 According to the rotating observer, shouldn't the tension in the string be twice as big as before (the tension from the centrifugal force "plus" the extra tension needed to provide the centripetal force of rotation)?The reason the rotating observer sees zero tension is because of yet another fictitious force in the rotating world, the Coriolis force, which depends on the velocity of a moving object.In this zero-tension case, according to the rotating observer the spheres now are moving, and the Coriolis force (which depends upon velocity) is activated.