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Moment of inertia of a rotated line

  1. Feb 16, 2010 #1


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    1. The problem statement, all variables and given/known data
    I have a drum that comprises of basically a cylinder and rotated line in the form y=mx+c

    http://img22.imageshack.us/img22/6108/druma.jpg [Broken]

    2. Relevant equations

    [tex]I=\int r^2 dm[/tex]

    3. The attempt at a solution

    I found the MOI for the cylinder as 1/2M(a2+b2) where a and b are the inner and outer radii respectively.

    I am getting trouble forming the MOI for the line. I am supposed to consider an elemental section, but I am not sure if I should consider a disc or some other shape.

    Disc: (Width dx and radius 'y')


    dm=ρ dV = ρ(πy2dx)

    dI=1/2ρπ y4 dx

    [tex]I_{line}= \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} y^4 dx[/tex]

    Since I have two lines, the inertia would be

    [tex]I = \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} (y_2^4 -y_1^4) dx[/tex]

    I can appropriately select x2 and x1 based on height of the cylinder and the height of the total drum. I can find the ρ for the material. But I'd need to calculate the equations for y2 and y1.

    So my MOI would be

    [tex]I = \frac{1}{2} M(a^2+b^2)+ \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} (y_2^4 -y_1^4) dx[/tex]

    Is this correct or did I make a mistake in my element or in the algebra?
    Last edited by a moderator: May 4, 2017
  2. jcsd
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