Shape of Slinky being twirled in 0 gravity

[phantomvommand]

Homework Statement

Find the shape of a Slinky inside the International Space Station (i.e. in weightless conditions) if it is rotating uniformly – like a skipping rope – with both ends of the spring twirled in unison.

Relevant Equations

F = ma

I am able to understand the textbook solution, except for its very first assumption:
We use the coordinate system shown in the figure, and find the shape ofthe spring (assumed to have already attained its stable configuration) in this frame.

Why is it fair to assume that the slinky will ever reach a stable configuration (ie equilibrium)? Why can't it keep spinning around like a skipping rope?

 

[haruspex]

Why is it fair to assume that the slinky will ever reach a stable configuration (ie equilibrium)? Why can't it keep spinning around like a skipping rope?

A stable configuration just means its shape is not changing as it spins.
From the diagram, it appears the ends are anchored.

 

[phantomvommand]

A stable configuration just means its shape is not changing as it spins.
From the diagram, it appears the ends are anchored.

Meaning to say, the x and y axes in the solution rotate, right? That clears it up. Thanks!

 

[haruspex]

Meaning to say, the x and y axes in the solution rotate, right? That clears it up. Thanks!

That's not how I would put it. The coordinate axes can stay where they are; the spring rotates as though rigid about the x axis.

 

[jbriggs444]

If the motion is like a jump rope then the $x$ axis (the line between the anchored end points) would remain in place. One then has a choice to make.

One could use an inertial coordinates where points on the rope are rotating in the $y$ and $z$ directions. With this choice, the bits of rope would be subject to centripetal acceleration.

Or one could use rotating coordinates where the coordinate system rotates with the rope. Each point on the rope would have a fixed $y$ position while its $z$ coordinate would always be zero. With this coice, the bits of the rope would be stationary, but subject to centrifugal force.

My preference would be to use rotating coordinates and centrifugal force. One less coordinate to worry about. This appears to match the choice made by the people who posed the problem.