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