Vertical circular motion with a changing radius

[ddddd28]

Hello,
I am curious to see how the mathematical analysis of a vertical circular motion with a sensible rope, that is to say, a rope that streches easily, looks like. k- constant of the rope.
thanks,

 

[vanhees71]

What have you done so far to get an idea? My tip: Use a Lagrangian!

 

[zwierz]

stretchable rope is of infinitely many degrees of freedom system even though the rope lies on a straight fixed line.

In addition to this OP says "vertical motion" this implies gravity. Then what is a circular motion in this context?
And how is mass distributed along the rope?