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
Related questions
Question
100%
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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, civil-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Structural Analysis (10th Edition)
Civil Engineering
ISBN:
9780134610672
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Principles of Foundation Engineering (MindTap Cou…
Civil Engineering
ISBN:
9781337705028
Author:
Braja M. Das, Nagaratnam Sivakugan
Publisher:
Cengage Learning
Structural Analysis (10th Edition)
Civil Engineering
ISBN:
9780134610672
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Principles of Foundation Engineering (MindTap Cou…
Civil Engineering
ISBN:
9781337705028
Author:
Braja M. Das, Nagaratnam Sivakugan
Publisher:
Cengage Learning
Fundamentals of Structural Analysis
Civil Engineering
ISBN:
9780073398006
Author:
Kenneth M. Leet Emeritus, Chia-Ming Uang, Joel Lanning
Publisher:
McGraw-Hill Education
Traffic and Highway Engineering
Civil Engineering
ISBN:
9781305156241
Author:
Garber, Nicholas J.
Publisher:
Cengage Learning