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-   -   Centroid of an ellipsoid (https://www.askmehelpdesk.com/showthread.php?t=708604)

  • Oct 12, 2012, 12:41 PM
    jimmy1843
    Centroid of an ellipsoid
    What is the centroid (or center of mass) of a homogenous half-ellipsoid in terms of its semi-axes (a, b, c)? Is it different from that of a half-ellipse?
  • Oct 12, 2012, 12:46 PM
    ebaines
    The center of mass is at the intersection of its semi-major and semi-minor axes.
  • Oct 12, 2012, 12:48 PM
    jimmy1843
    But where in the z axis? Remember this is half-ellipsoid.
  • Oct 12, 2012, 01:10 PM
    ebaines
    Quote:

    Originally Posted by jimmy1843 View Post
    Remember this is half-ellipsoid.

    Sorry - I missed that. You have values for a, b, c, for the equation



    Right? The center of mass along the z-axis can be found from



    where V = volume of the half elipsoid, which is , and A(z) is the cross-sectional area as a function of z. The cross-section is an ellipse - you will need to come up with equations for the lengths of the semi-major an semi-minor axes as a function of z, use that to find an expression for A as a function of z (hint - the area of an ellipse is times the lengths of the semi-major and semi-minor axes), and put that into the above integral. It works out pretty nicely - post back with what you get.
  • Oct 20, 2012, 10:21 PM
    jimmy1843
    Quote:

    Originally Posted by ebaines View Post
    Sorry - I missed that. You have values for a, b, c, for the equation



    Right? The center of mass along the z-axis can be found from



    where V = volume of the half elipsoid, which is , and A(z) is the cross-sectional area as a function of z. The cross-section is an ellipse - you will need to come up with equations for the lengths of the semi-major an semi-minor axes as a function of z, use that to find an expression for A as a function of z (hint - the area of an ellipse is times the lengths of the semi-major and semi-minor axes), and put that into the above integral. It works out pretty nicely - post back with what you get.

    In order to findan expression for A as a function of z, we consider:
    pi*x*y=A(z)

    We use:


    and


    solving for x and y in terms of z, we obtain:
    x = a*sqrt(1 - z^2/c^2)
    y = b*sqrt(1 - z^2/c^2)

    putting these two in the integral, we come up with


    Substituting x and y:


    After simplifying and integrating we get:





    Am I on the right track?
  • Oct 22, 2012, 06:59 AM
    ebaines
    Quote:

    Originally Posted by jimmy1843 View Post
    x = a*sqrt(1 - z^2/c^2)
    y = b*sqrt(1 - z^2/c^2)

    putting these two in the integral, we come up with


    Not quite - the formula is:



    Note the 'z' term that you left out. It results in the numerator becoming




    Quote:

    Originally Posted by jimmy1843 View Post



    Am I on the right track?

    Clearly the value for must lie somewhere between 0 and z. With the correction above you should get it now.
  • Oct 22, 2012, 08:31 AM
    jimmy1843
    Quote:

    Originally Posted by ebaines View Post
    Not quite - the formula is:



    Note the 'z' term that you left out. It results in the numerator becoming




    Clearly the value for must lie somewhere between 0 and z. With the corection above you should get it now.


    Oh yes! Sorry I missed it. My oversight. I think the answer should be:
    = 3/8c
    I hope this time I'm correct (?)
    Thanks.
  • Oct 22, 2012, 09:14 AM
    ebaines
    Quote:

    Originally Posted by jimmy1843 View Post
    I think the answer should be:
    = 3/8c
    I hope this time I'm correct (?)
    Thanks.

    Yes - I agree with your answer - nicely done!

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