A small particle of mass m moves in such a way that its potential...

The data given is

$U = \dfrac{1}{2}m{\omega}^{2}{r}^{2}$

From Bohr's quantization,

$mvr = \dfrac{nh}{2\pi} ⇒ {v}^{2} = {\left(\dfrac{nh}{2\pirm}\right)}^{2} . . . ( i )$

In an orbit the value of the kinetic energy is half of the potential energy,

$K = \dfrac{m{\omega}^{2}{r}^{2}}{4}$

Substituting the value of equation (i) in the kinetic energy,

$K = \dfrac{1}{2}m\left(\dfrac{{n}^{2}{h}^{2}}{4{\pi}^{2}{m}^{2}{r}^{2}}\right) = \dfrac{m{\omega}^{2}{r}^{2}}{4} ⇒ \dfrac{{n}^{2}{h}^{2}}{2{\pi}^{2}{m}^{2}{\omega}^{2}} = {r}^{4} ⇒ r ∝ \sqrt{n}$