A particle executes uniform circular motion with angular momentum 'L'. Its rotational kinetic energy becomes half, when the angular frequency is doubled. Its new angular momentum is

\(\mathrm{K}=\frac{1}{2} \mathrm{I} \omega^{2}, \quad \mathrm{~L}=\mathrm{I} \omega\)
\(\mathrm{K}^{\prime}=\frac{1}{2} \mathrm{I}^{\prime} \omega^{\prime 2}, \quad \omega^{\prime}=2 \omega\)
\(\therefore \mathrm{K}^{\prime}=\frac{1}{2} \mathrm{I}^{\prime}(2 \omega)^{2}, \quad \mathrm{~K}^{\prime}=\frac{\mathrm{K}}{2}\)
\(\therefore \frac{1}{2}\left(\frac{1}{2} \mathrm{I} \omega^{2}\right)=\frac{1}{2} \mathrm{I}^{\prime}(2 \omega)^{2}\)
\(\therefore \mathrm{I}^{\prime}=\frac{\mathrm{I}}{8}\)
\(\mathrm{~L}^{\prime}=\mathrm{I}^{\prime} \omega^{\prime}=\frac{\mathrm{I}}{8} \times 2 \omega=\frac{\mathrm{I} \omega}{4}=\frac{\mathrm{L}}{4}\)