Since friction with the horizontal table is negligible, by conservation of linear momentum, the center of mass (com) of the system bead/ring will travel at a constant velocity.
The position of the com of the system bead/ring is the barycenter of the bead (mass m) and the center of the ring (mass 5m).
This last statement implies the following:
In the frame of reference of the com, the bead will be constrained to traveling in a circle of radius 5R/6, centered on the com.
The initial velocity of the com in the frame of reference of the table is v/6.
The initial velocity of the bead in the frame of reference of the com is 5v/6.
Since friction between the bead and the ring is negligible, by conservation of angular momentum, in the frame of reference of the com, the circular motion of the bead will be at a constant...