A disk with moment of inertia ${\mathrm{I}}_1$ rotates about a frictionless, vertical axle with angular speed ${\mathrm{\omega }}_{\mathrm{i}}$.  A second disk, this one having moment of inertia ${\mathrm{I}}_2$ and initially not rotating, drops onto the first disk (Fig.). Because of friction between the surfaces, the two eventually reach the same angular speed ${\mathrm{\omega }}_{\mathrm{f}}$ The value of ${\mathrm{\omega }}_{\mathrm{f}}$ is