# The Bizarre Behavior of Rotating Bodies, Explained

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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
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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