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Friction of spheres on a rotating parabolic support

This article illustrates the role of friction on the motion of a rolling sphere on pedagogical example. We use a parabolic support rotating around it axis to study the static equilibrium positions of a single sphere. Due to the particular choice of the shape of support, some easy analytical calculations allow theoretical predictions. (i) In the frictionless case, there is an eigen frequency of rotation where the gravity balances the centrifugal force. All positions on the parabola are therefore in static equilibrium. At others rates of rotation, the sphere can go to the center or escape to infinity. It depends only on the sign of the detuning with the eigenfrequency. (ii) In contrast, we show that the static friction imposes a range of equilibrium positions at all rotating rates. These predictions can be compared to the maximum equilibrium radius measured on the experimental device. A reasonable estimate of the static friction between the support and spheres made of different materials can be extracted from this maximum equilibrium radius. To go further in understanding of the experimental results, we perform a stability analysis also in the case with friction. This analysis involves the rolling friction. We show that this rolling friction can be estimated in the same device just by the check the dissipation during the motion of the sphere.