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floridaMAO
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A very small cube of mass m is placed on the inside of a funnel rotating about a vertical axis (the central axis of the funnel) at a constant rate of v revolutions per second. The wall of the funnel makes an angle [tex]\theta[/tex] with the horizontal. The coefficient of static friction between cube and funnel is [tex]\mu_s[/tex] 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 v for which the cube will not move with respect to the funnel.
For reference, this is problem 53 chapter 6 of Halliday Resnick Krane vol. 1.
Am I correct in thinking that friction, gravity and normal force are the only forces acting on the box? And that maximum v corresponds to minimum friction force (0?)? And that minimum v corresponds to maximum friction force ([tex]\mu_sN[/tex])?
For reference, this is problem 53 chapter 6 of Halliday Resnick Krane vol. 1.
Am I correct in thinking that friction, gravity and normal force are the only forces acting on the box? And that maximum v corresponds to minimum friction force (0?)? And that minimum v corresponds to maximum friction force ([tex]\mu_sN[/tex])?
If it helps, the answers to the above are (a)[tex]\frac{1}{2\pi}\sqrt{\frac{g(\tan(\theta)+\mu_s)}{r(1-\mu_s\tan(\theta))}}[/tex] and (b) [tex]\frac{1}{2\pi}\sqrt{\frac{g(\tan(\theta)-\mu_s)}{r(1+\mu_s\tan(\theta))}}[/tex] (from the answers in the back of my book ). I'm hoping for an explanation of these answers. It might help to notice that the expressions inside the square roots are (a) [tex]\frac{g}{r}\tan(\theta+\tan^{-1}(\mu_s))[/tex] and (b) [tex]\frac{g}{r}\tan(\theta-\tan^{-1}(\mu_s))[/tex]