Locate the centroid y of the beam's cross-sectional area. Take a = 45 mm (Figure 1) Express your answer to three significant figures and include the appropriate units. μA ?
Locate the centroid y of the beam's cross-sectional area. Take a = 45 mm (Figure 1) Express your answer to three significant figures and include the appropriate units. μA ?
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![### Locating the Centroid of the Cross-Sectional Area of a Beam
**Part A**
Locate the centroid \( \bar{y} \) of the beam's cross-sectional area. Take \( a = 45 \, \text{mm} \).
**Explanation Figure 1**:
The figure illustrates a cross-sectional view of a beam. The beam consists of three rectangular sections:
- A top horizontal section with dimensions 400 mm (length) and 50 mm (height).
- Two vertical sections on either side with heights of 200 mm and unspecified width (represented by variable \( c \)).
**Dimensions:**
- Total length of the beam: 400 mm
- Height of the top horizontal section: 50 mm
- Height of each vertical side section: 200 mm
The goal is to find the \( \bar{y} \) coordinate of the centroid of this beam’s cross-sectional area.
**Input Form:**
Users are required to input the value of \( \bar{y} \) in the provided field, ensuring to:
1. Express the answer to three significant figures.
2. Include the appropriate units.
**Input Fields:**
\[ \bar{y} = \underline{\hspace{50px}} \text{ (Value)} \quad \underline{\hspace{50px}} \text{ (Units)} \]
**Example Interface:**
- Field for numeric value
- Field for unit
- Submit button
- Request Answer option for further assistance
Actively working through the problem will help develop a deeper understanding of centroid calculations in cross-sectional areas and enhance skills in applying theoretical concepts to practical engineering problems.
---
**Instructions for Calculation:**
1. Identify the geometric center of each rectangular section.
2. Calculate the areas of each section.
3. Use the formula for the centroid of composite areas, focusing on integrating the areas - making use of symmetry where applicable.
Submit your answers through the interface. Ensure accuracy and appropriate unit measures in your calculations.
**Key Formula:**
\[ \bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i} \]
where \( A_i \) is the area of each section, and \( y_i \) is the centroid of each section along the \( y \)-axis.
---
Utilize this educational tool to enhance your engineering and mathematical computation skills effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabbd02aa-e289-47d3-b3b4-87f7eeaaef64%2F79a56310-7861-4aa2-a225-306e56704583%2Ftx1312b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Locating the Centroid of the Cross-Sectional Area of a Beam
**Part A**
Locate the centroid \( \bar{y} \) of the beam's cross-sectional area. Take \( a = 45 \, \text{mm} \).
**Explanation Figure 1**:
The figure illustrates a cross-sectional view of a beam. The beam consists of three rectangular sections:
- A top horizontal section with dimensions 400 mm (length) and 50 mm (height).
- Two vertical sections on either side with heights of 200 mm and unspecified width (represented by variable \( c \)).
**Dimensions:**
- Total length of the beam: 400 mm
- Height of the top horizontal section: 50 mm
- Height of each vertical side section: 200 mm
The goal is to find the \( \bar{y} \) coordinate of the centroid of this beam’s cross-sectional area.
**Input Form:**
Users are required to input the value of \( \bar{y} \) in the provided field, ensuring to:
1. Express the answer to three significant figures.
2. Include the appropriate units.
**Input Fields:**
\[ \bar{y} = \underline{\hspace{50px}} \text{ (Value)} \quad \underline{\hspace{50px}} \text{ (Units)} \]
**Example Interface:**
- Field for numeric value
- Field for unit
- Submit button
- Request Answer option for further assistance
Actively working through the problem will help develop a deeper understanding of centroid calculations in cross-sectional areas and enhance skills in applying theoretical concepts to practical engineering problems.
---
**Instructions for Calculation:**
1. Identify the geometric center of each rectangular section.
2. Calculate the areas of each section.
3. Use the formula for the centroid of composite areas, focusing on integrating the areas - making use of symmetry where applicable.
Submit your answers through the interface. Ensure accuracy and appropriate unit measures in your calculations.
**Key Formula:**
\[ \bar{y} = \frac{\sum (A_i \cdot y_i)}{\sum A_i} \]
where \( A_i \) is the area of each section, and \( y_i \) is the centroid of each section along the \( y \)-axis.
---
Utilize this educational tool to enhance your engineering and mathematical computation skills effectively.
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