8-1. Determine y for the area shown in Figure P8-1. y 2" 1" 2" 2" 4" 4" FIGURE P8-1 X
8-1. Determine y for the area shown in Figure P8-1. y 2" 1" 2" 2" 4" 4" FIGURE P8-1 X
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![### Determining the Centroid (\( \bar{y} \)) for a Composite Area
**Problem Statement:** Determine \( \bar{y} \) for the area shown in Figure P8-1.
**Figure P8-1 Description:**
The diagram illustrates a composite area consisting of a large rectangular shape with a smaller rectangular cutout. The large rectangle has dimensions of 4 inches along the y-axis and 9 inches along the x-axis, composed of three segments: two 2-inch rectangles at the top and bottom separated by a 4-inch central rectangle. The cutout is placed vertically inside this large rectangle, aligned with the y-axis. The cutout's horizontal dimension is 2 inches and vertical dimension is 4 inches, centered within the larger rectangle.
**Objective:**
To calculate \( \bar{y} \), the centroid of the composite area along the y-axis.
**Approach:**
1. **Divide the Shape:** Split the composite area into simpler shapes (rectangles).
- Large rectangle: \( 8'' \times 5'' \) minus the cutout.
- Cutout area: \( 4'' \times 2'' \).
2. **Determine Areas and Centroid Locations:**
- Let the larger rectangle with no cutout be Area 1, and its centroid be \( (y_1)\).
- Let the cutout be Area 2, and its centroid be \( (y_2)\).
3. **Calculate Areas:**
- \( A_1 \) (large rectangle) = \(8 \times 5 = 40 \) square inches.
- \( A_2 \) (cutout) = \(4 \times 2 = 8 \) square inches.
4. **Find Centroid Locations:**
- Centroid of Area 1 (large rectangle without cutout) is at \( y_1 = 4 \) inches.
- Centroid of Area 2 (cutout) is at \( y_2 = 4 \) inches.
5. **Apply Centroid Formula:**
\[
\bar{y} = \frac{\sum (A_i \times y_i)}{\sum A_i}
\]
Here,
\[
\bar{y} = \frac{(40 \times 4) - (8 \times 4)}{40 -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a2e60b5-531b-408d-ad87-91178dbcae31%2F4b1b1b35-414a-4292-b56e-2b8a51103d4b%2F36vzxy_processed.gif&w=3840&q=75)
Transcribed Image Text:### Determining the Centroid (\( \bar{y} \)) for a Composite Area
**Problem Statement:** Determine \( \bar{y} \) for the area shown in Figure P8-1.
**Figure P8-1 Description:**
The diagram illustrates a composite area consisting of a large rectangular shape with a smaller rectangular cutout. The large rectangle has dimensions of 4 inches along the y-axis and 9 inches along the x-axis, composed of three segments: two 2-inch rectangles at the top and bottom separated by a 4-inch central rectangle. The cutout is placed vertically inside this large rectangle, aligned with the y-axis. The cutout's horizontal dimension is 2 inches and vertical dimension is 4 inches, centered within the larger rectangle.
**Objective:**
To calculate \( \bar{y} \), the centroid of the composite area along the y-axis.
**Approach:**
1. **Divide the Shape:** Split the composite area into simpler shapes (rectangles).
- Large rectangle: \( 8'' \times 5'' \) minus the cutout.
- Cutout area: \( 4'' \times 2'' \).
2. **Determine Areas and Centroid Locations:**
- Let the larger rectangle with no cutout be Area 1, and its centroid be \( (y_1)\).
- Let the cutout be Area 2, and its centroid be \( (y_2)\).
3. **Calculate Areas:**
- \( A_1 \) (large rectangle) = \(8 \times 5 = 40 \) square inches.
- \( A_2 \) (cutout) = \(4 \times 2 = 8 \) square inches.
4. **Find Centroid Locations:**
- Centroid of Area 1 (large rectangle without cutout) is at \( y_1 = 4 \) inches.
- Centroid of Area 2 (cutout) is at \( y_2 = 4 \) inches.
5. **Apply Centroid Formula:**
\[
\bar{y} = \frac{\sum (A_i \times y_i)}{\sum A_i}
\]
Here,
\[
\bar{y} = \frac{(40 \times 4) - (8 \times 4)}{40 -
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