a
Moment of inertia of the system about the axis of rotation (through point \(P\)) is

\(I = {m_1}{x^2} + {m_2}{\left( {L - x} \right)^2}\)

By work energy theorem,

Work done to set the rod rotating with angular

velocity \({{\omega _0}}\) = Increase in rotational kinetic energy 

\(W = \frac{1}{2}I\omega _0^2 = \frac{1}{2}\left[ {{m_1}{x^2} + {m_2}{{\left( {L - x} \right)}^2}} \right]\omega _0^2\)

For \(W\) to be mimimum,\(\frac{{dW}}{{dx}} = 0\)

\(i.e.,\,\frac{1}{2}\left[ {2{m_1}x + 2{m_2}\left( {L - x} \right)\left( { - 1} \right)} \right]\omega _0^2 = 0\)
or \({m_1}x - {m_2}\left( {L - x} \right) = 0 ( {{\omega _0} \ne 0}\)
or \({{m_1} + {m_2}} x = {m_2}L\,or\,x = \frac{{{m_2}L}} {{{m_1} + {m_2}}}\)