All Problems Dynamics of a solid body
Problem 1.278
Two horizontal discs rotate freely about a vertical axis passing through their centres. The moments of inertia of the discs relative to this axis are equal to \(I_{1}\) and \(I_{2},\) and the angular velocities to \(\omega_{1}\) and \(\omega_{2} .\) When the upper disc fell on the lower one, both discs began rotating, after some time, as a single whole (due to friction). Find: (a) the steady-state angular rotation velocity of the discs; (b) the work performed by the friction forces in this process.
(a) \(\omega=\frac{I_{1} \omega_{1}+\dot{I}_{2} \omega_{2}}{I_{1}+I_{2}}\) (b) \(A=-\frac{I_{1} I_{2}}{2\left(I_{1}+I_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}\)