= Solution
To avoid contact with the ceiling, require $Y(\omega)<h$. Writing $X=r^2\omega^2$ gives
$$
X^2-2ghX+g^2r^2<0,
$$
whose roots are $g(h\pm\sqrt{h^2-r^2})$. Because $h>r$, the lower root is less than $gr$, while the assumed branch has $X>gr$. The lower inequality is therefore automatic on this branch, leaving
$$
\boxed{r^2\omega^2<gh+g\sqrt{h^2-r^2}.}
$$
Equality is the first grazing contact. This branch check is essential: retaining both algebraic roots as independent lower and upper speed limits would introduce a spurious lower limit on avoiding the ceiling.
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