Question

A particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the nth power of $R$. If the period of rotation of the particle is $T$ then -

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

Answer 1

Answer: (C)

$T \propto R^{(n + 1) / 2 }$

Answer 2

$m \omega^{2} R=k R^{-n}=\frac{k}{R^{n}}$
$\Rightarrow \frac{1}{T^{2}} \propto \frac{1}{R^{n+1}}$
$\Rightarrow T \propto R^{\left(\frac{n+1}{2}\right)}$