Take an element of length $dx$ and mass $dm$ at a distance $x$ from the rotating end.

Here mass of elemental part $\left(dm\right)=\rho dx$

Centrifugal force on this small element $\left(dF\right)=\left(dm\right) x{\omega }^2$

Total centrifugal force,

$F=\int dF={\int }_0^l\rho x{\omega }^{2}dx$

$\Rightarrow F=\frac{\rho {\omega }^2{l}^2}{2}$