From: veritasium
The phenomenon known as the Dzhanibekov effect, also referred to as the tennis racket theorem or the intermediate axis theorem, describes the counterintuitive behavior of rotating objects [00:00:13].
Discovery and Initial Secrecy: The Dzhanibekov Effect
In 1985, Soviet cosmonaut Vladimir Dzhanibekov observed this effect while on a mission to rescue the Salyut 7 space station [00:00:37]. After unpacking supplies, he noticed a wing-nut spinning off a bolt [00:00:54]. The wing-nut maintained its orientation for a short time, then flipped 180 degrees, and continued to flip back and forth at regular intervals without any external forces or torques acting on it [00:01:05]. This strange and counterintuitive phenomenon was kept secret by the Russians for ten years [00:01:30]. The mission itself was so dramatic that Russia made a movie about it in 2017 [00:00:47].
Public Recognition: The Tennis Racket Theorem
Six years later, in 1991, a paper titled “The Twisting Tennis Racket” was published in the Journal of Dynamics and Differential Equations [00:01:39]. This paper described how a tennis racket, when flipped in the air, not only rotates as intended but also performs a half-turn around an axis passing through its handle, resulting in the side originally facing the flipper facing away upon catching [00:01:52].
Although the paper suggested this twisting phenomenon seemed new and was not mentioned in general classical mechanics texts [00:04:43], it was actually present in the Landau and Lifshitz textbook cited by the authors [00:04:54]. An understanding of the intermediate axis theorem dates back at least 150 years to Louis Poinsot’s book “The New Theory of Rotating Bodies” [00:05:01]. The effect is particularly striking in microgravity [00:05:18].
Physics of Rotational Dynamics
Understanding this effect requires basic rotational dynamics concepts related to an object’s moment of inertia.
Principal Axes and Moment of Inertia
An object, like a tennis racket, has three principal axes around which it can spin [00:02:12]:
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Smallest Moment of Inertia (Easiest to Spin): An axis running through the handle of the racket [00:02:17]. Spinning around this axis is easiest, achieving the greatest angular velocity for a given torque [00:02:36]. This is because the mass is distributed closest to this axis [00:02:46]. Spins about this axis are stable [00:03:15].
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Greatest Moment of Inertia (Hardest to Spin): An axis running perpendicular to the head of the racket [00:02:27]. Spinning about this axis is slowest because the mass is distributed farthest from it [00:03:06]. This is the maximum moment of inertia axis [00:03:12]. Spins about this axis are also stable [00:03:15].
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Intermediate Moment of Inertia (Unstable Spin): An axis running parallel to the head of the racket [00:02:22]. When spinning about this axis, the intermediate axis, the half-twist phenomenon occurs [00:03:26]. This effect is not unique to tennis rackets but can be observed with cell phones or discs [00:03:42]. The key requirement for the intermediate axis effect is an object possessing three different moments of inertia about its three principal axes [00:03:56]. Objects with spherical symmetry or only two different moments of inertia (like a spinning ring) will not demonstrate this theorem [00:04:08]. The objects that demonstrate this phenomenon are called “asymmetric tops” [00:04:32].
Intuitive Explanation by Terry Tao
Richard Feynman, a famous physicist, reportedly stated there was no intuitive way to understand the intermediate axis theorem [00:05:51]. However, mathematician Terry Tao, a Fields Medal winner, provided an intuitive explanation in 2011 on Math Overflow [00:06:19].
Imagine a thin, rigid, massless disc with heavy point masses on opposite edges of the x-axis and light point masses on opposite edges of the y-axis [00:06:44].
- Rotating around the x-axis (small masses moving) yields the smallest moment of inertia [00:07:12].
- Rotating about the z-axis (all four masses moving) yields the greatest moment of inertia [00:07:18].
- Rotating about the y-axis (intermediate moment of inertia) is where the effect occurs [00:07:23].
When the disc spins perfectly about the y-axis, centripetal forces keep the large masses in uniform circular motion [00:07:28]. If we switch to a rotating frame of reference, centrifugal forces appear, pushing masses away from the rotation axis proportional to their distance [00:07:41]. In a perfect y-axis spin, only the big masses experience centrifugal force, balanced by centripetal forces [00:08:09].
If the disc is bumped slightly off the y-axis [00:08:26]:
- The small masses now experience some centrifugal force due to their distance from the y-axis [00:08:29].
- Tension forces within the disc ensure the small masses remain orthogonal to the big masses [00:08:36].
- The large masses, due to their significant inertia, constrain the small masses to largely remain in the y-z plane [00:08:41].
- The centrifugal forces on the small masses accelerate them, causing them to move further from the y-axis, increasing these forces [00:08:52].
- This acceleration continues until the small masses flip to opposite sides [00:09:03].
- For the first half of the flip, centrifugal forces accelerate the masses; in the second half, they slow them down, causing them to come to rest on the opposite side [00:09:07].
- This pattern repeats indefinitely, with the disc flipping back and forth at regular intervals [00:09:25].
Implications and Misconceptions: Could the Earth Flip?
The secrecy surrounding the Dzhanibekov effect might have stemmed from Dzhanibekov’s subsequent experiment: he attached modeling clay to the wing-nut and observed the same periodic flipping [00:09:55]. This led to speculation that if a spinning ball in space could flip, the Earth might too [00:10:05]. Amidst the Mayan prophecies of the end of the world in 2012, conspiracy theorists and media outlets found this idea irresistible [00:10:18]. The official Russian federal space agency, Roscosmos, even published an article in 2012, acknowledging the “astonishment and simultaneous danger to a certain part of the scientific world” caused by the Dzhanibekov effect and the hypothesis of the Earth’s potential overturn [00:10:29].
However, experiments by astronaut Don Pettit aboard the space station provide clues to the Earth’s stability [00:11:07]. A solid object (like a book or a cylinder) spins stably about its smallest or largest moment of inertia axes [00:11:11]. But a liquid-filled cylinder spinning about its smallest moment of inertia axis is unstable and will eventually rotate about its largest moment of inertia axis [00:11:23].
This occurs because, while an isolated object’s angular momentum remains constant, its kinetic energy can be converted into other forms, such as heat [00:11:38]. Spinning about the axis with the smallest moment of inertia corresponds to the greatest kinetic energy [00:11:59]. As this energy dissipates (e.g., through liquid sloshing), the object tends towards the state of minimum kinetic energy, which is spinning about the axis with the largest moment of inertia [00:12:06]. This is the lowest energy state for a given angular momentum [00:12:19].
The U.S. learned this with Explorer One, their first satellite [00:13:06]. Designed for spin stabilization about its long axis, it rotated end-over-end within hours [00:13:17]. Its flexible antennas allowed energy dissipation, forcing it to rotate about the axis that maximized its moment of inertia [00:13:26].
Like Explorer One, Earth has internal mechanisms for dissipating energy [00:13:06]. Over time, it has come to spin about the axis with its maximum moment of inertia [00:13:11]. Most astronomical objects behave similarly; for example, Mars’s Tharsis Rise, a major mass concentration, is located at its equator, maximizing its distance from the rotation axis and ensuring Mars rotates with its maximum moment of inertia [00:13:20]. Therefore, the Earth will not flip; it is stably spinning about its axis of maximum moment of inertia [00:13:46].
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