A uniform horizontal disk of mass \(M\) and radius \(R\) is rotating with an initial angular velocity \(\omega_0\) about a fixed vertical axis through its center. The rotational inertia of the disk is \(I_{disk} = \dfrac{1}{2}MR^2\). A small piece of clay of mass \(m\) is dropped vertically onto the disk and sticks to it at a distance \(r\) from the center. Friction in the axle is negligible. Which of the following expressions represents the final angular velocity \(\omega_f\) of the disk-clay system?