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Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio
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Verified Answer
The correct answer is:
$1: \sqrt{2}$
The equation for angular momentum is
$\mathrm{L}=\sqrt{2 \mathrm{~K}_{\mathrm{Rot}} \times \mathrm{I}}$
So, $\mathrm{L} \propto \sqrt{\mathrm{I}}$
$\therefore \quad$ The ratio of angular momentum of the two bodies is
$\frac{\mathrm{L}_1}{\mathrm{~L}_2}=\sqrt{\frac{\mathrm{I}_1}{\mathrm{I}_2}}$
$\frac{\mathrm{L}_1}{\mathrm{~L}_2}=\sqrt{\frac{\mathrm{I}}{2 \mathrm{I}}} \quad \ldots . .\left(\right.$ given $\left.\mathrm{I}_2=2 \mathrm{I}\right)$
$\therefore \quad \frac{\mathrm{L}_1}{\mathrm{~L}_2}=\frac{1}{\sqrt{2}}$
$\mathrm{L}=\sqrt{2 \mathrm{~K}_{\mathrm{Rot}} \times \mathrm{I}}$
So, $\mathrm{L} \propto \sqrt{\mathrm{I}}$
$\therefore \quad$ The ratio of angular momentum of the two bodies is
$\frac{\mathrm{L}_1}{\mathrm{~L}_2}=\sqrt{\frac{\mathrm{I}_1}{\mathrm{I}_2}}$
$\frac{\mathrm{L}_1}{\mathrm{~L}_2}=\sqrt{\frac{\mathrm{I}}{2 \mathrm{I}}} \quad \ldots . .\left(\right.$ given $\left.\mathrm{I}_2=2 \mathrm{I}\right)$
$\therefore \quad \frac{\mathrm{L}_1}{\mathrm{~L}_2}=\frac{1}{\sqrt{2}}$
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