Consider the shaded area shown in (Figure 1). Suppose that a = 100 mm , b = 200 mm , and r = 80 mm Determine the moment of inertia of the shaded area about the y axis. Express your answer to three significant figures and include the appropriate units.
Consider the shaded area shown in (Figure 1). Suppose that a = 100 mm , b = 200 mm , and r = 80 mm Determine the moment of inertia of the shaded area about the y axis. Express your answer to three significant figures and include the appropriate units.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Consider the shaded area shown in (Figure 1). Suppose that a = 100 mm , b = 200 mm , and r = 80 mm
Determine the moment of inertia of the shaded area about the y axis.
Express your answer to three significant figures and include the appropriate units.
![### Determining the Moment of Inertia
#### Problem Statement:
Consider the shaded area shown in (Figure 1). Suppose that \( a = 100 \ \text{mm} \), \( b = 200 \ \text{mm} \), and \( r = 80 \ \text{mm} \).
Figures and overall layout:
1. **Description**
- This problem involves calculating the moment of inertia of a shaded area about the y-axis.
- The shaded area is defined geometrically, incorporating parameters \(a\), \(b\), and \(r\).
2. **Figure Details**
- The figure consists of a composite area with specific dimensions:
- \( a \): The horizontal distance shown in the figure (100 mm).
- \( b \): The vertical distance shown in the figure (200 mm).
- \( r \): The radius of a semicircle cut out from the shape (80 mm).
- The figure shows a geometrical shape with a semicircular cut-out.
- The distance \( b \) is marked on both the vertical and horizontal axis.
- The overall shape resembles a trapezoid with a semicircle removed from one of its vertical sides.
- The coordinate axes have their origin \( O \) at the bottom left corner of the figure.
3. **Question**
- **Part A:** Determine the moment of inertia of the shaded area about the y-axis.
- You need to express your answer to three significant figures and include the appropriate units.
4. **Input Fields and Submission**
- \( I_y = \) [Value Field] [Units Field]
- There are submission buttons, including reloading, undo, and checkmark functionalities.
5. **Feedback**
- Feedback is provided for incorrect submissions with the note: "Incorrect; Try Again; 4 attempts remaining."
### Solution Approach:
1. **Break Down the Composite Area**
- Calculate the moments of inertia for each elementary area (the trapezoid and the semicircular cut-out).
2. **Applying Parallel Axis Theorem**
- Adjust the moments of inertia for any shape not oriented or located at the axis of interest.
3. **Combine Moments of Inertia**
- Summing individual contributions while considering subtractions for the cut-out shapes.
4. **Units and Reporting**
- Express results in square millimeters (\(\text{mm}^4\)) or](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F225ea937-ec91-4f60-983d-e6a97001b794%2F2b5aae37-0efc-4633-ac7c-e59cc52c96d8%2Ffbwr59v_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the Moment of Inertia
#### Problem Statement:
Consider the shaded area shown in (Figure 1). Suppose that \( a = 100 \ \text{mm} \), \( b = 200 \ \text{mm} \), and \( r = 80 \ \text{mm} \).
Figures and overall layout:
1. **Description**
- This problem involves calculating the moment of inertia of a shaded area about the y-axis.
- The shaded area is defined geometrically, incorporating parameters \(a\), \(b\), and \(r\).
2. **Figure Details**
- The figure consists of a composite area with specific dimensions:
- \( a \): The horizontal distance shown in the figure (100 mm).
- \( b \): The vertical distance shown in the figure (200 mm).
- \( r \): The radius of a semicircle cut out from the shape (80 mm).
- The figure shows a geometrical shape with a semicircular cut-out.
- The distance \( b \) is marked on both the vertical and horizontal axis.
- The overall shape resembles a trapezoid with a semicircle removed from one of its vertical sides.
- The coordinate axes have their origin \( O \) at the bottom left corner of the figure.
3. **Question**
- **Part A:** Determine the moment of inertia of the shaded area about the y-axis.
- You need to express your answer to three significant figures and include the appropriate units.
4. **Input Fields and Submission**
- \( I_y = \) [Value Field] [Units Field]
- There are submission buttons, including reloading, undo, and checkmark functionalities.
5. **Feedback**
- Feedback is provided for incorrect submissions with the note: "Incorrect; Try Again; 4 attempts remaining."
### Solution Approach:
1. **Break Down the Composite Area**
- Calculate the moments of inertia for each elementary area (the trapezoid and the semicircular cut-out).
2. **Applying Parallel Axis Theorem**
- Adjust the moments of inertia for any shape not oriented or located at the axis of interest.
3. **Combine Moments of Inertia**
- Summing individual contributions while considering subtractions for the cut-out shapes.
4. **Units and Reporting**
- Express results in square millimeters (\(\text{mm}^4\)) or
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