Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. $(a)$ What is the angular speed of the two-disc system? $(b)$ Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$.

(N/A) Part $(a)$: According to the law of conservation of angular momentum,the total angular momentum of the system remains constant as no external torque acts on the system.
Initial angular momentum $L_{i} = I_{1}\omega_{1} + I_{2}\omega_{2}$.
When the discs are joined,they rotate with a common angular speed $\omega$. The total moment of inertia of the system becomes $I = I_{1} + I_{2}$.
Final angular momentum $L_{f} = (I_{1} + I_{2})\omega$.
Equating $L_{i} = L_{f}$,we get: $I_{1}\omega_{1} + I_{2}\omega_{2} = (I_{1} + I_{2})\omega$.
Therefore,the angular speed of the system is $\omega = \frac{I_{1}\omega_{1} + I_{2}\omega_{2}}{I_{1} + I_{2}}$.
Part $(b)$: Initial kinetic energy $E_{i} = \frac{1}{2}I_{1}\omega_{1}^{2} + \frac{1}{2}I_{2}\omega_{2}^{2}$.
Final kinetic energy $E_{f} = \frac{1}{2}(I_{1} + I_{2})\omega^{2} = \frac{1}{2}(I_{1} + I_{2})\left(\frac{I_{1}\omega_{1} + I_{2}\omega_{2}}{I_{1} + I_{2}}\right)^{2} = \frac{(I_{1}\omega_{1} + I_{2}\omega_{2})^{2}}{2(I_{1} + I_{2})}$.
The loss in kinetic energy is $\Delta E = E_{i} - E_{f} = \frac{1}{2}I_{1}\omega_{1}^{2} + \frac{1}{2}I_{2}\omega_{2}^{2} - \frac{(I_{1}\omega_{1} + I_{2}\omega_{2})^{2}}{2(I_{1} + I_{2})}$.
Simplifying this expression,we get $\Delta E = \frac{I_{1}I_{2}(\omega_{1} - \omega_{2})^{2}}{2(I_{1} + I_{2})}$.
Since $I_{1}, I_{2} > 0$ and $(\omega_{1} - \omega_{2})^{2} > 0$ (as $\omega_{1} \neq \omega_{2}$),$\Delta E > 0$,which implies $E_{i} > E_{f}$.
This loss in kinetic energy is due to the work done against the frictional force that acts between the surfaces of the discs until they attain a common angular velocity.