OK, thanks. I think I've got it. There is actually nothing wrong with the equation I derived.
As the prevois poster pointed out, ω is not arbitrary. In fact ω just has a lower limit given by \omega=\sqrt{\frac{g}{l}}, which corresponds to the frequency of a pendulum oscillating at low amplitude.
Suppose a mass m suspended from a string of length l is undergoing uniform circular motion in a horizontal plane, with angular velocity ω. Calculate the centripetal acceleration a.
If T is the tension in the string, and θ the angle the string makes with the vertical, then the vertical and...