Given central force law

The central force varies inversely as the $n^{\text{th}}$ power of radius: $F \propto \frac{1}{R^n}$

Let $F = \frac{k}{R^n}$ where $k$ is a constant.

Use condition for circular motion

For uniform circular motion, the required centripetal force is: $\frac{mv^2}{R} = \frac{k}{R^n}$

Rearranging, $mv^2 = kR^{1-n}$ $v^2 \propto R^{1-n}$ $v \propto R^{\frac{1-n}{2}}$

Relate speed to time period

The time period of revolution is $T = \frac{2\pi R}{v}$

Substituting the proportionality of $v$, $T \propto \frac{R}{R^{\frac{1-n}{2}}}$

$T \propto R^{1-\frac{1-n}{2}}$

Simplify the exponent: $1-\frac{1-n}{2} = \frac{2-(1-n)}{2} = \frac{n+1}{2}$

Therefore, $T \propto R^{\frac{n+1}{2}}$

Match with options

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

Conclusion

The correct option is D.