Question

A particle of mass $m$ moves in a circular orbit under the central potential field, $U\left(r\right)=\frac{-c}{r}$, where $c$ is positive constant. The correct radius$\left(r\right)$-velocity$\left(v\right)$ graph of the particle's motion is :

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Given, potential field,
$U\left(r\right)=\frac{-c}{r}$

$\Rightarrow F=-\frac{dU}{dr}=\frac{c}{r^2}$

We know that the centripetal force is given by,

$F_c=\frac{m{v}^2}{r}$

$\therefore \frac{m{v}^2}{r}=\frac{c}{r^2}$

$\Rightarrow r=\frac{c}{m{v}^2}$

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

This situation is best represented by the graph of option $\left(A\right)$.