Question

A body of mass $m$ experiences a central force $\overrightarrow{F}=-k\overrightarrow{r}$. Here $\overrightarrow{r}$ is the position from centre of the force and $k$ is a positive constant. It moves in a closed circular path such that it's angular momentum about the centre of force is quantized $L=\frac{nh}{2\pi }$. Take potential energy = 0 at $r=0$.

• A. It's possible that radius is quantized such that $r\propto n^{1/2}$.
• B. It's possible that mechanical energy is quantized such that $E\propto n$.
• C. It's speed is quantized such that $v\propto n^{1/2}$.
• D. It's angular velocity is quantized such that $\omega \propto n$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A It's possible that radius is quantized such that $r\propto n^{1/2}$.
B It's possible that mechanical energy is quantized such that $E\propto n$.
C It's speed is quantized such that $v\propto n^{1/2}$.

For circular motion,
$\frac{m{v}^2}{r}=kr\Rightarrow v^2\propto r^2\Rightarrow v\propto r\text{or}v=cr$

$\text{Angular momentum}mvr=\frac{nh}{2\pi }$
$\Rightarrow c{r}^2\propto n\text{or}r\propto \sqrt{n}$

$\Rightarrow v\propto r\propto \sqrt{n}$

Angular velocity $\omega =v/r\propto n^0$

Potential energy $PE=\int \overrightarrow{F}.d\overrightarrow{r}=\frac{k{r}^2}{2}$

$TE=KE+PE=\frac{1}{2}m{v}^2+\frac{1}{2}k{r}^2$

i.e. $TE\propto r^2\propto (\sqrt{n}{)}^2\propto n$