where \(I=\) moment of inertia 

\(\omega= \) Angular velocity \(=\frac{L}{I}\)

\(L=\) Angular momentum

\(I=\frac{1}{2}\left(m_{1} r_{1}^{2}+m^{2} r_{2}^{2}\right)\)

Thus, \(E=\frac{1}{2}\left(m_{1} r_{1}^{2}+m_{2} r_{2}^{2}\right) \omega^{2}\)  .... \((i)\)

\(E=\frac{1}{2}\left(m_{1} r_{1}^{2}+m_{2} r_{2}^{2}\right) \frac{L^{2}}{I^{2}}\)

\(L=n \frac{n h}{2 n}\) (According Bohr's Hypothesis )

\(E=\frac{1}{2}\left(m_{1} r_{1}^{2}+m_{2} r_{2}^{2}\right) \frac{L^{2}}{\left(m_{1} r_{1}^{2}+m_{2} r_{2}^{2}\right)^{2}}\)

\(E = \frac{1}{2}\frac{{{L^2}}}{{\left( {{m_1}r_1^2 + {m_2}r_2^2} \right)}}\) \( = \frac{{{n^2}{h^2}}}{{8{\pi ^2}\left( {{m_1}r_1^2 + {m_2}r_2^2} \right)}}\)

\(E = \frac{{\left( {{m_1} + {m_2}} \right){n^2}{h^2}}}{{8{\pi ^2}{r^2}{m_1}{m_2}}}\) \(\left[ {\because {r_1} = \frac{{{m_2}r}}{{{m_1} + {m_2}}}\,;{r_2} = \frac{{{m_2}r}}{{{m_1} + {m_2}}}} \right]\)