This is cool, it’s sort of like the hanging chain problem. Simplest solution probably uses the Lagrangian. Basically you want to minimize the potential energy, which is found by adding the gravitational potential and the spring energy as a function of where on the spring the weight is hanging. Then integrate the Euler-Lagrange equation to get the shape of the uncoiling spring.
See https://en.wikipedia.org/wiki/Catenary for info about the hanging chain. There are good online tutorials about how to solve it but a quick web search didn’t find the one I remembered.



Wonder if they can trigger cosmic inflation in their lab.