A uniform thin rod of mass \(M\) and length \(L\) is suspended vertically from a frictionless pivot at its top end. A small ball of clay with mass \(m\) is traveling horizontally with speed \(v_0\) when it strikes the rod at a distance \(y = \dfrac{2}{3}L\) below the pivot. The clay sticks to the rod, and the rod-clay system begins to rotate. Which of the following correctly characterizes the conservation of linear momentum and angular momentum for the rod-clay system during the collision?