Dec 5, 2020
Why a Tennis Racket Led Russians to Believe the Earth Was Going to Flip
The basic physics behind the Intermediate axis theorem.
If you are reading this on your phone, have one near you, or have a tennis racket with you, flip it 4–5 times as shown below, trying to rotate it perfectly about its horizontal axis and catching it at the edge.
Unless you are a very coordinated person, you likely observed that when you caught whatever object you rotated, it not only rotated about the axis you turned it about but also did a small little turn along another axis. That abnormal behavior that breaks our intuition was described in the tennis racket theorem or more commonly known as the intermediate axis theorem.
In essence, this theorem states that rotation about the first and third principal axes of an object is stable while the rotation about the second or intermediate axis is not. But what does that mean?
Laws of Angular Motion
To understand why the tennis racket effect baffled scientists, we must first understand the basics of how physicists today believe objects in motion behave.
Newton’s 3 laws of motion tell us that objects that begin at rest tend to stay at rest unless acted upon by a force, and will remain in motion until again acted upon by a force. What isn’t always clear about these laws is that the motion Newton describes is directional. That means that if a force is horizontally applied to an object, that object will only begin to move in the horizontal direction the force is applied. In order to move the object vertically, a separate force would have to act on the object. The same is true in rotational motion. A torque (which is a force applied at a certain distance) can only cause an object to rotate along the axis that it was applied.
So, you might be wondering — If I only apply a torque along one axis to this object, then why does the object rotate along another?
Moments of Inertia
To understand the theorem we must first understand what an intermediate axis is. Any 3-dimensional object has a number of principal axes of rotation, usually one along each of its three coordinate axes (x,y,z). For an object like a sphere, due to symmetry, there is only one principal axis. For an iPhone, they are the axis along the direction where the charger is plugged in from the bottom, an axis perpendicular to the screen at its center, and finally the intermediate axis, which is the axis perpendicular to the side face of the phone and passing through its center. These three axes can be identified by their moments of inertia.
The moment of inertia of a rotational object is basically how hard it is to move that object. For example, if you had a meter tall ball of aluminium, it would require much more torque (Force applied at a radius) to get it rolling as compared to a small ping pong ball. Different shapes have different moments of inertia, as shown in the table below which displays some common rotational objects.
The 1st axis of a rotational object is generally the one with the smallest moment of inertia, which is the one going through the bottom of the phone (in the direction a charger plugs in). Rotating around this axis requires the least amount of force to accelerate it because the mass distribution is close to the axis. Along the 3rd axis, which goes through the screen, the mass distribution is furthest away, and therefore spinning about this axis has the greatest moment of inertia. In these two axes, any disturbances or rotations in other axes are mitigated and result in stable rotation about that axis.
An Intuitive Description of the Theorem, and Rotoational Forces
While we earlier questioned why the object flips when given no torque along the intermediate axis, we forgot about a major force that appears in rotational motion: The centripetal force.
You know those amusement park rides where you are in a giant pirate ship and it flips around? Well, it turns out the harness isn’t what really keeps you from falling.
When an object is spinning in a circle, it has a net force on it, called the centripetal force. This net force is always pointed towards the center of the circle and allows the object to rotate at a constant speed. In the case of the flipping pirate ship, the seat you sit in and the long support beams apply a force onto you, keeping you moving along a circular path.
In the reference frame of your body, what you experience is called the centrifugal force. You feel this force as pressure from the seat as it pushes your body along the circular path of the ship.
If we treat an object, such as your phone, as a collection of a bunch of tiny particles, or point-masses, then we can understand that when spinning it, each of these point-masses will experience a centrifugal force. Well, it turns out that, according to simulation, the intermediate axis is the only one in which the centrifugal force will cause the edge point-masses to move further away from the natural plane they began in. This is what causes the flip that we observe.
It’s really that simple, but yet super weird. Why only this axis? Physicists are still uncovering more details, but for now, we accept this explanation to be true because it agrees with the mathematical proof that has been around for over a hundred.
The Math
Using conservation of angular momentum and Euler’s equations, the unstable rotation can be observed mathematically. When rotating around its first axis, it can be assumed that the angular velocity is greatest for that axis while the other axes have minimal angular velocities as they are small disturbances. By taking the derivative of the second equation and substituting, it is observed that rotation about the other axes is opposed and cannot be magnified. The same follows for the third axis.
For the second axis, however, when differentiating the first equation and substituting, it is evident that small disturbances around other axes cause the object to flip. I suggest reading the papers themselves, this Wikipedia article, or this mathoverflow thread.
How was this discovered?
In 1985 cosmonaut Vladimir Dzhanibekov was sent to save the Salyut-7 Soviet Space Station, which needed a great rescue. The mission was so incredible that a movie, “Salyut 7” was made from it. When unpacking supplies, Vladimir Dzhanibekov spun off a wing nut that was attached to a bolt, and as he watched it move through space, he observed the strange behavior of the nut flipping around periodically even though no force was applied to it. Years later, in 1991, a paper was published called “The Twisting Tennis Racket” and explained that a tennis racket makes a half turn through an unstable axis. This paper had no mention of the behavior Vladimir Dzhanibekov observed, mainly because the Russians hid his findings and observations for almost 10 years. Why did they hide it?
The End of the World
Cosmonauts and the Russians believed that since the Earth was a rotating body, it too could flip over during its orbit. Thankfully, this hypothesis was disproved. In space, astronauts rotated objects like cylinders, which, as expected, rotated stably around their first and third axis. However, a liquid-filled cylinder rotated about its first axis is not stable. Although some would expect that it should because kinetic energy and angular momentum are conserved, energy in a liquid-filled cylinder is dissipated in forms such as heat and results in the object rotating about its axis with the largest moment of inertia.
Conclusion
The Intermediate axis is one of the great phenomena of rotational motion. Its discovery (which was actually a rediscovery) confused modern scientists, but in reality, the mathematical explanation had been worked out and proven many years prior to the events that took place on the Russian Salyut-7. The history of the theorem serves as a reminder to the scientific and mathematics community that even though solving equations may appear trivial, it can hold the answers to future questions.
Sources
A lot of good information was from Veritasium’s youtube video, I highly suggest watching it: https://youtu.be/1VPfZ_XzisU
Explanations of the physics and math behind this theorem: https://thatsmaths.com/2019/12/12/the-intermediate-axis-theorem/
