Centroid of Composite Bodies - Statics of Rigid Bodies

In summary, the centroid of composite bodies refers to the geometric center of a shape that is formed by combining multiple simple shapes. In the context of statics of rigid bodies, the centroid is crucial for analyzing the equilibrium and stability of structures. The centroid's location can be determined by calculating the weighted average of the centroids of the individual components based on their areas or volumes. Understanding the centroid helps in assessing how forces and moments affect the entire composite body, ensuring accurate predictions of behavior under load.
  • #1
Nova_Chr0n0
16
3
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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  • #2
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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  • #3
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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FAQ: Centroid of Composite Bodies - Statics of Rigid Bodies

What is the centroid of a composite body?

The centroid of a composite body is the point at which the entire area or volume of the body can be considered to be concentrated. It is the weighted average of the centroids of the individual components that make up the composite body, taking into account their respective areas or volumes.

How do you find the centroid of a composite body?

To find the centroid of a composite body, you need to:1. Divide the body into simpler shapes whose centroids are known or can be easily determined.2. Calculate the area or volume of each of these simpler shapes.3. Determine the coordinates of the centroid of each shape.4. Use the formula for the centroid of a composite body, which is the weighted average of the centroids of the individual shapes, weighted by their areas or volumes: \[ \bar{x} = \frac{\sum (A_i \cdot x_i)}{\sum A_i} \] \[ \bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i} \] Here, \( A_i \) is the area (or volume) of the i-th shape, and \( (x_i, y_i) \) are the coordinates of its centroid.

What is the significance of the centroid in statics?

In statics, the centroid is crucial because it is the point through which the resultant of all the gravitational forces (weight) of the body acts. Knowing the centroid helps in analyzing and solving problems related to balance, stability, and support reactions of structures and mechanical systems.

Can the centroid of a composite body be outside the physical boundaries of the body?

Yes, the centroid of a composite body can be located outside its physical boundaries. This typically occurs in cases where the body has a non-uniform shape or contains voids (holes). The centroid is a theoretical point that represents the average position of the distributed area or volume, not necessarily a point within the material itself.

What is the difference between centroid and center of mass?

The centroid and center of mass are often used interchangeably, but they are not always the same. The centroid refers to the geometric center of an area or volume, assuming uniform density. The center of mass, on the other hand, is the point at which the mass of an object is considered to be concentrated, taking into account the actual distribution of mass. If the density is uniform, the centroid and center of mass coincide; otherwise, they may differ.

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