A thin rod of mass $M$ and length $L$ is struck at one end by a ball of clay of mass $m,$ moving with speed $v$ as shown in figure. The ball sticks to the rod. After the collision, the angular momentum of the clay-rod system about $A,$ the midpoint of the rod, is

A thin ring of mass $2 m$ and radius $R$ is pivoted at $P$ on a frictionless table, as shown in figure. A bug of mass $m\mathit{.}$ runs along the ring with speed $v$ with respect to the ring. The bug starts from the pivot with the ring at rest. When the bug reaches the diametrically opposite point on the ring (point $X$), it is moving with velocity $\text{'}v\text{'}$ with respect to the table. The value of $\frac{v}{v^{\text{'}}}$ is