\(L _{1}= I \omega\)

\(L _{1}=\frac{ ML ^{2}}{12} \omega\)

Final angular momentum

\(L _{2}=\left(\frac{ ML ^{2}}{12}+\frac{ M }{3} \frac{ L ^{2}}{4}+\frac{ M }{3} \frac{ L ^{2}}{4}\right) \omega^{\prime}\)

From Momentum conservation theorm

\(\frac{M L^{2}}{12} \omega=\left(\frac{M L^{2}}{12}+\frac{M L^{2}}{12}+\frac{M L^{2}}{12}\right) \omega^{\prime}\)

\(\omega=3 \omega^{\prime}\)

\(\omega^{\prime}=\frac{\omega}{3}\)