A uniform rod of length l is being rotated in a horizontal plane...

The mass of the rod at a distance $x$ from the axis is $dm = \dfrac{M}{L} . dx$
$T = ( dm ) {\omega}^{2}x$

Integrating the above equation from $x$ to $L$
$= {∫}_{x}^{L}\dfrac{M}{L}{\omega}^{2}xdx$
$= \dfrac{M}{L}{\omega}^{2}{\left[\dfrac{{x}^{2}}{2}\right]}_{x}^{L}$
$\dfrac{M}{2L}{\omega}^{2}\left[{L}^{2} - {x}^{2}\right]$
The above equation depicts the relation between tension and length of rod as downward opening parabola.