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A particle performs uniform circular motion with an angular momentum L. If the frequency of the particle's motion is doubled and its kinetic energy is halved, then its angular momentum becomes
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$\frac{\mathrm{L}}{4}$
Angular momentum, $\mathrm{L}=\mathrm{I} \omega$ and rotational kinetic energy $\mathrm{K}=\frac{1}{2} \mathrm{I} \omega^2 \therefore \mathrm{K}=\frac{1}{2} \mathrm{~L} \omega$
$\therefore \frac{\mathrm{L}_2}{\mathrm{~L}_1}=\frac{\mathrm{K}_2}{\mathrm{~K}_1} \times \frac{\omega_1}{\omega_2}=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$
$\therefore \frac{\mathrm{L}_2}{\mathrm{~L}_1}=\frac{\mathrm{K}_2}{\mathrm{~K}_1} \times \frac{\omega_1}{\omega_2}=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$
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