A long cylindrical volume contains a uniformly distributed charge...

Electric field due to a long cylinder is,

$E = \dfrac{\lambda}{2\pi{ε}_{0}r} = \dfrac{ρA}{2\pi{ε}_{0}r}$

If a charge $q$ is rotating in a circle of radius $r$

$\dfrac{m{v}^{2}}{r} = qE$ $⇒ \dfrac{m{v}^{2}}{r} = q\dfrac{ρA}{2\pi{ε}_{0}r} ⇒ \dfrac{1}{2}m{v}^{2} = \dfrac{qρ\pi{R}^{2}}{4\pi{ε}_{0}} = \dfrac{qρ{R}^{2}}{4{ε}_{0}}$