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Determine the moment of inertia of the area about the x-axis shown on the figure. -3 in.3 in.→ B A tatyta 1 in. 2 in. 2 in. 2 in. 4 in. X

Determine the moment of inertia of the area about the x-axis shown on the figure. -3 in.3 in.→ B A tatyta 1 in. 2 in. 2 in. 2 in. 4 in. X

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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11. Determine the inertia about the x-axis. Determine the centroid of the shaded area along the y-axis.
**Title: Determining the Moment of Inertia of a T-Shape Area about the X-Axis** **Objective:** Determine the moment of inertia of the given area about the x-axis. **Diagram Analysis:** The provided figure illustrates a T-shaped area with the following dimensions: - The horizontal part of the T is 6 inches wide (3 inches from the center line y to the right and 3 inches from the center line y to the left) and 2 inches thick. - The vertical part of the T is 1 inch wide and 6 inches tall (extending 4 inches below the joint and 2 inches above the joint where it meets the horizontal part). **Dimensions Detailed:** - The top horizontal segment of the T-shape extends 3 inches on either side of the y-axis, resulting in a total horizontal length of 6 inches. - The vertical segment has a total height of 6 inches: 4 inches below the joint (point A) and 2 inches above the joint. - The width of the vertical segment is 1 inch. - The bottom of the vertical segment extends 2 inches to the left, followed by a 1-inch wide vertical segment, and extends another 2 inches to the right. **Organization of the Diagram:** - An x-y coordinate system is used, where the y-axis is vertical and the x-axis is horizontal. - Point A is located at the bottom of the vertical segment on the y-axis. - Point B is at the intersection of the vertical and horizontal segments on the y-axis. - There are arrows showing dimensions along the T-shaped structure, specifying the lengths and distances in inches. **Procedure to Calculate Moment of Inertia:** 1. **Split T-Shape into Two Rectangles:** - Horizontal Rectangle (Top Segment): - Width: 6 inches - Height: 2 inches - Centroid distance from x-axis: 1 inch - Vertical Rectangle (Bottom Segment): - Width: 1 inch - Height: 6 inches - Centroid distance from x-axis: 4 inches 2. **Use the Parallel Axis Theorem:** The moment of inertia \( I \) for a composite shape is calculated by summing the moments of inertia of each individual shape about a common axis (x-axis in this case). \[ I_{\text{total}} = I_1
Transcribed Image Text:**Title: Determining the Moment of Inertia of a T-Shape Area about the X-Axis** **Objective:** Determine the moment of inertia of the given area about the x-axis. **Diagram Analysis:** The provided figure illustrates a T-shaped area with the following dimensions: - The horizontal part of the T is 6 inches wide (3 inches from the center line y to the right and 3 inches from the center line y to the left) and 2 inches thick. - The vertical part of the T is 1 inch wide and 6 inches tall (extending 4 inches below the joint and 2 inches above the joint where it meets the horizontal part). **Dimensions Detailed:** - The top horizontal segment of the T-shape extends 3 inches on either side of the y-axis, resulting in a total horizontal length of 6 inches. - The vertical segment has a total height of 6 inches: 4 inches below the joint (point A) and 2 inches above the joint. - The width of the vertical segment is 1 inch. - The bottom of the vertical segment extends 2 inches to the left, followed by a 1-inch wide vertical segment, and extends another 2 inches to the right. **Organization of the Diagram:** - An x-y coordinate system is used, where the y-axis is vertical and the x-axis is horizontal. - Point A is located at the bottom of the vertical segment on the y-axis. - Point B is at the intersection of the vertical and horizontal segments on the y-axis. - There are arrows showing dimensions along the T-shaped structure, specifying the lengths and distances in inches. **Procedure to Calculate Moment of Inertia:** 1. **Split T-Shape into Two Rectangles:** - Horizontal Rectangle (Top Segment): - Width: 6 inches - Height: 2 inches - Centroid distance from x-axis: 1 inch - Vertical Rectangle (Bottom Segment): - Width: 1 inch - Height: 6 inches - Centroid distance from x-axis: 4 inches 2. **Use the Parallel Axis Theorem:** The moment of inertia \( I \) for a composite shape is calculated by summing the moments of inertia of each individual shape about a common axis (x-axis in this case). \[ I_{\text{total}} = I_1
### Educational Content #### Understanding Beam Deflection In the study of beam deflection, understanding the moment of inertia (I) is crucial. A sample problem might help in grasping the concept better. ##### Sample Problem: **Diagram Information:** The diagram depicts a beam section where a vertical load is applied at point A. The diagram provides measurements for different segments of the beam: - The beam extends 2 inches to the left of point A. - The beam extends 2 inches to the right of point A. - The section within 1 inch of point A is also marked. **Question:** What is the moment of inertia of the beam section described above, at point A? ##### Choices: - \( \circ \) 62.7 \( \text{in}^4 \) - \( \circ \) 58.7 \( \text{in}^4 \) - \( \circ \) 347 \( \text{in}^4 \) - \( \circ \) 14.7 \( \text{in}^4 \) - \( \circ \) 359 \( \text{in}^4 \) - \( \circ \) 26.7 \( \text{in}^4 \) Exploring multiple-choice questions like this one can help solidify your understanding of moments of inertia in beam sections.
Transcribed Image Text:### Educational Content #### Understanding Beam Deflection In the study of beam deflection, understanding the moment of inertia (I) is crucial. A sample problem might help in grasping the concept better. ##### Sample Problem: **Diagram Information:** The diagram depicts a beam section where a vertical load is applied at point A. The diagram provides measurements for different segments of the beam: - The beam extends 2 inches to the left of point A. - The beam extends 2 inches to the right of point A. - The section within 1 inch of point A is also marked. **Question:** What is the moment of inertia of the beam section described above, at point A? ##### Choices: - \( \circ \) 62.7 \( \text{in}^4 \) - \( \circ \) 58.7 \( \text{in}^4 \) - \( \circ \) 347 \( \text{in}^4 \) - \( \circ \) 14.7 \( \text{in}^4 \) - \( \circ \) 359 \( \text{in}^4 \) - \( \circ \) 26.7 \( \text{in}^4 \) Exploring multiple-choice questions like this one can help solidify your understanding of moments of inertia in beam sections.
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