Stable Rod Rotation

Consider the following mapping from Cartesian coordinates to spherical coordinates:

$x = r \, cos \theta \, sin \phi \\ y = r \, sin \theta \, sin \phi \\ z = r \, cos \phi$ In this coordinate system, $\theta$ is the angle in the $x y$ plane with respect to the $+x$ axis. $\phi$ is the angle with respect to the $+z$ axis.

Consider a uniform rigid rod of mass $M$ and length $L$ , with one end hinged at the origin, and the other end free to move. The hinge permits any combination of $(\theta, \phi)$ , subject to the natural constraints on each parameter. There is also an ambient gravitational acceleration $g$ in the $-z$ direction.

The following scenario represents a stable rotation that will continue indefinitely:

$\dot{\phi} = 0 \\ \ddot{\phi} = 0 \\ \dot{\theta} = \sqrt{\frac{- A \, g}{B \, L \, cos \phi}} \\ \ddot{\theta} = 0 \\ \frac{\pi}{2} < \phi < \pi$

If $A$ and $B$ are positive coprime integers, determine $A + B$ .

Notes:
- You may want to try this one first