Orientation guess

[axlamelas]

For alpha > 0 => h > 2 sqrt(R). In the limit were h / (2 sqrt(R)) >> 1, the rings will seam like two opposite points in the axis. To have the lowest potential energy, the mass needs to be as close as possible, therefore, the orientation with the dumb-bell align with the vertical axis will be the lowest in energy. Following the same logic, for alpha < 0 => h < 2 sqrt(R), the lowest energy configuration will be with the dumb-bells parallel to the rings. For alpha = 0, the intermidiate configuration should have the lowest energy.

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