The figure shows a schematic of a simple Watt governor mechanism with the spindle $O_{1}O_{2}$ rotating at an angular velocity $\omega$ about a vertical axis. The balls at $P$ and $S$ have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot $O_{1}$ is denoted by $h$. The value of $h = 400$ $\text{mm}$ at a certain $\omega$. If $\omega$ is doubled, the value of $h$ will be _________________ $\text{mm}$.

1. $50$
2. $100$
3. $150$
4. $200$