A uniform rod of length l   is being rotated in a horizontal plane with a constant angular speed about…

The mass of the rod at a distance $x$ from the axis is $\mathrm{d}\mathrm{m}=\frac{\mathrm{M}}{\mathrm{L}}.\mathrm{d}\mathrm{x}$
$\mathrm{T}=\left(\mathrm{d}\mathrm{m}\right){\mathrm{\omega }}^2\mathrm{x}$

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