What is the tension of the rope?

[hutchphd]

My guess is that losses due to non-ideal flexibility and extensibility of the rope are much greater than to air resistance and so it wouldn't make a lot of difference.

Again one needs to be precise here. Neither lack of flexibility nor extensibility is necessarily a "loss" mechanism. This same problem with a good spring ( or a mass on a spring ) will self-evidently produce a system that will oscillate forever. This system will do the same.
The details of the loss mechanism depend upon which other degrees off freedom are coupled...for me that is not particularly interesting

 

[Chestermiller]

Again one needs to be precise here. Neither lack of flexibility nor extensibility is necessarily a "loss" mechanism. This same problem with a good spring ( or a mass on a spring ) will self-evidently produce a system that will oscillate forever. This system will do the same.
The details of the loss mechanism depend upon which other degrees off freedom are coupled...for me that is not particularly interesting

I totally agree. I am formulating a catenary version of this model in which the rope is extensible, but has no bending rigidity (negligible moment of inertia). Anyone interested in participating and discussing such a model (which is non-dissipative)?

 

[pbuk]

Again one needs to be precise here. Neither lack of flexibility nor extensibility is necessarily a "loss" mechanism.

OK, I should have said:

My guess is that the practical limitation in peak kinetic energy due to non-ideal flexibility and extensibility of the rope is much greater than due to air resistance and so it wouldn't make a lot of difference.​

This same problem with a good spring ( or a mass on a spring ) will self-evidently produce a system that will oscillate forever. This system will do the same.

With an ideal (inextensible) rope what is there to oscillate?

 

[hutchphd]

Where do I find an inextensible rope?
Seriously in the limit of unobtainable materials there will be an infinite tension at the bottom and the tip of the rope will snap back up to the original position and the cycle begins again. In the real world this is of course fabulously unlikely to happen. Absent other "frictional" degrees of freedom this thing will just keep wiggling.
If the rope is a perfect spring it is easy to see that this oscillation is correct.

 

[haruspex]

My guess is that losses due to non-ideal flexibility and extensibility of the rope are much greater than to air resistance and so it wouldn't make a lot of difference. Or are you thinking that a vacuum would make it possible to use something really thin and flexible like an aramid or Vectran fibre? OK, this would be interesting

The energy-conserving equation says the initial GPE ends up concentrated in a vanishingly small element of rope.
Air resistance rises with speed, quadratically even. That makes it very effective at capping the speed. Non-ideal flexibility should be speed-independent, I would think, so only absorbs a certain amount of energy.