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Two discs have moments

Question:

Two discs have moments of intertia $I_{1}$ and $I_{2}$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega_{1}$ and $\omega_{2}$ respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

1. $\frac{\mathrm{I}_{1} \mathrm{I}_{2}}{\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$

2. $\frac{\left(\mathrm{I}_{1}-\mathrm{I}_{2}\right)^{2} \omega_{1} \omega_{2}}{2\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right)}$

3. $\frac{\mathrm{I}_{1} \mathrm{I}_{2}}{2\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$

4. $\frac{\left(\omega_{1}-\omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$

Correct Option: , 3

Solution:

From conservation of angular momentum we get

$\mathrm{I}_{1} \omega_{1}+\mathrm{I}_{2} \omega_{2}=\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega$

$\omega=\frac{\mathrm{I}_{1} \omega_{1}+\mathrm{I}_{2} \omega_{2}}{\mathrm{I}_{1}+\mathrm{I}_{2}}$

$\mathrm{k}_{\mathrm{i}}=\frac{1}{2} \mathrm{I}_{1} \omega_{1}^{2}+\frac{1}{2} \mathrm{I}_{2} \omega_{2}^{2}$

$\mathrm{k}_{\mathrm{f}}=\frac{1}{2}\left(\mathrm{I}_{1}+\mathrm{I}_{2}\right) \omega^{2}$

$\mathrm{k}_{\mathrm{i}}-\mathrm{k}_{\mathrm{f}}=\frac{1}{2}\left[\mathrm{I}_{1} \omega_{1}^{2}+\mathrm{I}_{2} \omega_{2}^{2}-\frac{\left(\mathrm{I}_{1} \omega_{1}+\mathrm{I}_{2} \omega_{2}\right)^{2}}{\mathrm{I}_{1}+\mathrm{I}_{2}}\right]$

Solving above we get

$\mathrm{k}_{\mathrm{i}}-\mathrm{k}_{\mathrm{f}}=\frac{1}{2}\left(\frac{\mathrm{I}_{1} \mathrm{I}_{2}}{\mathrm{I}_{1}+\mathrm{I}_{2}}\right)\left(\omega_{1}-\omega_{2}\right)^{2}$