Why does this model fall apart when angular velocity is small?

Question

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?

Answer 1

I have reproduced your calculation. If $\theta$ is the angle from the central vertical axis of the hemisphere to the ball, the tangential downward force is $g \sin \theta = g\frac rR$ The tangential upward force due to rotation is $\omega ^2 r \cos \theta=\omega ^2 r \sqrt {1-\frac {r^2}{R^2}}$ When $\omega \lt \sqrt{\frac gR}$ the downward force is always greater and the ball will sit at the bottom of the bowl. When $\omega \gt \sqrt{\frac gR}$ you get the right answer.