• # question_answer 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 Main 10-4-2019 Morning] A) $\frac{{{I}_{1}}\omega _{1}^{2}}{12}$                        B) $\frac{3}{8}{{I}_{1}}\omega _{1}^{2}$C) $\frac{{{I}_{1}}\omega _{1}^{2}}{6}$                          D) $\frac{{{I}_{1}}\omega _{1}^{2}}{24}$

Solution :

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

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