If the tension in the string is T, resolving horizontally and vertically and using F=ma (where a is the acceleration towards the centre):

Tcosq = mg

Tsinq = mlsinqw²

Eliminating T:

lw² cosq = g

and hence the angle q is

cos^-1

lw²

If we increase the angular velocity then cosq decreases and angle q increases and hence the mass rises up and the radius of the circle it moves on also increases.

As cosq £ 1 we have

g
lw²
£ 1

so

w ł ć
ú
Ö

g
l

and hence the smallest angular speed with which the ball can whirl in a circle on this string is

ć
ú
Ö

g
l

.