A uniform rod of mass m and length l has one end clamped. It rests...

The center of mass of the uniform rod is at a distance $r_{cm} = \dfrac{l}{2}$ from the clamped end.

The rod is rotating with a uniform angular velocity $\omega$ on a smooth horizontal surface.

The centripetal acceleration of the center of mass is $a_{cm} = \omega^2 r_{cm} = \omega^2$\left(\dfrac{l}{2}\right).

The net horizontal force on the rod is provided by the clamp.

Using Newton's second law, the horizontal component of the force applied by the clamp is $F = m a_{cm} = m \omega^2 \left(\dfrac{l}{2}\right) = \dfrac{1}{2}$m \omega^2 l$.$