Centroid of Composite Bodies - Statics of Rigid Bodies

[Nova_Chr0n0]

Homework Statement

Locate the centroid (x̄,ȳ) of the shaded area.

Relevant Equations

Formulas for Centroid of Composite Bodies

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

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:

Now I will solve for ȳ

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.

 

[erobz]

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.

 

[Lnewqban]

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.