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Frequency of oscillating cylinder

  1. Oct 20, 2008 #1
    1. The problem statement, all variables and given/known data

    The problem is this: Find the angular frequency of the system in the figure when it's displaced at a small angle from equlibrium, given that ρ_0 < ρ_1. There is friction with the ground, so the motion is a rolling motion, without slipping.


    2. Relevant equations

    I used the following equations, and got a wrong answer:

    f_s = friction
    m = mass of small cylinder
    M = mass of large cylinder
    x = linear displacement from equilibrium
    θ - angular displacement from equilibrium

    for linear forces (here the tag stands for d/dt):

    f_s = (m+M)x''

    since it's a rolling motion:

    x = Rθ
    so: x'' = Rθ''

    And the equation for the moments, from the center of the large cylinder (I think this is wrong):

    f_sR - mgsinθ*R/2 = Iθ''

    also:

    m = (π ρ_1 R^2)/4
    M = π ρ_0 R^2 - (π ρ_0 R^2)/4
    I = π ρ_0 R^4 - (π ρ_0 R^4)/2 + (π ρ_1 R^4)/2

    3. The attempt at a solution

    Using all the above equations and getting the ODE for x gives (unless I got the factors wrong):

    ω = (ρ_1/(10ρ_0 + 6ρ_1))*(g/R)

    This is not right - the right answer is:

    ω = (10(ρ_1 - ρ_0)/(7(ρ_1 + 31ρ_0))*(g/R)

    Which is of course a lot more sensible since there shouldn't be an oscillation for ρ_1 = ρ_0, and for ρ_1 < ρ_0 the model is wrong.

    So what's the right answer? And what am I doing wrong?

    Thanks,

    Lior

    BTW - why isn't the latex working?
     
  2. jcsd
  3. Oct 20, 2008 #2
    Oh, the figure...

    4 characters
     

    Attached Files:

  4. Oct 20, 2008 #3
    And also -

    ρ_0, ρ_1 are the mass densities of the respective parts of the cylinder.
     
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