Setup
A ball of mass $m$ is attached to a string of length $L$. It is made to rotate at speed $\omega$ as a conical pendulum making an angle $\theta$ with the vertical.
Question
This post shows how the angle the ball makes is
$cos (\theta) = \frac g {L \omega^2}$
One would expect the angle $\theta$ to approach zero as $\omega$ approaches zero, but that is not what this solution shows. Can someone help me understand the above solution?
Thanks!
When the angle $\theta$ gets small then $$ \omega\to \sqrt{\frac{g}{L}} $$ which is the angular frequency of small amplitude swinging of a simple pendulum with length $L$. The angular velocity $\omega$ cannnot get smaller than this value. The small-angle conical pendulum motion is just independant equal-amplitude $x$ and $y$ direction swings that are 90 degrees out of phase: $$ x(t)= a \cos\omega t\\ y(t)= a \sin\omega t $$
If $\omega$ is too small then the ball cannot “rotate upwards” (as asked in the linked question), i.e. the ball will execute the usual motion about the lowest point.
This question contains additional technical details, but roughly speaking there is a critical minimum frequency so that the rotating pendulum is stable about some non-zero $\theta$ angle. This is the gist of the linked question. As your manipulations show, if you are blow this critical frequency, there is no angle $\theta$ for which $\cos\theta=g/(L\omega^2)$ because the right hand side is greater than 1.
When there is no solution the effect of the rotation is simply to reduce (but not overcome) the effect of $g$. When $\omega^2$ is large enough, the centripetal force can overcome gravity and the pendulum can oscillate about some angle that is not at the bottom.
the equations This
$$T\,\sin(\theta)=m\,\,r\,\omega^2=m\,L\,\sin(\theta)\,\omega^2\tag 1$$ $$T\,\cos(\theta)=m\,g\tag 2$$
if $~\omega\ne 0~$ then you obtain
$$T=m\,L\,\omega^2\quad,\cos(\theta)=\frac {g}{L\,\omega^2}$$
and if $~\omega=0~$ from equation (1)
$$T\,\sin(\theta)=0\quad\Rightarrow \theta=0$$
and from equation (2)
$$T=m\,g$$
