Andre from Tudor Vianu National College, Bucharest, Romania and Shaun from Nottingham High School both sent in excellent solutons to this problem.
forces
After drawing the picture, I observed that the forces
(the gravitational force acting on the ball, the tension in the wire and the
centrifugal force) keep the body in equilibrium. Considering the
centrifugal force, I work in a non-inertial frame of reference,
i.e. in the frame centred on the ball, which is in an
accelerated movement in respect to Earth.

In terms of the vectors we have

m _
g
 
 + _
F
 
c 
 + _
T
 
 = 0.

Resolving horizontally and vertically and using F=ma (where a is the acceleration towards the centre and T is the magnitude of the tension in the string):
Tcosq = mg

Tsinq = mlsinqw2
Eliminating T:
lw2 cosq = g
and hence the angle q is
cos-1 g
lw2

.

I know that cos-1 is a decreasing function in the interval of interest for the problem. As w increases, the angle q also increases and the whirling ball rises up, the radius of its circular path also increasing.

The ball can whirl in a circle while q > 0. So, as I explained above, to find the smallest angular velocity, I have to find the smallest angle. For q® 0,
g
lw2
 ® 1

and so
w®   æ
 ú
Ö
g
l
 
 .
This is the smallest angular velocity with which the ball could rotate in a circle.

The period of this movement is:
T = 2p
w
 = 2p   æ
 ú
Ö
l
g
 
 .
It is interesting to observe that this is also the period of isochronal oscillations of the mathematical pendulum, i.e. the period of oscillation of a material point of mass m attached to an inextensible string without mass deviated from the vertical by angles less than 5o.