Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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solve for problem 6.2

### Finding the Centroid of Given Cross-Sections

In this exercise, we aim to determine the centroid of the two provided cross-sections. The centroid (geometric center) is a crucial parameter in structural engineering as it helps in the analysis of bending, shear, and deformation characteristics of structures.

#### Cross-Section 6.1

This cross-section consists of a shape resembling a block letter "U". Below are the dimensions:

- Total width at the top: 10 inches
- Width of the vertical legs: 2 inches each
- Inner width between the vertical legs: 6 inches
- Height of the vertical leg: 8 inches
- Height of the horizontal section at the top: 2 inches

Steps to locate the centroid:
1. Divide the cross-section into simpler shapes whose centroids are easier to find.
2. Calculate the area of each sub-shape.
3. Determine the centroid of each sub-shape relative to a common reference point.
4. Use the principle of moments to find the combined centroid of the complex shape.

#### Cross-Section 6.2

This cross-section forms a truncated shape, consisting of a rectangular section with an isosceles trapezoid attached. Additionally, there appears to be a cutout rectangular hole within the trapezoid. The dimensions are as follows:

- Height of the entire section: 11 inches
- Width at the bottom: 7 inches
- Width at the top: 4 inches
- Height of the rectangular hole: 1 inch
- Width of the rectangular hole: 2 inches
- Bottom height excluding the trapezoidal extra portion: 2 Inches

Steps to locate the centroid:
1. Divide the cross-section into simpler shapes, excluding the area of the hole.
2. Compute the area of each sub-shape and the hole.
3. Find the centroid of each sub-shape and the hole.
4. Use the principle of moments, accounting for the subtraction due to the hole, to find the centroid of the entire section.

### Detailed Steps to Calculate Centroid for Each Shape

**Step 1: Divide the shape:**
   - For Section 6.1, we can split the shape into three rectangular sections.
   - For Section 6.2, it can be considered as a combination of a rectangle and a trapezoid, then subtract the hole area.

**Step 2: Calculate Areas:**
   - Compute
Transcribed Image Text:### Finding the Centroid of Given Cross-Sections In this exercise, we aim to determine the centroid of the two provided cross-sections. The centroid (geometric center) is a crucial parameter in structural engineering as it helps in the analysis of bending, shear, and deformation characteristics of structures. #### Cross-Section 6.1 This cross-section consists of a shape resembling a block letter "U". Below are the dimensions: - Total width at the top: 10 inches - Width of the vertical legs: 2 inches each - Inner width between the vertical legs: 6 inches - Height of the vertical leg: 8 inches - Height of the horizontal section at the top: 2 inches Steps to locate the centroid: 1. Divide the cross-section into simpler shapes whose centroids are easier to find. 2. Calculate the area of each sub-shape. 3. Determine the centroid of each sub-shape relative to a common reference point. 4. Use the principle of moments to find the combined centroid of the complex shape. #### Cross-Section 6.2 This cross-section forms a truncated shape, consisting of a rectangular section with an isosceles trapezoid attached. Additionally, there appears to be a cutout rectangular hole within the trapezoid. The dimensions are as follows: - Height of the entire section: 11 inches - Width at the bottom: 7 inches - Width at the top: 4 inches - Height of the rectangular hole: 1 inch - Width of the rectangular hole: 2 inches - Bottom height excluding the trapezoidal extra portion: 2 Inches Steps to locate the centroid: 1. Divide the cross-section into simpler shapes, excluding the area of the hole. 2. Compute the area of each sub-shape and the hole. 3. Find the centroid of each sub-shape and the hole. 4. Use the principle of moments, accounting for the subtraction due to the hole, to find the centroid of the entire section. ### Detailed Steps to Calculate Centroid for Each Shape **Step 1: Divide the shape:** - For Section 6.1, we can split the shape into three rectangular sections. - For Section 6.2, it can be considered as a combination of a rectangle and a trapezoid, then subtract the hole area. **Step 2: Calculate Areas:** - Compute
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