Friction + centripetal acceleration problem

[Anisotropic Galaxy]

A small cube of mass m is placed on the inside of a funnel rotating about a vertical axis at a constant rate of w revolutions per second. The wall of the funnel makes an angle theta with the horizontal. The coefficient of static friction between cube and funnel is u and the center of the cube is at a distance r from the axis of rotation. Find the (a) largest and (b) smallest values of w for which the cube will not move with respect to the funnel.

I of course try to draw a free body diagram that looks pretty weird.

does w_min look like sqrt(g(sin(theta)-ucos(theta))/r(cos(theta)+usin(theta)))? And wmax the same except for the fact that you add ucos)theta_ on the top except for subtracting.

The problem is also Halliday volume 1 chapter 5 problem 18

Thanks!

 

[Galileo]

Ugh, let's put that in https://www.physicsforums.com/showthread.php?t=8997":
$$\omega_{min}=\sqrt{g(\sin(\theta)-\mu\cos(\theta))/r(\cos(\theta)+\mu \sin(\theta))}$$
or, dividing top and bottom of the fraction by cos(theta) and using some cosmetics:
$$\omega_{min}=\sqrt{\frac{g}{r}\cdot \frac{\tan \theta-\mu}{1+\mu \tan \theta}}$$

Much prettier

By the way, I got the same answer for $\omega_{min}$, for $\omega_{max}$ I get:

$$\omega_{max}=\sqrt{\frac{g}{r}\cdot \frac{\tan \theta+\mu}{1-\mu \tan \theta}}$$
so there's a change in the denominator too.