A uniform rod of length \(L\) and mass \(M\) is resting on a horizontal, frictionless surface and is attached to a fixed, frictionless pivot at one end. A small piece of clay of mass \(m\) slides across the surface with velocity \(v\) perpendicular to the rod and strikes the tip of the rod, sticking to it. Which of the following correctly characterizes the conservation of momentum for the clay-rod system during the brief collision, and provides a correct justification?