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 inches2. **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 DeflectionIn 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.

Find the moment of inertia about x and y axis

Answered: Determine the moment of inertia of the…

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

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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Concept explainers

Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provide resistance in changing its state of velocity or motion for coming to a state of rest. In other words, Newton's first law of motion describes the law of inertia of a rigid body.

Question

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

### 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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