess I was hoping for “clear to a random person on the street.” But ZenBeam’s reply might be, as s/he said, “about as clear as you’re going to get.”

Shade said, “no-one produced a way you could see why. It was a matter of ‘here’s what it is, and here’s a matrix that’s representative of it, and Lo! it has three axes, two stable and one unstable…’”

So I’ll just be content to know that some people, somewhere, really do understand the physics behind this curious twist. In the meanwhile I’ll just sit back, have a beer, and read your responses. I hope y’all are having a bit of fun with my question……and thanks for taking the time to respond. It’s fun.

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** Tennis Racquet Theorm**