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Finding the combined centroid of two regions given the centroids of both regions

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data
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    3. The attempt at a solution
    I tried to do this problem by taking the average of the y-bar centroid values but that gave me the wrong answer. I am only interesting in knowing why this method is incorrect.

    Thanks!
     
  2. jcsd
  3. Feb 7, 2013 #2

    SteamKing

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    What is the definition of centroid? Your method would be valid if the areas of the two regions were equal.
     
  4. Feb 7, 2013 #3

    HallsofIvy

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    It can be shown that the centroid of the union of two (non-overlapping) regions is the weighted average of the two centroids, weighted by the areas of the regions. That is, if the two regions have centroid [itex](x_1, y_1)[/itex] and [itex](x_2, y_2)[/itex] and have areas [itex]A_1[/itex] and [itex]A_2[/itex], respectively, then the centroid of the combined regions is at
    [tex]\left(\frac{A_1x_1+ A_2x_2}{A_1+ A_2}, \frac{A_1y_1+ A_2y_2}{A_1+ A_2}\right)[/tex]
     
  5. Feb 7, 2013 #4

    haruspex

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    Re: Finding the combined centroid of two regions given the centroids o

    As your grandmother should have taught you, don't take an average of averages. If an elephant and a flea get on opposite ends of a symmetric seesaw, will they balance? So where's their combined centroid?
     
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