Explanation:
The centripetal force acting on a particle is given by :
[tex]F=\dfrac{mv^2}{r}[/tex]
Since, [tex]v=r\omega[/tex]
[tex]F=mr\omega^2[/tex]
[tex]F\propto r\omega^2[/tex]
Case 1.
If [tex]T\propto \sqrt{r}[/tex]
Since, [tex]T=\dfrac{2\pi}{\omega}[/tex]
So, [tex]\omega\propto \dfrac{1}{\sqrt{r} }[/tex]
[tex]\omega^2\propto \dfrac{1}{r}[/tex]
So, the force becomes,
[tex]F\propto r\times \dfrac{1}{r}[/tex]
F is independent of r
Case 2.
If [tex]T\propto r^{3/2}[/tex]
[tex]\omega\propto \dfrac{1}{r^{3/2}}[/tex]
[tex]\omega^2 \propto \dfrac{1}{r^3}[/tex]
So, [tex]F\propto r\times \dfrac{1}{r^2}[/tex]
So, the force is inversely proportional to the square of radius.
Case 3.
If T is independent of r, the force will be directly proportional to the radius of orbit.
Hence, this is the required solution.
