Finding Maximum and Minimum Values of T for a Rotating Block in an Inverted Cone

[ExpoDecay]

Homework Statement

A small block with mass m is placed inside an inverted cone that is rotating about a vertical axis such that the time for one revolution of the cone is T. The walls of the cone make an angle β with the vertical. The coefficient of static friction between the block and the cone is $\mu_{s}$. If the block is to remain at a constant height H above the apex of the cone, what are the maximum and minimum values of T?

Homework Equations

$\Sigma$F=ma

a$_{rad}$=$\frac{V^{2}}{R}$

V=$\frac{2\pi R}{T}$

f$_{s}$=$\mu_{s}$n

R=H tan$\beta$ (From diagram)

The Attempt at a Solution

As far as I can tell, the only problem I'm having is with my diagram. I first placed the x-axis along the side of the cone with the friction force parallel to it, and then moved it clockwise until the weight was parallel to the y-axis. The angles that form can be seen in the attached file, along with my work.

What I come up with is T$_{max}$=2$\pi$$\sqrt{\frac{h tanβ(cosβ-\mu_{s}sinβ)}{g(sinβ+\mu_{s}cosβ)}}$ and T$_{min}$=2$\pi$$\sqrt{\frac{h tan(cosβ+\mu_{s}sinβ)}{g(sinβ-\mu_{s}cosβ)}}$

The answer that the book gives is T$_{max}$=2$\pi$$\sqrt{\frac{h tanβ(sinβ+\mu_{s}cosβ)}{g(cosβ-\mu_{s}cosβ)}}$ and T$_{min}$=2$\pi$$\sqrt{\frac{h tanβ(sinβ-\mu_{s}cosβ)}{g(cosβ+\mu_{s}sinβ)}}$

I can only come up with this solution if I switch the angles around that the normal and friction forces make.

Also, the question comes from Young and Freedman 11th edition. Chapter 5, problem 5.119

 

[Spinnor]

In the first force diagram β is in the wrong place?

 

[ExpoDecay]

Oh man, you're right! Thank you.