Why a Tennis Racket Led Russians to Believe the Earth Was Going to Flip
By: Sajiv Shah
Tags: Mechanics, Physics, Science, Astrophysics, Earth

When doing so, unless you are a very coordinated person, you likely realized then 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 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?

What’s an Intermediate Axis?

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 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.

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.

Different moment of inertia calculations for different shapes. Image from http://hyperphysics.phy-astr.gsu.edu/

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. On the other hand, in the 2nd axis, or intermediate axis, any disturbance is amplified and results in the object’s angular momentum being transferred to other axes.

Brief Description of the Physics

Image from David Muller’s Youtube Channel, Veritasium

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 are opposed and cannot be magnified. The same follows for the third axis.

Euler’s equations

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 rescuing. 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

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/