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

In summary, the conversation discusses the incorrect method of finding the centroid of two non-overlapping regions by taking the average of the y-bar centroid values. It is explained that the correct method is to find the weighted average of the two centroids, based on the areas of the regions. Taking an average of averages is not a reliable method and is compared to the example of an elephant and a flea on a seesaw.
  • #1
theBEAST
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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!
 
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  • #2
What is the definition of centroid? Your method would be valid if the areas of the two regions were equal.
 
  • #3
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


theBEAST said:
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?
 

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

What is the definition of a centroid?

A centroid is the point at the center of any shape or region, where all the mass of that shape or region is evenly distributed. It is also known as the center of gravity.

How is the centroid of a region calculated?

The centroid of a region is calculated by finding the average of the x-coordinates and the average of the y-coordinates of all the points within that region.

What is the significance of finding the combined centroid of two regions?

Finding the combined centroid of two regions is important in various fields such as engineering, physics, and mathematics. It helps in determining the overall center of mass of a system or object composed of multiple regions.

What information is needed to find the combined centroid of two regions?

To find the combined centroid, the individual centroids of both regions and their respective areas are needed.

Can the combined centroid be outside of both regions?

Yes, it is possible for the combined centroid to be outside of both regions. This can occur if one region has a significantly larger area or if the individual centroids are located at opposite ends of the regions.

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