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Homework Statement:

It is a mass doing circular motion on a smooth horizontal table.
For a given maximum tension ##F_(max)## , I can solve for a maximum angular speed ##\omega_c## of circular motion. But I don't understand how Under some conditions the system can only achieve a maximum angular frequency ##ω_i < ω_c##.
I don't understand part (d)
Relevant Equations:

Hooke's law $$F=k(RR_0)$$
Tension provides as centripetal force $$k(RR_0)=m\omega^2R$$
and this is my solution
for question (d), it may seems that $$R=(k)/(km\omega^2)R_0$$ so that $$\omega ≠ \omega_i =√(k/m)$$
but $$\omega_c <\sqrt{k/m}$$ is always true, ##\omega_i## corresponds to the limit case when ##F_max## is infinitely large
Besides, I don't know other Physics prevents ##\omega## reaches ##\omega_c##