I'm doing a physics problem in which a marble spins around a spinning bowl and both have angular velocity $\omega$. It rotates with radius $r$ around the central axis and the hemispherical bowl has radius $R$. I've solved for the radius $r$ in terms of $\omega$:
$$r=\sqrt{R^2-\frac{g^2}{\omega^4}}$$
But I can't figure out what this means when $\omega$ is really small ($<\sqrt{\frac{g}{R}}$ ).
Some hypotheses:
- Something to do with friction
- It falls through the center of the bowl (if it were hollow)
- We're missing something in this model
Are any of these right? Is this more complex than it seems?