A small particle of mass m moves in such a way that its potential energy U = 1 2 m ω 2 r 2 where ω…

The data given is

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

From Bohr's quantization, 

$$\begin{gathered} mvr=\frac{nh}{2\pi } \\ \Rightarrow v^2={\left(\frac{nh}{2\pi rm}\right)}^2 ...\left(i\right) \end{gathered}$$

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

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

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

$$\begin{gathered} K=\frac{1}{2}m\left(\frac{n^2{h}^2}{4{\pi }^2{m}^2{r}^2}\right)=\frac{m{\omega }^2{r}^2}{4} \\ \Rightarrow \frac{n^2{h}^2}{2{\pi }^2{m}^2{\omega }^2}=r^4 \\ \Rightarrow r\propto \sqrt{n} \end{gathered}$$