Hula-Hoop-Like Motion and Subcritical Bifurcation of the Rotation of an Elliptical Paper Disk Coated with Camphor

When an elliptical camphor-coated paper disk, with the center of mass fixed at the axis of rotation, is placed on the surface of water, it rotates spontaneously with a constant angular velocity. In experiments involving different water depths, we found that a subcritical bifurcation occurs; i.e., the camphor-coated disk remains at rest when the water depth is shallow, rotates when sufficiently deep, and exists in both the rotational and rest states at the intermediate depth. The camphor-coated disk exhibits a hula-hoop-like motion as it rotates. We propose a phenomenological model of this phenomenon that includes the effect of friction at the rotation axis. The model produces behaviors similar to the experimental results, such as the hula-hoop-like motion of the camphor-coated disk and the subcritical bifurcation. These results show that the friction is important for the emergence of both the subcritical bifurcation and the hula-hoop-like motion.