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If alpha is greater than 0 the rings are small and far apart, so I expect their gravitational energy to pull dumbbell along z-axis, it attracts the masses of the dumbbell and they are as close to the masses of the small rings if they are along z.
If alpha is smaller than 0 the rings are large and close so the gravitation of the ring is coming from a direction under a high angle theta, meaning that the average origin of the potential of the rings (above and below the dumbbell) lies in the xy-plane, meaning that to be as close as possible to this the dumbbells will be in this plane too, theta=pi/2.
alpha equal to 0 will then be the point where the two previous cases go go over in each other, meaning no preferred direction.
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