Why is angular momentum conserved when radius changes?

[kyin01]

Homework Statement

Why is angular momentum conserved?

Homework Equations

$$\vec{\tau}= \vec{r} x \vec{F}$$
magnitude of $$\tau = I \alpha$$

The Attempt at a Solution

I first don't agree on why angular momentum is conserved, because if the radius change doesn't that change the torque?

And by definition angular momentum of a system is conserved when there are no external torque. While for particles, angular moment is conserved when there is no change in torque.

So the torque changes when radius is shortened, why is it that angular momentum is conserved?

 

[Dick]

r and F are parallel. They are both directed towards the hole. The angle between them is zero so Fxr vanishes, regardless of the magnitude of r changing.