A particle of mass $m$ moves in a circular orbit under the central potential
Question:

A particle of mass $m$ moves in a circular orbit under the central potential

field, $U(r)=\frac{-C}{r}$, where $C$ is a positive constant. The correct radius –

velocity graph of the particle’s motion is :

Correct Option: 1

Solution:

(1)

$\mathrm{U}=-\frac{\mathrm{C}}{\mathrm{r}}$

$\mathrm{F}=-\frac{\mathrm{dU}}{\mathrm{dr}}=-\frac{\mathrm{C}}{\mathrm{r}^{2}}$

$|\mathrm{F}|=\frac{\mathrm{mv}^{2}}{\mathrm{r}}$

$\frac{\mathrm{C}}{\mathrm{r}^{2}}=\frac{\mathrm{mv}^{2}}{\mathrm{r}}$

$\mathrm{v}^{2} \propto \frac{1}{\mathrm{r}}$