Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Problem -08 Moment of InertiaDetermine the polar moment of inertia of the area shown with respect to (a) point O, (b) the centroid of the area.
![### Figure 8
**Description:**
The image presents a shaded segment of a circle centered at point O. This segment forms a quarter arc (90 degrees) of a larger circle. Detailed measurements are provided to describe the inner and outer radii of the quarter arc.
**Diagram Analysis:**
- The inner radius of the shaded segment is 3 inches. This distance is shown from the center point O to the beginning of the arc.
- The outer radius of the shaded segment is 4.5 inches. This measurement is taken from the center point O to the outer edge of the arc.
The shaded area represents the space between these two radii, forming an annular sector.
### Educational Context
In geometry, sectors and segments of circles are important concepts. This figure specifically demonstrates an annular sector, which is part of an annulus (the region between two concentric circles). It can be useful for problems involving area calculations, understanding circular motion, and applications in engineering and design.
**Key Points:**
- **Quarter Arc**: Indicates how a 90-degree segment looks between two concentric circles.
- **Inner Radius (3 inches)**: Important for calculating inner edges and areas.
- **Outer Radius (4.5 inches)**: Vital for outer edge dimensions and areas.
**Useful Formulas:**
1. **Area of an annular sector**:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi (R^2 - r^2)
\]
where:
- \(\theta\) is the central angle in degrees (here, 90°),
- \(R\) is the outer radius (4.5 inches),
- \(r\) is the inner radius (3 inches).
2. **Arc Length of an annular sector**:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi R
\]
where \(\theta\) and \(R\) are as defined above.
This figure aids in visualizing and solving problems related to sectors of circles, particularly annular sectors.](https://content.bartleby.com/qna-images/question/79826512-1148-4736-b4a7-6e7f4ec13f28/2ecb7613-ea81-4403-94f5-9917e990d0b9/8i5pqga_processed.png)
Transcribed Image Text:### Figure 8
**Description:**
The image presents a shaded segment of a circle centered at point O. This segment forms a quarter arc (90 degrees) of a larger circle. Detailed measurements are provided to describe the inner and outer radii of the quarter arc.
**Diagram Analysis:**
- The inner radius of the shaded segment is 3 inches. This distance is shown from the center point O to the beginning of the arc.
- The outer radius of the shaded segment is 4.5 inches. This measurement is taken from the center point O to the outer edge of the arc.
The shaded area represents the space between these two radii, forming an annular sector.
### Educational Context
In geometry, sectors and segments of circles are important concepts. This figure specifically demonstrates an annular sector, which is part of an annulus (the region between two concentric circles). It can be useful for problems involving area calculations, understanding circular motion, and applications in engineering and design.
**Key Points:**
- **Quarter Arc**: Indicates how a 90-degree segment looks between two concentric circles.
- **Inner Radius (3 inches)**: Important for calculating inner edges and areas.
- **Outer Radius (4.5 inches)**: Vital for outer edge dimensions and areas.
**Useful Formulas:**
1. **Area of an annular sector**:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi (R^2 - r^2)
\]
where:
- \(\theta\) is the central angle in degrees (here, 90°),
- \(R\) is the outer radius (4.5 inches),
- \(r\) is the inner radius (3 inches).
2. **Arc Length of an annular sector**:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi R
\]
where \(\theta\) and \(R\) are as defined above.
This figure aids in visualizing and solving problems related to sectors of circles, particularly annular sectors.
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