I want to thank the sponsor of this episode, LastPass, which remembers your passwords so you don't have to More about them at the end of the show 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 and 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 6 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 spent 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 spend 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 access effect work is an object that has three different moments of inertia about its three principal axes and 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 a 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 it random intervals on social media, these videos crop up to frenzied questions of, "Is this real?" and "What's going on?", "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 and 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 disks 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. 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Terence Tao's original post can be found here, and is a pleasure to read.
It's important to realize that the moment of inertia is a rank 2 tensor. During a flip, the angular velocity changes, but so does the moment of inertia about the axis of rotation. The exterior product of the two (which is angular momentum), is still conserved.
Treating the moment of inertia as a 3x3 matrix, we can see that the possibility for such a "flip" and the impulse required to excite one depends on the number of eigenvalues the matrix has and much they differ from each other.
EDIT: angular momentum, not angular velocity
they almost got me worried the earth was going to flip over! :)
First they say the Earth won't flip over, but if you combine this with tectonic motion moving mountains around I get all worried again. There is a shit load of mass on Antarctica too.
Ok, so maybe the Earth won't flip, but it seems entirely possible that tectonic shifts could cause unexpected shifts in rotation.
It makes sense that the Earth rotates around it's 3rd axis since there is a slight bulge at the equator. I assume this bulge is itself caused by centrifugal forces, but it's interesting to note the self-reinforcing nature here: Rotation causes centrifugal forces to push out around the equator. The the bulge at the equator keeps the equator stationary.
You can apply the same explanation of instability for the third principal axis as well. Probably something is missing here. Can someone explain?
The explanation begins at 6:44.
The most interesting part begins at 9:43. It talks about spinning objects that dissipate their energy through viscosity, flexible antennas, etc., and concludes that such an object will lose energy until it spins stably around its axis of greatest moment of inertia, which is what requires the least energy for the given angular momentum. That settles, in the negative, the question regarding whether we could expect the Earth to flip. It won't because it has already arrived at a state where it spins stably in this way.
To say "the greatest mathematician alive" is disrespectful to so many living great mathematicians, some of them even passing since the posting of this video (Tate).
So if most astronomical objects like to spin on their third axis. Given that a massive spinning object produces a frame dragging effect on gravity, thus warping space time and as a result increasing relative mass along that plane... Is this effect why planets align with the rotation of their host stars, and galaxies form as disks around supermassive black holes?