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Moment of Inertia / Mass Element proof

  1. Dec 27, 2006 #1
    1. The problem statement, all variables and given/known data

    A thin square plate of side a has one corner at the origin and two sides along the positive x and y axes. If the density of the plate is given by p(x,y) = xy show that its mass is M=(1/4)a^4
    If the distance of the mass element dM = pdS from the origin is r the moment of inertia of the plate is I= integral [r^2 (pdS)] where S is the surface of the plate. Prove that I=Ma^2

    2. Relevant equations

    3. The attempt at a solution

    I've proved the first half by showing

    M = pdV and taking a small mass element, then integrating

    (0->a)int(dx) (0->a)int(dy) xy = (a^2)/2 * (a^2)/2 = (1/4)a^4

    However I can't seem to get started with the second part...
  2. jcsd
  3. Dec 27, 2006 #2


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    Just use the given definition:

    [tex]I=\int_0^a \int_0^a r^2\rho dxdy[/tex].

    What is r^2 in terms of x and y?
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