Need help proving MoI is negligible

[Chiborino]

I did an experiment to measure the spring constant of a torsion spring (lab set-up pictured here:

I have no problems calculating anything, but in the calculations, we assumed the inertia of the spring and axis to be negligibly small and it worked nicely. In the lab report I need to be able to prove this claim but I have no idea how to do it without taking the entire apparatus apart and measuring everything, which I'm not allowed to do anyways.

How do I go about proving that only the masses (2, 3, and 4 in the picture) provide non-negligible contributions to the moment of inertia of the system?

In case it helps, the total inertia calculated is:
$\frac{1}{12}(M_{rod}L^{2}+M_{cyl}[6r_{o}^2+6r_{i}^2+2h^2+24d^2]$

 

[Chiborino]

My current idea is to use our data for the equation of motion of this thing without the masses so ONLY that negligible inertia is there and use our calculated spring constant and angular frequency to calculate what the inertia actually is. But using something we calculated by ignoring inertia to calculate the ignored inertia seems circular to me.

 

[AlephZero]

You should be able to show the MOI of the shaft is negligible by estimating its dimensions and density. You don't need to calculate it to 6 decimal places just to show it doesn't matter.

For the torsion spring, I guess the outer end is fixed so it doesn't rotate.