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Friction + centripetal acceleration problem

  1. Oct 19, 2005 #1
    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

  2. jcsd
  3. Oct 19, 2005 #2


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    Ugh, let's put that in LateX:
    [tex]\omega_{min}=\sqrt{g(\sin(\theta)-\mu\cos(\theta))/r(\cos(\theta)+\mu \sin(\theta))}[/tex]
    or, dividing top and bottom of the fraction by cos(theta) and using some cosmetics:
    [tex]\omega_{min}=\sqrt{\frac{g}{r}\cdot \frac{\tan \theta-\mu}{1+\mu \tan \theta}}[/tex]

    Much prettier o:)

    By the way, I got the same answer for [itex]\omega_{min}[/itex], for [itex]\omega_{max}[/itex] I get:

    [tex]\omega_{max}=\sqrt{\frac{g}{r}\cdot \frac{\tan \theta+\mu}{1-\mu \tan \theta}}[/tex]
    so there's a change in the denominator too.
    Last edited: Oct 19, 2005
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