How does the velocity of a mass spun around a pencil change?

[Happiness]

Suppose a mass $m$ is attached to the end of a string whose other end is attached to a cylindrical pencil. The mass is then spun around the pencil in a circle (whose centre coincides with the centre of the pencil) such that the string wraps around the outer surface of the pencil, thereby decreasing the distance $r$ between the mass and the pencil.

By the conservation of angular momentum $L$, as $r$ decreases, the speed $v$ of the pencil increases. ($L = rmv$) However, the tension exerted on the mass is always perpendicular to its velocity and hence to its displacement. Thus, the tension does no work and should not change the speed of the mass. We have a contradiction.

If we consider the thickness of the pencil, and so more accurately say that the tension is not exactly directed towards the centre of the pencil, but rather it is directed at a point a distance $a$ away from the centre of the pencil, where $a$ is the radius of the pencil. Then for the string to be wrapped around the pencil, the string would have be directed a little "backwards" with respect to the velocity of the mass (instead of being perpendicular). In this case, the tension would be doing negative work, decreasing the speed of the mass. Again, we have a contradiction.

 

[DrGreg]

But the mass isn't following a perfectly circular path, it's on a spiral path...

 

[nasu]

What angular momentum is conserved? In respect to what point?