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  4. Two coaxial discs, having moments of inertia I1 and (I1/2) , are rotating with respective angular velocities ω1 and (ω1/2) , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If Ef and Ei are the final and initial total energies, then (Ef - Ei) is :

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Q. Two coaxial discs, having moments of inertia $I_1 $ and $\frac{I_1}{2}$ , are rotating with respective angular velocities $\omega_1$ and $\frac{\omega_1}{2}$ , about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies, then $(E_f - E_i)$ is :

JEE MainJEE Main 2019System of Particles and Rotational Motion

Solution:

$E_{i} = \frac{1}{2} I_{1} \times\omega^{2}_{1} + \frac{1}{2} \frac{I_{1}}{2} \times\frac{\omega_{1}^{2}}{4} $
$ = \frac{I_{1} \omega_{1}^{2}}{2} \left(\frac{9}{8}\right) = \frac{9}{16} I_{1}\omega^{2}_{1} $
$ I_{1}\omega_{1} + \frac{I_{1}\omega_{1}}{4} = \frac{3I_{1}}{2}\omega $
$ \frac{5}{4} I_{1}\omega_{1} = \frac{3I_{1}}{2} \omega$
$ \omega = \frac{5}{6} \omega_{1} $
$E_{f} = \frac{1}{2} \times\frac{3I_{1}}{2} \times\frac{25}{36} \omega^{2}_{1} $
$= \frac{25}{48} I_{1} \omega_{1}^{2}$
$ \Rightarrow E_{f} - E_{i} = I_{1} \omega_{1}^{2} \left(\frac{25}{48} - \frac{9}{16}\right) = \frac{-2}{48} I_{1} \omega_{1}^{2} $
$ = \frac{-I_{1} \omega^{2}_{1}}{24} $