JEE MAIN - Physics (2023 - 11th April Evening Shift - No. 20)
A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity $$\omega$$. A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If E be the initial Kinetic energy of the system, then final Kinetic energy will be $$\frac{E}{x}$$. The value of $$x$$ is
답변
3
설명
The conservation of angular momentum states that the angular momentum (L) remains constant. The relation between kinetic energy (KE), angular momentum (L), and moment of inertia (I) is given by:
$$ \mathrm{KE}=\frac{\mathrm{L}^2}{2 \mathrm{I}} $$
Using this relation, we can find the ratio of the final kinetic energy ($$\mathrm{KE}_{\text{final}}$$) to the initial kinetic energy ($$\mathrm{KE}_{\text{initial}}$$ or E):
$$ \frac{\mathrm{KE}_{\text{final}}}{\mathrm{KE}_{\text{initial}}}=\frac{\mathrm{I}_{\text{initial}}}{\mathrm{I}_{\text{final}}} $$
Since the moment of inertia triples, we have $$\mathrm{I}_{\text{final}} = 3\mathrm{I}_{\text{initial}}$$. Therefore,
$$ \frac{\mathrm{KE}_{\text{final}}}{\mathrm{E}}=\frac{\mathrm{I}_{\text{initial}}}{3\mathrm{I}_{\text{initial}}}=\frac{1}{3} $$
This means that the final kinetic energy of the system is:
$$ \mathrm{KE}_{\text{final}}=\frac{E}{3} $$
So, the value of $$x$$ is 3.
$$ \mathrm{KE}=\frac{\mathrm{L}^2}{2 \mathrm{I}} $$
Using this relation, we can find the ratio of the final kinetic energy ($$\mathrm{KE}_{\text{final}}$$) to the initial kinetic energy ($$\mathrm{KE}_{\text{initial}}$$ or E):
$$ \frac{\mathrm{KE}_{\text{final}}}{\mathrm{KE}_{\text{initial}}}=\frac{\mathrm{I}_{\text{initial}}}{\mathrm{I}_{\text{final}}} $$
Since the moment of inertia triples, we have $$\mathrm{I}_{\text{final}} = 3\mathrm{I}_{\text{initial}}$$. Therefore,
$$ \frac{\mathrm{KE}_{\text{final}}}{\mathrm{E}}=\frac{\mathrm{I}_{\text{initial}}}{3\mathrm{I}_{\text{initial}}}=\frac{1}{3} $$
This means that the final kinetic energy of the system is:
$$ \mathrm{KE}_{\text{final}}=\frac{E}{3} $$
So, the value of $$x$$ is 3.