Question

The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $V(r)=k{r}^2/2$, where $k$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $R$ about the point $O$. If $v$ is the speed of the particle and $L$ is the magnitude of its angular momentum about $O$, which of the following statements is (are) true?

• A. $v=\sqrt{\frac{k}{2m}}R$
• B. $v=\sqrt{\frac{k}{m}}R$
• C. $L=\sqrt{mk}{R}^2$
• D. $L=\sqrt{\frac{mk}{2}}{R}^2$

Answer 1

Answer: (B), (C)

$V=\frac{k{r}^2}{2}$
$F=-kr$ (towards centre) $\left[F=\frac{dV}{dr}\right]$
At r = R
$kR=\frac{m{v}^2}{R}$ [Centripetal force]
$v=\sqrt{\frac{k{R}^2}{m}}=\sqrt{\frac{k}{m}}R$
$L=m\sqrt{\frac{k}{m}}{R}^2$