# Angular velocity: why does decreasing the radius affect the tangential velocity?

I'm given the following very simple problem:

There is an object of mass $$m$$ attached to a string (whose mass can be neglected) on a table, whereby the string goes through the hole in the middle of the table and the particle moves on this frictionless horizontal table in a circular motion. It's initial angular velocity is $$\omega_0$$ and the initial radius/distance to the hole is $$r_0$$. Now you pull the string s.t. the distance to the hole gets smaller and is now $$r$$. What is the angular velocity $$\omega_1$$ now?

By simply applying conservation of angular momentum one obtains $$\omega_1 = \omega_0 \left(\dfrac{r_0}{r}\right)^2$$. However, I also thought about solving this by using the fact that the magnitude of the tangential velocity $$v$$ doesn't change during this process and hence I obtained $$\omega_0 r_0 = \omega_1 r \iff \omega_1 = \omega_0 \left(\dfrac{r_0}{r}\right)$$ which is clearly wrong. Why is the assumption that the tangential velocity doesn't change wrong? Intuitively I would assume that since I'm exerting a force parallel to the string that this doesn't affect the tangential component of the objects velocity.