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Moment of Inertia

  1. Apr 30, 2007 #1


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    1. The problem statement, all variables and given/known data
    There is a rectangular prismof uniform mass distribution with lengths of [itex]a[/itex], [itex]b[/itex], and [itex]c[/itex] (b>a>c). Calculate it's rotational inertia about an axis through one cornet and perpendicular to the large faces.

    2. Relevant equations
    [tex] I = \int r^2 dm [/tex]
    [tex] r^2 = x^2 + y^2 + z^2 [/tex]
    [tex] \rho = \frac{M}{V} [/tex]
    [tex] V = abc [/tex]

    3. The attempt at a solution

    I am examining a cubic differential mass of [itex]dm[/itex]'s contribution on the moment of inertia about the axis of rotation. The radius between [itex]dm[/itex] and the axis of rotation is [itex]r^2 = x^2 + y^2 + z^2[/itex]. The density, [itex]\rho[/itex], is constant which is [itex]\frac{M}{V}[/itex], so [itex] dm = \rho dV [/itex].

    [tex] I = \int r^2 dm = \int (x^2 + y^2 + z^2) \rho dV [/tex]
    [tex] I = \rho \iiint_V x^2 dV + y^2 dV + z^2 dV = \int^a_0 \int^b_0 \int^c_0 x^2 dzdydx + \int^a_0 \int^b_0 \int^c_0 y^2 dzdydx + \int^a_0 \int^b_0 \int^c_0 z^2 dzdydx [/itex]
    [tex] I = \frac{\rho}{3} ( a^3 bc + ab^3 c + abc^3)[/tex]
    [tex] I = \frac{M}{3abc} ( a^3 bc + ab^3 c + abc^3)[/tex]
    [tex] I = \frac{M}{3} (a^2 + b^2 + c^2) [/tex]

    This problem looked cool so I did it, but it was an even one so there is no answer in the back of the book. Does this look right?
    Last edited: Apr 30, 2007
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