Questions about Mach's principle and empty universe has been asked many times but nowhere can I find it's cross examination with centripetal force. From my point of view the simple logic of centripedal force disproves Mach's principle.
You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?
So let's take and empty universe and put the rotating spheres inside. If Mach's principle was right then the spheres can either rotate or not rotate without any effect on the string, meaning that in empty universe the inertia of the linear motion of the spheres can be changed by the string without generating any centripetal/centrifugal force. This is direct violation of Newton's laws of motion.
An object ... continues to move at a constant velocity, unless acted upon by an external force.