A uniform horizontal disk of mass \(M\) and radius \(R\) rotates freely about a vertical, frictionless axle through its center at an initial angular velocity \(\omega_0\). A thin, non-uniform rod of mass \(M\) and length \(R\) has a linear mass density given by \(\lambda(r) = C r\), where \(r\) is the distance from one end of the rod and \(C\) is a constant. The rod, initially at rest, is dropped vertically onto the spinning disk such that the end with \(r = 0\) lands precisely at the disk's center and the rod lies radially along the disk surface. Friction between the disk and the rod quickly brings the rod to rest relative to the disk. What is the final angular velocity of the disk-rod system?