Length of a helix vs composition of forces

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roll up a spring

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The device

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My logic:

1/ I suppose the device conserved the energy without the spring (the center of gravity of the helix moves).

2/ I add the spring, compared to the case 1/, I have the works from the torques from F on the axis 'x' and 'y' and I win the difference of the length of the spring: I give these calculations in the image.

I drew it but, note the disk has inside it a nut (fixed on the disk), when the screw thread rotates (relatively to the disk) then the screw thread moves up (60°)

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I updated the images

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I can cancel the centripetal forces and ONLY them. My spring don't use the centripetal forces, it uses the rotation of the screw-thread relatively to the disk.

My device don't use the centripetal forces but at least with the device to cancel the centripetal forces I can compare :

1/ without the spring, the energy is conserve, the SUM = 0

2/ with the spring, I gave all the new energies the NEW_SUM = SUM + new energies I gave. Like the SUM = 0, the additional energies must be at zero if the energy is conserved, it seems here it is not