When two masses are rotating around a center point but connected by a spring, what happens?

Question

Let's say the masses are each 1 kg, the initial velocity in a circular path clockwise around the center point is 1 m s^-1, and the massless spring connecting them has k= 2 N m^-1, with a resting length of 0.5 m. The supporting surface is frictionless. What path will the masses follow, in detail? Will the system radially oscillate indefinitely?

Answer 1

Wow interesting question. Not sure I've seen a problem like this, but I'll give it a try:

My first thought is that the exact motion will depend on the initial conditions, which are a little vague here. There is a "sweet spot," which you can find by identifying that the centripetal force here is provided by the restorative force of the spring. This yields a quadratic equation that yields a definitive number for R. If the system somehow starts with the given velocity of 1 m/s at this exact distance, then it seems that it would stay at this condition indefinitely - in uniform circular motion - in the absence of friction.

If, however, the 1m/s speed is somehow initiated while the spring is at any other length, then yes, I suppose it would radially oscillate indefinitely, keeping in mind that it would conserve angular momentum the whole time (which can only really be calculated if you have an initial condition). So, the rotation would cause the spring to stretch up until some maximum point, at which time the rotational speed will reach its minimum. As the spring then begins to contract, the rotation will speed up until it reaches its maximum rotational speed when the spring is compressed to its minimum length. I can't think of a good reason why this stretching and un-stretching wouldn't go on forever.