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Determine the mass moment of inertia of a quarter of an annulus

  1. Apr 25, 2012 #1
    1. The problem statement, all variables and given/known data
    B.1

    The quarter ring has mass m and was cut from a thin uniform plate. Knowing that r1 = 1/2r2 determine the mass moment of inertia of the quarter ring with respect to A. axis AA' B. The centroidal axis CC' that is perpendicular to the plane of the quarter ring.

    CC' is located possibly where the center of mass is? (See attached file). EDIT: obviously it is the centroid.. the problem says centroidal axis, I'm dumb..

    2. Relevant equations

    IAA' = 1/4mr2
    ICC' = 1/2mr2

    Parallel Axis Theorem
    I = I(bar) + md2

    3. The attempt at a solution

    I originally tried to just use the mass moments of inertia to calculate it. I then realized that the center of mass of the quarter annulus will not be at the origin O in this case so I probably will have to use the parallel axis theorem.

    I am really lost on this and I originally calculated
    IAA' = (1/16)m(r22 - (1/2)r22)
    ICC' = (1/8)m(r22 - (1/2)r22)
     

    Attached Files:

    Last edited: Apr 25, 2012
  2. jcsd
  3. Apr 25, 2012 #2
    I've calculated I(bar)AA' = (1/4)[(1/4)(mr22 - (1/2)mr22)] and simplified this to:
    I(bar)AA' = (1/32)mr22

    Then adding that to md2 where d = (1/2)r1 → (1/4)r2 from the parallel axis theorem to get
    (1/32)mr22 + m(1/4)r22 = (9/32)mr22

    The answer given by my professor is (5/16)mr22
    Did I make a mistake somewhere or is the provided answer wrong?
     
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