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

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Homework Statement


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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!
 

Answers and Replies

  • #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.
 
  • #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]
 
  • #4
haruspex
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I tried to do this problem by taking the average of the y-bar centroid values but that gave me the wrong answer.
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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