Skipping-rope in space On a space station a person wants to play skipping-rope. Although they cannot jump, they turn the rope at a constant angular velocity.
What would the shape of the skipping rope be, and could we set up an equation by taking a generalised case?
 A: For obvious reasons the curve is called the Skipping  Rope Curve. It is also known as the Troposkein . A mathematical expression for it is more difficult to derive than in the case of a stationary rope sagging under gravity, for which the result is the Catenary. 
For the catenary, the external force on each element of the rope is gravity, which is the same on each element. For the rotating rope, in the absence of gravity, the "external force" is the fictitious centrifugal force, which is proportional to the distance of the element from the rotation axis. The result is the Jacobi Elliptic Function denoted 'sn'. 
A: The situation is not quite the same as a catenary. In a catenary, the rope is in a uniform gravitational field. In this situation, once steady state is reached, the rope could have a centripetal force outwards, which would mean it's "gravity" increases linearly with distance from the center.
Consider the derivation given in the wikipedia article. You'd need to change $T\sin\phi = \lambda gs$. But it changes to a number that's proportional to the distance of the rope from the astronaut's hand. That's actually a hard number to parameterize, because it's needs the solution we are trying to obtain!
I'll leave it to others to figure out the solution, but qualitatively, you would expect it to be a little more pointy than a normal catenary, since the middle of the rope would be effectively heavier.
