Engineering Centroid of Composite Bodies - Statics of Rigid Bodies

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The discussion focuses on locating the centroid of a composite body using area calculations. The user has computed areas for a triangle, a square minus a quarter circle, and a larger quarter circle minus a smaller one. While the x̄ value is correct, the user is struggling with the ȳ value, which they found to be inconsistent with online solutions. It is suggested that the user may have overlooked the impact of the hollow areas in their calculations, which affects the centroid's position, indicating that the resultant centroid should be higher than calculated. Understanding the geometry of the composite body is crucial for accurate centroid determination.
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Homework Statement
Locate the centroid (x̄,ȳ) of the shaded area.
Relevant Equations
Formulas for Centroid of Composite Bodies
1698042147756.png
1698042973332.png
1698043116611.png

The figure and formulas is shown above. My strategy of cutting the areas/shapes is shown below:
1698042252346.png

Area 1 = Area of Triangle
Area 2 = Area of the square - Area of the quarter circle
Area 3 = Area of the larger quarter circle - Area of the smaller quarter circle

Computing for the areas, I got:
1698042494429.png


Now I will solve for ȳ
1698042762082.png


Now my problem is, when I've tried to search for the answers in the internet, ȳ = 1.386 or 1.39 inches. I've got the x̄ right which is x̄=-1.175 or -1.18 inches. I cannot find my mistake here, can someone enlighten me? The solutions I've found on the internet used this cutted areas.
1698043377121.png
 
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Nova_Chr0n0 said:
Homework Statement: Locate the centroid (x̄,ȳ) of the shaded area.
Relevant Equations: Formulas for Centroid of Composite Bodies

View attachment 334022View attachment 334031View attachment 334032
The figure and formulas is shown above. My strategy of cutting the areas/shapes is shown below:
View attachment 334025
Area 1 = Area of Triangle
Area 2 = Area of the square - Area of the quarter circle
Area 3 = Area of the larger quarter circle - Area of the smaller quarter circle

Computing for the areas, I got:
View attachment 334026

Now I will solve for ȳ
View attachment 334030

Now my problem is, when I've tried to search for the answers in the internet, ȳ = 1.386 or 1.39 inches. I've got the x̄ right which is x̄=-1.175 or -1.18 inches. I cannot find my mistake here, can someone enlighten me? The solutions I've found on the internet used this cutted areas.
View attachment 334034
You appear to have forgotten about the quarter circles taken out in the ##\bar y## computation for each area with regards to the component centroids.
 
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The posted math does not match the actual geometry of the composite body.

Calculated values of y of the centroid for A2 and A3 are less than the actual values, as those correspond to solid square and quarter of circle of reduced area (after substracting hollow areas).

The hollow areas at the bottom of those shapes make the resultant centroid to relocate to a higher position than for the solid shape.
 
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