MHT CET · Physics · Rotational Motion
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
- A \(1: 2\)
- B \(\sqrt{2}: 1\)
- C \(2: 1\)
- D \(1: \sqrt{2}\)
Answer & Solution
Correct Answer
(D) \(1: \sqrt{2}\)
Step-by-step Solution
Detailed explanation
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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