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Two coaxial discs, having moments of inertia $I_1$ and $\frac{I_1}{2}$ are a rotating with respectively angular velocities $\omega_1$ and $\frac{\omega_1}{2}$, about their common axes. 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
$\frac{{{I_1}\omega _1^2}}{6}$
$\frac{3}{8}{I_1}\omega _1^2$
$ - \frac{{{I_1}\omega _1^2}}{{12}}$
$ - \frac{{{I_1}\omega _1^2}}{{24}}$
Solution
${E_i} = \frac{1}{2}{I_1} \times \omega _1^2 + \frac{1}{2}\frac{I}{2} \times \frac{{\omega _1^2}}{4}$
$ = \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\frac{{25}}{{49}} – \frac{{ – 2}}{{48}}{I_2}\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}}$
