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Homework Help: Finding the moment(s) of inertia

  1. Mar 27, 2013 #1
    1. The problem statement, all variables and given/known data

    A wedge of a sphere of radius 14 cm (similar to one
    segment of an orange) is oriented so that the axis is aligned
    with the z-axis, one face is in the xz plane, and the other
    is inclined at an angle of α = 29o
    , as shown. The wedge is
    made of metal having a density of 4500 kg/m3
    . In the coordinate system shown, compute a) Ixx, b) Iyy, and c) Izz.

    The picture is attached

    2. Relevant equations

    I know that

    I_xx= ∫(y^2+z^2)ρ dV
    I_yy= ∫(x^2+z^2)ρ dV
    I_zz= ∫(y^2+y^2)ρ dV

    3. The attempt at a solution

    I'm having trouble thinking of how to replace the dV with something in terms of x y and z

    The first thing I did was find the mass, but I'm not sure if it would help at all

    I did V=4/3 π r^3

    and then multiplied by 29/360 to find the volume of the wedge and the multiplied by the density.

    Also, just thought of this now:

    r = (x^2 + y^2 + z^2)^1/2

    could I differentiate the volume equation to find dV in terms of r and dr (4 pi r^2 dr)
    and substitute in the above expression for r and just tack on dx dy and dz? SOrry if that breaks all rules of physics

    Attached Files:

  2. jcsd
  3. Mar 27, 2013 #2


    User Avatar
    2017 Award

    Staff: Mentor

    That does not help.
    You can write dV = dx dy dz in cartesian coordinates. There are similar formulas for spherical coordinates and other coordinate systems.

    y^2 + z^2 and similar expressions are not constant for constant r, this does not work. You will need more than 1 integral.

    For I_zz, you can use the symmetry of the problem, if you know the moment of inertia of a ball.
  4. Mar 27, 2013 #3
    After posting this I converted it to spherical coordinates and used triple integration and found the correct answers. Thanks though!
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