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Finding the angular momentum using the inertia tensor/matrix

  1. Nov 18, 2015 #1
    1. The problem statement, all variables and given/known data
    A thin ring of radius r is constrained to rotate with constant angular velocity ω as shown in attached picture. Let the linear mass density of the ring be ρ(θ)=ρ0(2+sin2θ) where ρ0 is constant.

    a) Find the angular momentum L of the ring about O, at the instant the ring is in the xy plane as shown. Answer this part twice: (i) by using the moments and products of inertia Iij, and (ii) by directly integrating L=∫dmr x v.
    image.jpeg

    2. Relevant equations
    The moment of inertia tensor/ matrix.
    L==ntransposeIn

    3. The attempt at a solution
    Not sure if I am starting this problem properly, attached below is my attempt.
    image.jpeg

    Knowing that I must plug these moments and products of India inside the tensor matrix.

    Please help
     
  2. jcsd
  3. Nov 18, 2015 #2

    SteamKing

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    When you calculated the moment of inertia Ixx, you treated the density of the ring, ρ, as a constant. According to the OP, ρ(θ) = ρo(2+sin2θ), where ρo is a constant. If you are going to calculate the MOI matrix for the ring, you must take this arbitrary density function into account. This extends even to calculating the mass of the ring.
     
  4. Nov 18, 2015 #3
    Thank you.

    But are my integral limits correct considering it is a ring? Moreover, shall I integrate with respect to y, x, and/or θ, or just one variable?
     
  5. Nov 18, 2015 #4

    SteamKing

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    I don't think so.

    You seem to have integrands which use cartesian coordinates while the limits appear to be expressed in polar coordinates (what does x = 2πr mean? Isn't that the circumference of the ring?)
     
  6. Nov 18, 2015 #5
    Thanks.
     
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