Q. The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $𝑉(𝑟)=𝑘𝑟^2 /2$, 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?
Solution:
$V = \frac{kr^2}{2}$
$F = - kr $ (towards centre) $\left[ F = \frac{dV}{dr} \right]$
At r = R
$kR = \frac{mv^2}{R} $ [Centripetal force]
$v = \sqrt{\frac{kR^2}{m}} = \sqrt{\frac{k}{m} } R$
$L = m \sqrt{\frac{k}{m}} R^2$
