Proof for conservation of angular momentum in rotating frame of reference

Question

imagine a simple situation such that an object is rotating on a table about a rope as seen in the below figure

now if i pull on the rope from the center , the radius of circular motion of the object decreases so the angular velocity must increase due to conservation of linear momentum.

my doubt is , if i were to evaluate the situation in rotating frame of reference which was rotating along the ball

the increase in tension resulting only a increase force in radial direction so what causes my angular velocity to increase then? (since there is no force in the tangential direction)

Answer 1

since there is no force in the tangential direction

In the rotating reference frame, the Coriolis force $\vec{F}_\text{cor} = 2 m \vec{v} \times \vec{\Omega}$ on an object moving inwards acts in the tangential direction. Specifically, if $\vec{v} = - v \hat{r}$, we have $$ \vec{F}_\text{cor} = 2 m (-v \hat{r}) \times (\Omega \hat{z}) = 2 m v \Omega \hat{\theta} $$ and so the ball will move in the tangential direction according to a rotating observer.