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Setting up a Double Integral for Moment of Inertia

  1. Sep 11, 2015 #1
    1. The problem statement, all variables and given/known data

    >Problem:<br>Find the Moment of Inertia of a circular disk of uniform density about an axis which passes through the center and makes an angle of $\dfrac{\pi}{6}$ with the plane of the disc.

    2. Relevant equations

    Moment of Inertia ($I$) is $$\int r^2dm$$ where $r$ is the perpendicular distance from the chosen axis (and can vary) and $dm$ is an elemental mass.

    3. The attempt at a solution





    I interpreted the disk as a 2 Dimensional structure, hence it has mass per unit area ($\sigma$)

    $$dm=\sigma dA$$

    Thus, $$I=\int r^2dm=\int \int_A \sigma dA$$
    $$$$
    Using a change of variables to Cylindrical Co-ordinates,

    $$dA=rdrd\theta$$
    $$$$
    $$\Longrightarrow I=\int\int \sigma r^3 dr d\theta$$
    $$$$
    However, this is the Integral for finding the Moment of Inertia for an axis perpendicular to the plane of the body. I cannot understand how to modify it for the given question.
    $$$$
    I would be very grateful if somebody could please solve this question without resorting to differential equations.
    Many thanks in anticipation!

    PS. This is not a homework question. I came across it while browsing the Web for problems on Moment of Inertia.

    PPS. I don't know why the Tex code isn't working on top of the page. I'm really sorry for that.
     

    Attached Files:

  2. jcsd
  3. Sep 11, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Cylindrical coordinates are tricky, as your object does not have this symmetry around the rotation axis. If you want to keep the system where it has this symmetry, you have to modify the r2 factor. It will depend on θ.
    Alternatively, work in the system of the axis. Cartesian coordinates there could work.

    If you know about tensors, calculate the inertia tensor and do a coordinate transformation.
     
  4. Sep 11, 2015 #3
    Thank you for your reply, Sir. Sir, I haven't learnt Tensor Analysis (I'm yet in the early months of grade 11). Please could you show me how else to approach this problem? Also, could you please explain this line again: "If you want to keep the system where it has this symmetry"? I was unable to understand. How should I modify the r2 factor, Sir?
     
  5. Sep 11, 2015 #4

    SteamKing

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    Staff Emeritus
    Science Advisor
    Homework Helper

    You can calculate the MOI of a figure about an arbitrary axis by transforming the MOI properties which are determined about the standard coordinate axes.

    This article shows how:

    http://www.eng.auburn.edu/~marghitu/MECH2110/staticsC4.pdf [Broken]

    BTW, such work is usually found in first or second year college courses in calculus or mechanics. I'm not sure you have the requisite math background, being only about halfway thru high school, to understand some of the concepts you need to solve this problem.
     
    Last edited by a moderator: May 7, 2017
  6. Sep 11, 2015 #5
    Thanks Sir. I'll go through the site when I wake up (it is really late here in the night in India).
    Actually Sir, Maths (specifically Calculus) is my passion and I've spent a lot of time on it ; I've done till Multivariable Calculus - till Vector Fields and Line Integrals (the basics from Apostol's Books on Calculus). Unfortunately though, I haven't done Differential Equations properly, a drawback I've been trying to rectify...
     
    Last edited by a moderator: May 7, 2017
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