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Whirling a conker on a string

A ball of mass $m$ is attached to a light inextensible string of length $l$. The ball is whirled around so that it moves in a horizontal circle with constant angular speed $\omega$.

Find the angle $\theta$ between the string and the vertical. Show that the angle $\theta$ is given by $\cos^{-1}{g\over l\omega^2}$.

Now increase the angular velocity. What happens to the ball?

What is the smallest angular velocity with which the ball can whirl in a circle on the end of the string in this way?