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Step-by-Step Solution
Step 1: Express the central force and centripetal force requirement
When a particle moves in a circular orbit under the influence of a central force, the required centripetal force is provided by that central force. Mathematically, for a particle of mass $m$ moving in a circle of radius $R$ with angular speed $\omega$, the centripetal force is:
$F_{\text{centripetal}} = m\,\omega^2\,R.$
According to the problem, the central force $F$ is inversely proportional to $R^n$, i.e.,
$F \propto \dfrac{1}{R^n}.$
Step 2: Set up the proportional relationship
We can write:
$m\,\omega^2\,R \propto \dfrac{1}{R^n}.$
Or equivalently:
$m\,\omega^2\,R = \dfrac{k}{R^n},$
where $k$ is a constant of proportionality.
Step 3: Solve for $ \omega $
From the above equation:
$\omega^2 = \dfrac{k}{m\,R^{n+1}}.$
Taking the square root on both sides:
$\omega \propto \dfrac{1}{R^{\tfrac{n+1}{2}}}.$
Step 4: Relate angular speed to the period of revolution
The period of rotation $T$ is related to the angular speed $\omega$ by:
$T = \dfrac{2\pi}{\omega}.$
Hence,
$T \propto \dfrac{1}{\omega}.$
Substituting $\omega \propto \dfrac{1}{R^{\tfrac{n+1}{2}}}$ into this relationship:
$T \propto R^{\tfrac{n+1}{2}}.$
Step 5: Conclude the proportionality
Therefore, the period of rotation $T$ varies with the radius $R$ as:
$T \propto R^{\tfrac{n+1}{2}}.$
This matches Option 4 from the given choices.
