Considering tetta the angle between z axis and the dumbbell in the x-z plane (so that if tetta=90 the dumbbell is in the x-y plane), we can describe the lowest energy orientations in each case:
-If alpha > 0 , we have the minimum energy at tetta=0 or 180, so the dumbbell will be oriented parallel to the z axis, and perpendicular to the ring..
-If alpha < 0 , we have the minimum energy at tetta=90, so the dumbbell will be oriented in the x-y plane, parallel to the rings and perpendicular to the z plane.
-If alpha = 0 , then the energy doesn’t depend on tetta, so we will find the dumbbell in any position, because all of them will be equal in therms of energy.