Given: ● X ● x1 x1 = 15.35 mm . x2 = 16.74 mm x3 = 16.10 mm y1 = 19.32 mm y2 = 20.55 mm • y3 = 16.37 mm x2 x3 y3 y2 y1 Find x & y centroid
Given: ● X ● x1 x1 = 15.35 mm . x2 = 16.74 mm x3 = 16.10 mm y1 = 19.32 mm y2 = 20.55 mm • y3 = 16.37 mm x2 x3 y3 y2 y1 Find x & y centroid
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![### Finding the Centroid of a Polygon
To find the centroid (geometric center) of the given polygon, we'll use the dimensions depicted in the drawing and the provided data. The centroid can be found using the coordinates (x̄, ȳ).
#### Given Dimensions:
- **x1** = 15.35 mm
- **x2** = 16.74 mm
- **x3** = 16.10 mm
- **y1** = 19.32 mm
- **y2** = 20.55 mm
- **y3** = 16.37 mm
#### Steps to Find the Centroid:
1. Divide the polygon into simpler shapes (typically triangles or rectangles).
2. Calculate the centroid for each simpler shape.
3. Use the area-weighted method to find the overall centroid coordinates.
#### Diagram Explanation:
The diagram shows a polygon divided into various segments. The relevant dimensions are marked as follows:
- **x1, x2,** and **x3** indicate the horizontal distances from the origin (left side).
- **y1, y2,** and **y3** indicate the vertical distances from the bottom to certain points of the polygon.
### Applying the Centroid Formula:
The general centroid formula for a composite shape is given by:
\[ \bar{x} = \frac{\sum (A_i \cdot x_i)}{\sum A_i} \]
\[ \bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i} \]
where:
- \( x_i \) and \( y_i \) are the centroids of individual shapes,
- \( A_i \) is the area of individual shapes,
- \( \bar{x} \) and \( \bar{y} \) are the coordinates of the overall centroid.
### Calculation (Example Process):
1. **Identify Individual Shapes**: Divide the given polygon into triangles or rectangles.
2. **Calculate the Areas**: Use basic geometric area formulas.
3. **Find Individual Centroids**: Use centroid formulas for each shape (\(x_i, y_i\)).
4. **Apply the Centroid Formula**: Combine using the area-weighted method.
**Note**: For complex shapes, more detailed step-by-step calculation may be needed, which usually involves integrating over the shape or using geometric decomposition as shown in the diagram.
Finally, substitute](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0062023f-64ec-498a-8360-7c79f327cf33%2Fa37a1aa1-eff0-43df-83ac-ddad87d10d70%2F3hwnoii_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Centroid of a Polygon
To find the centroid (geometric center) of the given polygon, we'll use the dimensions depicted in the drawing and the provided data. The centroid can be found using the coordinates (x̄, ȳ).
#### Given Dimensions:
- **x1** = 15.35 mm
- **x2** = 16.74 mm
- **x3** = 16.10 mm
- **y1** = 19.32 mm
- **y2** = 20.55 mm
- **y3** = 16.37 mm
#### Steps to Find the Centroid:
1. Divide the polygon into simpler shapes (typically triangles or rectangles).
2. Calculate the centroid for each simpler shape.
3. Use the area-weighted method to find the overall centroid coordinates.
#### Diagram Explanation:
The diagram shows a polygon divided into various segments. The relevant dimensions are marked as follows:
- **x1, x2,** and **x3** indicate the horizontal distances from the origin (left side).
- **y1, y2,** and **y3** indicate the vertical distances from the bottom to certain points of the polygon.
### Applying the Centroid Formula:
The general centroid formula for a composite shape is given by:
\[ \bar{x} = \frac{\sum (A_i \cdot x_i)}{\sum A_i} \]
\[ \bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i} \]
where:
- \( x_i \) and \( y_i \) are the centroids of individual shapes,
- \( A_i \) is the area of individual shapes,
- \( \bar{x} \) and \( \bar{y} \) are the coordinates of the overall centroid.
### Calculation (Example Process):
1. **Identify Individual Shapes**: Divide the given polygon into triangles or rectangles.
2. **Calculate the Areas**: Use basic geometric area formulas.
3. **Find Individual Centroids**: Use centroid formulas for each shape (\(x_i, y_i\)).
4. **Apply the Centroid Formula**: Combine using the area-weighted method.
**Note**: For complex shapes, more detailed step-by-step calculation may be needed, which usually involves integrating over the shape or using geometric decomposition as shown in the diagram.
Finally, substitute
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