Question

A particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the $n^{\text{th}}$ power of $R$. If the period of rotation of the particle is $T,$ then

• A. $T\propto R^{(n+1)/2}$
• B. $T\propto R^{n/2}$
• C. $T\propto R^{3/2}$ for any $n$
• D. $T\propto R^{\frac{n}{2}+1}$

Answer 1

The correct option is A $T\propto R^{(n+1)/2}$

$F=\frac{k}{R^n}=m{\omega }^{2}R$
${\omega }^2\propto \frac{1}{R^{n+1}}\Rightarrow T=\frac{2\pi }{\omega }$
So, $T\propto R^{\frac{n+1}{2}}$