Tennis Racquet Flip

What it shows:

A simple and convincing demonstration of the intermediate axis theorem. Consider an object (a tennis racquet in this case) with three unequal principle moments of inertia. If the racquet is set into rotation about either the axis of greatest moment or least moment and is thereafter subject to no external torques, the resulting motion is stable. However, rotation about the axis of intermediate principle moment of inertia is unstable — the smallest perturbation grows and the rotation axis does not remain close to the initial axis of rotation.

How it works:

The lowest rotational inertia of the tennis racquet is associated with the axis of rotation that runs down the length of the handle (z-axis in the illustration) and it is thus easiest to spin it about that axis. The highest rotational inertia has the axis of rotation perpendicular to the plane of the racquet and passing through the COM (y-axis), and it requires the greatest torque to get it spinning about that axis. The third axis (x-axis) is in the plane of the racquet, perpendicular to the other two axes, with an intermediate rotational inertia. The racquet is set in motion to rotate about any of these three axes by simply orienting it properly and flipping it into the air. The subsequent rotation is totally stable about the axis involving the lowest or highest rotational inertia — the rotation is unaffected by any extraneous motion of the hand that might perturb the pure rotation. On the other hand, rotation about the intermediate axis is unstable and very sensitive to any accidental motion about the other two axes — the smallest perturbation grows rapidly and the rotation axis changes; e.g., the racquet "flips over."

tennis racquet

To demonstrate this and make the instability obvious, one side of the racquet is covered with red tape and the other side with green tape. Holding the racquet by its handle, it is thrown it into the air in such a way as to make it rotate once about the intermediate axis before catching it again by its handle. If one starts with the red side up before the throw, it will be caught with the green side up (and vice versa). Not so for the other two axes.

Setting it up:

The set-up is trivial — just supply the lecturer with the racquet.


The intermediate axis theorem is a consequence of Euler's equations for the force-free motion of a rigid body, but it is by no means physically obvious and we don't have an intuitive understanding of the motion of a rigid body with three unequal principle moments of inertia. But we're in good company. For example, John Mallinckrodt (CSU Pomona) relates the story of a student asking Richard Feynman if there is any intuitive way to understand the result; Feynman went into deep thought for about 10 or 15 seconds and answered, "no." If you know of a plausibility argument why it makes sense that rotation is not stable about the intermediate axis, let us know! Meanwhile, problem 9.14 (p. 417) and exercise 9.33 (p. 421) in David Morin's book, Classical Mechanics, (Cambridge University Press, 2007) will guide you through the mathematics.