Spring with mass problem

1. The problem statement, all variables and given/known data
A slinky, a helical spring (used as a toy which is able to travel down a flight of steps) whose unstretched length is negligibly small, obeys Hooke’s law with a good approximation, and it has a considerable elongation due to its own weight.

a) The slinky of mass m resting on the table is slowly raised at its top end until its lower end just raised from the table. The length of the spring at this position is L. How much work was done during lifting the spring?

b) If the slinky is released from this position, interestingly its lowermost turn does not move until the whole spring reaches its totally compressed length, (see the figure). What is the initial speed of the slinky at which it begins to fall right after reaching its totally compressed position?

2. Relevant equations

3. The attempt at a solution
I have never dealt with springs having mass so I am not sure how to start with the problems.

If the spring would have been massless, the work done is given by ##\frac{1}{2}kL^2## but it isn’t and even if this formula applies to the current situation, I don’t have the value of ##k##. How to proceed?

Any help is appreciated. Thanks!


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