(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 [itex]\mu_{s}[/itex]. 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?

2. Relevant equations

[itex]\Sigma[/itex]F=ma

a[itex]_{rad}[/itex]=[itex]\frac{V^{2}}{R}[/itex]

V=[itex]\frac{2\pi R}{T}[/itex]

f[itex]_{s}[/itex]=[itex]\mu_{s}[/itex]n

R=H tan[itex]\beta[/itex] (From diagram)

3. 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[itex]_{max}[/itex]=2[itex]\pi[/itex][itex]\sqrt{\frac{h tanβ(cosβ-\mu_{s}sinβ)}{g(sinβ+\mu_{s}cosβ)}}[/itex] and T[itex]_{min}[/itex]=2[itex]\pi[/itex][itex]\sqrt{\frac{h tan(cosβ+\mu_{s}sinβ)}{g(sinβ-\mu_{s}cosβ)}}[/itex]

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

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

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