QJEEM23Phd098: JEE Main 2023 Physics - Class 11 JEEM23Phd098 Solution: A circular plate is rotating in horizontal plane, about an axis …

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.