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Sphere rolling down an incline

  1. May 4, 2008 #1
    1. The problem statement, all variables and given/known data
    A hollow spherical shell with mass 2.50 kg rolls without slipping down a slope that makes an angle of 32.0 degrees with the horizontal.
    a. Find the magnitude of the acceleration [tex] a_c[/tex] of the center of mass of the spherical shell.
    b. Find the magnitude of the frictional force acting on the spherical shell.
    c. Find the minimum coefficient of friction [tex] \mu [/tex] needed to prevent the spherical shell from slipping as it rolls down the slope.


    2. Relevant equations
    For part a.
    Since its pure roll, [tex] a_c = \alpha * R
    \alpha = a_c/R [/tex]
    [tex] \tau = R*Friction = I (moment-of-inertia) * \alpha [/tex]
    [tex]Friction = (I*\alpha)/R = (I*a_c)/R^2 [/tex]
    [tex]Ma_c = Mgsin(\theta)-Friction[/tex]
    [tex]Ma_c = Mgsin(\theta)-Ia_c/R^2[/tex]
    [tex]a_c = (MR^2*g*sin(\theta))/(MR^2+I)[/tex]


    3. The attempt at a solution
    I for sphere =[tex] 2/3 MR^2[/tex]
    so, [tex]a_c = (MR^2*g*sin(\theta))/(MR^2+2/3*MR^2)[/tex]
    MR^2 cancels..
    [tex]a_c = 3/5*g*sin(\theta)[/tex]
    for a_c i got [tex]a_c = 3.12m/s^2[/tex] i think im right unless i made a mathematical error some where.
    and substituting a_c, in [tex]Ma_c = Mgsin(\theta)-Friction[/tex]
    i got Friction = 5.19 N.
    And c,
    this where I'm kind of stuck. I'm assuming since they are asking for minimum [tex]\mu[/tex] Friction is 0 in [tex] Ma_c = Mgsin(\theta)-Friction [\tex]
    [tex]a_c = gsin(\theta) [/tex].
    [tex]Friction = (I*\alpha)/R = (I*a_c)/R^2 [/tex], and
    [tex] Friction = \mu*mg*sin(\theta) [/tex]
    [tex] /mu= ((I*a_c)/R^2)/mg*sin(\theta)
    idk if I'm right in assuming Friction is 0 in one part and not in other.. Any hints/guides and help would greatly be appreciated.
     
    Last edited: May 4, 2008
  2. jcsd
  3. May 4, 2008 #2

    alphysicist

    User Avatar
    Homework Helper

    Hi dk214,

    I don't think this is right; you've already found the force of friction. Now they want the minimum [itex]\mu[/itex] that can supply that force; in other words they want the coefficient for which that frictional force is a maximum. What does that give?
     
  4. May 5, 2008 #3
    I dont know if I'm understanding the question right. Are they just asking for the [tex] \mu [/tex] for the friction I found.?
    which would just be Friction/Normal
    [tex] \mu = 5.12/Mgcos(\theta) [/tex]
    [tex] \mu = .246[/tex]
     
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