Centripetal Force.. Help!

We did the following experiment in class:

A piece of string had a rubber bung tied to one end, an approx 15 cm copper tube around it, and then a heavier mass tied to the other end. (looked like this http://demolab.phys.virginia.edu/demos/pictures/whirligig.jpg)

When you held the tube horizontally and made the rubber bung swing around with a high enough velocity, the heavier mass rose and the bung's radius increased.

So we thought that if you swung it at a constant velocity, nothing would happen and the mass would remain where it is, and that it's only when you apply more force to swinging the bung around that the mass starts to move. So is it that you're increasing the velocity and so the centripetal therefore increases, which pulls the mass up? That's what my teacher suggested but then I'm unsure about how that would work since the centripetal is towards the circle of rotation's centre.

Would it be because if the centripetal is acting on the bung, then an equal and opposite force would act on the other end of the string (in the same horizontal plane) and as a result pull the mass on the other end up?

Thanks in advance ​