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

Solution :
$$\eqalign{ & m{\omega ^2}R = {\text{Force}}\, \propto \frac{1}{{{R^n}}}\,\,\,\,\,\,\left( {{\text{Force}} = \frac{{m{v^2}}}{R}} \right) \cr & \Rightarrow {\omega ^2} \propto \frac{1}{{{R^{n + 1}}}} \Rightarrow \omega \, \propto \frac{1}{{{R^{\frac{{n + 1}}{2}}}}} \cr & {\text{Time period }}T = \frac{{2\pi }}{\omega } \cr & {\text{Time period, }}T\, \propto \,{R^{\frac{{n + 1}}{2}}} \cr} $$