lets neglect gravity and consider the rope weightless,
No, lets not. Why? Because the rope is
not weightless. Once you have explained how you can neglect gravity, then we can proceed to the next part of your story. Such as imaging how your balls can travel in this way....
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"I tell you, we are here on Earth to fart around, and don't let anybody tell you any different." Kurt Vonnegut, Jr.