Question

A small particle of mass $m$ moves in such a way that $P.E=-\frac{1}{2}mk{r}^2$, where $k$ is a constant and $r$ is the distance of the particle from origin. Assuming Bohr's model of quantization of angular momentum and circular orbit, $r$ is directly proportional to:

• A. $n^2$
• B. $n$
• C. $\sqrt{n}$
• D. none of these

Answer 1

The correct option is C $\sqrt{n}$

We know,
$-\frac{1}{2}\times P.E=K.E$
$=-\frac{1}{2}(-\frac{1}{2}mk{r}^2)=\frac{1}{2}m{v}^2\text{and}mvr=\frac{nh}{2\pi },$
$v^2=\frac{n^2{h}^2}{4{\pi }^2{m}^2{r}^2};$

$r^2=\frac{n^2{h}^2}{2{\pi }^2{m}^2{k}^2}$
$r\propto \sqrt{n}$

Hence, the correct option is $C$