Angular momentum conservation and constant velocity as expla

[Soren4]

I'm confused about situations involving rotating frames in which the angular momentum is conserved and the initial velocity does not change. I'll make an example.

Take a rotating carousel (constant angular velocity) with no friction on it and a ball. At the initial time instant the ball has the same velocity of the carousel and is far from the center $O$. It is given a radial impulse, so that it gains a radial velocity also.

In a inertial frame the path of the ball is a straight line, while in the rotating frame it is deviated by Coriolis force.

The angular momentum of the ball is conserved with respect to $O$ in the inertial frame. Nevertheless I read many times that the "deviation" in rotating frame can be explained if we think about the fact that the ball has a greater tangential velocity than the rest of the carousel, so it appears to go faster.

So on the one side we have that, taking the center of the turntable $O$ as pivot $L=mrv_{\theta}=mr^2\omega$ is conserved, on the other hand that the initial velocity $v= \omega r$ does not change.

Neglecting $m$, saying that $r^2 \omega$ and $r \omega$ are constant is definitely not the same thing. But are both of them conserved in this case?

In this kind of situation are the two description (the one that uses the fact of greater velocity and the other that is based on the conservation of angular momentum) equivalent?
Is there one more correct to use?

 

[andrewkirk]

So on the one side we have that, taking the center of the turntable $O$ as pivot $L=mrv_{\theta}=mr^2\omega$ is conserved, on the other hand that the initial velocity $v= \omega r$ does not change.

Neglecting $m$, saying that $r^2 \omega$ and $r \omega$ are constant is definitely not the same thing. But are both of them conserved in this case?

$r\omega$ is not conserved, because that is the tangential component of velocity, which does not remain constant. As the ball, after release, moves further away from the carousal centre, its linear velocity $v$ remains constant, but the radial component increases and the tangential component decreases, as they are functions of position.

$r^2\omega$ is conserved however. Angular momentum is still meaningful, and is still conserved, when a body is not in circular motion.