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> In a centrifuge, $a\_c$ should be constant. If $m$ increases, the $r$ will increase in order to maintain a constant $a\_c$.
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Constant centrieptal acceleration is given by
$a\_c={ v^2 \over r}$

and $a\_c = \omega^2 \* r$

But the conflict between these two equations arises when we increase $r$.
In the first equation - if we increase $r$, $v$ should increase so that $a\_c$ remains constant.
But in the second equation - since $\omega$ is constant in a centrifuge, if we increase $r$ then $a\_c$ is no longer constant.

I dont understand how this is happening and would appreciate any help.I have also looked at other similar questions on this site but they do not answer to this conflict. This is not a duplicate.

Also, does anyone know how to mathematically show that radius increases when mass increases?