The Bizarre Behavior of Rotating Bodies

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What you are looking at is known as the Dzhanibekov effect, or the tennis racket theorem, or the intermediate axis theorem, but we'll get to that. Now, you may have seen clips like this one before, but in this video, I will provide the best intuitive explanation of how this effect works, or at least, that's my goal. Now, it involves arguably the best mathematician alive, Soviet era secrets, and the end of the world.

So, in 1985, cosmonaut Vladimir Dzhanibekov was tasked with saving the Soviet space station Salyut 7, which had completely shut down. The mission was so dramatic that the Russians made a movie out of it in 2017. After rescuing the space station, Dzhanibekov unpacked supplies sent up from Earth which were locked down with a wing-nut, and as the wing-nut spun off the bolt, he noticed something strange: the wing-nut maintained its orientation for a short time, and then it flipped, 180 degrees.

And as he kept watching, it flipped back a few seconds later, and it continued flipping back and forth at regular intervals. This motion wasn't caused by forces or torques applied to the wing-nut: there were none. And yet, it kept flipping. It was a strange and counterintuitive phenomenon—one that the Russians kept secret for 10 years. Why the secrecy? Well, that is what we're gonna find out.

Six years later, in 1991, a paper was published in the Journal of Dynamics and Differential Equations called "The Twisting Tennis Racket," and although it was related, it of course makes no mention of the secret Dzhanibekov effect. The paper says if you hold a tennis racket facing you, and then flip it in the air like this, it not only rotates the way you intend it to, it also makes a half turn around an axis that passes through its handle, so the side that was originally facing you will be facing away when you catch it.

Now, to understand this, we need to go through some basics. Like there are three ways to spin a tennis racket about its three principal axes. The first is about an axis that runs through the handle, like this. The second is the way we were spinning it before, with an axis that runs parallel to the head of the racket, and the third is about an axis that runs perpendicular to the head of the racket.

Now, it's easier to spin the racket around some of these axes than others. That is, you get more angular velocity for a given amount of torque. It's easiest to spin the racket around this first axis; it gets going really fast, and that is because the mass is distributed closer to this axis than to any of the others. We say its moment of inertia is the smallest when spinning in this orientation. Spinning about the third axis has the greatest moment of inertia, and so the racket gets spinning pretty slowly, and that's because this mass is distributed as far from this axis as possible; so this is the maximum moment of inertia axis.

Now, what you'll notice with spins about these axes is that they're stable. There's no rotation happening about any of the other axes when you try to rotate around the first or third axes. But rotating about the second axis, the intermediate axis, where the moment of inertia is in between the other two, well that is where you get this half twist, and there's virtually nothing you can do to stop it.

And it's not just tennis rackets, of course. I've done this before with cell phones and with a disc with a hole in it. I took this disc on an ice rink and in a zero-g plane. I have been obsessed with the intermediate axis theorem, and what you need to make the intermediate axis effect work is an object that has three different moments of inertia about its three principal axes.

Well, that's not every object. This object, for example, a spinning ring, has only two different moments of inertia for rotation like that, and then rotations like this. Spinning things is not a specialty. Wow, I feel like it should be—rotations like that. That's the one I was looking for. Anything with spherical symmetry has only one moment of inertia, so these objects will not demonstrate the tennis racket theorem. For that, you need what's called an asymmetric top—something with three different moments of inertia in its three different principal axes.

Now, the tennis racket paper claims the twisting phenomenon seems to be new; it is not mentioned in general texts on classical mechanics amongst other sources that they've checked, but it is actually. It's even in the textbook they've cited—Landau and Lifschitz. In fact, an understanding of the intermediate axis theorem goes back at least another hundred and fifty years to a book called "The New Theory of Rotating Bodies" by Louis Poinsot. So this is old physics, but in space, the phenomenon looks like something new.

In microgravity, the effects are just so much more striking than a half twist of a tennis racket, and at random intervals on social media, these videos crop up to frenzied questions of, "Is this real?" and "What's going on?" and "How does this work?" Well, a number of simulations and animations have been made, but if you really want to understand what's happening, most people resort to the math, including me in the past.

Well, the mathematics is kind of complicated and boy is there a lot of math. There's this story of a student who asked famous physicist Richard Feynman if there was any intuitive way of understanding the intermediate axis theorem, and as the story goes, he thought about it carefully and deeply for ten or fifteen seconds, and then said... "No."

Well, the goal of this video is to prove Feynman wrong—to provide an intuitive explanation of the intermediate axis theorem, but the explanation is not mine; it actually comes from one of the greatest living mathematicians, Terry Tao. He has won the Fields Medal amongst a host of other awards, and for this video, I actually asked him for an interview, but he declined because he's busy solving centuries-old math problems, so, you know, fair enough.

But that's okay, because we have the explanation he posted to Math Overflow in 2011. It goes like this: Imagine we have a thin rigid massless disc centered in our coordinate system. Now add some heavy point masses to opposite edges of the disk on the x-axis. Even though they're point masses, I'll put some large cubes around them to remind us of their significant mass.

Then, add some light point masses on opposite edges of the disc on the y-axis. Now, this disc has three different moments of inertia about its three principal axes. Rotating around the x-axis has the smallest moment of inertia, since only the light masses are moving. Rotating about the z-axis has the greatest moment of inertia, since all four masses are going around, and rotating about the y-axis has the intermediate moment of inertia.

Rotating like this, the only forces in the disc are centripetal forces which accelerate the big masses towards the center. This keeps them turning in uniform circular motion. Now, what if we change reference frames? So, now we're rotating with the disc. Well, then we see centrifugal forces appear. Normally, I don't like talking about centrifugal forces because, well, if you analyze things in inertial frames of reference, you never have to deal with them.

But, if you're in a rotating frame of reference, then centrifugal forces do appear in the analysis, pushing any masses away from the rotation axis, and those forces are proportional to their distance from the axis—in this case, the y-axis. So here, there is no centrifugal force on the small masses because they're located right on the y-axis. So the only centrifugal force acts on the big masses outwards, and that's balanced by the centripetal forces pushing inwards.

Now, this is all fine and good, but what if the disc is bumped so that it's no longer rotating perfectly about the y-axis? Well, now the small masses will experience some centrifugal force proportional to their distance from the y-axis. Tension forces within the disc ensure that these small masses remain orthogonal to the big masses, and since the big masses are still spinning in roughly the same positions as they were before, with lots of inertia, they constrain the small masses to lie more or less in the y-z plane.

The little centrifugal forces on these small masses start accelerating them, and those forces get bigger as the masses move further and further from the y-axis, and they keep accelerating until they flip onto opposite sides. Now, for the first half of this flip, the centrifugal forces are accelerating the small masses, but in the second half, the centrifugal forces slow the masses down, reversing all the previous acceleration so that they basically come to rest when they reach the opposite side.

The pattern then repeats indefinitely, with the disc flipping back and forth at regular intervals, and there you have it—an intuitive explanation for the intermediate axis theorem, or tennis racket theorem, or Dzhanibekov effect, or whatever you want to call it.

So if this is well-established classical physics, why did the Soviets make it classified for ten years? Well, possibly because of what Dzhanibekov did after observing the strange behavior of the wing-nut. He attached a ball of modeling clay or plasticine to it and tried spinning that. And sure enough, he found that just like the wing-nut, this ball flipped over periodically, and the implication was that maybe since the Earth is a spinning ball in space, it too could flip over.

I mean, we know the Earth's magnetic poles have reversed in the past, so could this be related? In 2012, with the Mayan prophecies of the end of the world, speculation about the Dzhanibekov effect proved irresistible for some conspiracy theorists and people in the media. Plus, on May 13th, 2012, the official site of the Russian federal space agency, RusCosmos, posted an article in honor of Dzhanibekov's 70th birthday, and in it, they said the spinning nut of Dzhanibekov caused astonishment and simultaneous danger to a certain part of the scientific world.

A hypothesis was proposed that our planet, in the course of its orbital motion, can execute the same overturn. So, how do we assess the validity of this hypothesis? I mean, is the Earth actually going to flip over? Well, we can get some clues from simple experiments performed by astronaut Don Pettit aboard the space station. He shows that a book will spin stably about its first or third axis, just as we'd expect, and a solid cylinder will also spin stably around its first or third axis.

But a liquid-filled cylinder spinning about the first axis—that's the one with the smallest moment of inertia—it's unstable, and it'll end up rotating about its axis with the largest moment of inertia. Why is this? For an isolated object spinning in space, you'd probably think both its angular momentum and its kinetic energy would be constant, but that's only half true. Angular momentum stays constant, but kinetic energy can be converted into other forms of energy, like heat.

So, in this case, as the liquid's sloshing around inside, the energy can be dissipated, and spinning about the axis with the smallest moment of inertia also means spinning with the greatest kinetic energy. And as this kinetic energy is dissipated, the cylinder has no other option but to spin about the axis that achieves the minimum kinetic energy, and that is the one with the largest moment of inertia.

So when it's rotating end-over-end for a given amount of angular momentum, then rotating with the maximum moment of inertia is the lowest energy state. So that is the state that all bodies will tend towards if they have any way of dissipating their energy. The U.S. learned this the hard way with their first satellite—the Explorer One. It was designed to spin about its long axis and be spin stabilized, but within hours of achieving orbit, it was rotating end over end.

But, what happened? I mean, it seems like a rigid cylinder. Well, the problem was these flexible antennas. They allowed the satellite to dissipate energy as they swung back and forth, gradually reducing the kinetic energy of the satellite until it had to rotate by the axis that maximized its moment of inertia. Now, the Earth is just like this. It has ways of dissipating energy internally, so over time, it has come to spin about the axis with maximum moment of inertia, and most astronomical objects do the same.

Mars, for example, has a mass concentration, or major positive gravity anomaly called the Tharsis Rise, and it is located, not coincidentally, at the equator because that puts it as far as possible from the axis of rotation and ensures that Mars is rotating with the maximum moment of inertia. Most asteroids, far from rotating about random axes, they spin—almost all of them around the axis with the maximum moment of inertia.

So the Earth won't flip. It's spinning about the axis with the maximum moment of inertia, and that is stable. Hey, this part of the video is sponsored by LastPass. If you were relieved to learn that the Earth is not going to flip over, think about the relief you'd feel if you no longer had to remember all of your passwords. LastPass can do that for you.

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