College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Question
7
![**Determining Moments of Inertia for a Trapezoidal Area**
In this lesson, we will learn how to calculate the moments of inertia about the \( x \)- and \( y \)-axes for a trapezoidal area using the provided method.
### Diagram Explanation
Below is a diagram of a symmetric trapezoid with the following dimensions labeled:
- The top width of the trapezoid is 16 inches (8 inches from the center on both left and right sides).
- The bottom width of the trapezoid is 26 inches (13 inches from the center on both left and right sides).
- The height of the trapezoid is 11 inches.
- The origin, \( O \), is located at the center of the trapezoid at the intersection of the axes.
\(\begin{array}{c}
\quad \quad \quad \quad y \quad \quad \quad \quad \quad \quad y
\xrightarrow{ x}
\\ 8" \quad \quad \quad \quad \quad \quad 8"
\xrightarrow{\xrightarrow{}}
\\ \quad \quad \quad \quad ε \quad \quad \quad \quad \quad \quad \quad \quad 11"
\\ \quad \quad \quad \quad
\\ \quad \quad 13" \quad \quad \quad \quad 13"
\end{array}\\
### Calculations
The moments of inertia \( I_x \) and \( I_y \) are calculated using the integral calculus methods outlined in this section. For a composite shape like this trapezoid, the moments of inertia can be determined using predefined formulas or by applying the parallel and perpendicular axis theorems.
### Answers Section
Below, you will input the calculated values for the moments of inertia:
- \( I_x \) (moment of inertia about the \( x \)-axis):
\[ \text{I}_x =\_\_\_\_\_\_ \, \text{in}^4 \]
- \( I_y \) (moment of inertia about the \( y \)-axis):
\[ \text{I}_y =\_\_\_\_\_\_ \, \text{in}^4 \]
To find these values, break down the trapezoid into simpler shapes, calculate their individual moments of inertia, and then sum them using the parallel axis theorem](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59a6d07-a993-40f7-b57e-cdb2ad079d78%2F2213ccaf-1992-4fc9-8cc3-5da131fe1138%2F3zv48l5_processed.png&w=3840&q=75)
Transcribed Image Text:**Determining Moments of Inertia for a Trapezoidal Area**
In this lesson, we will learn how to calculate the moments of inertia about the \( x \)- and \( y \)-axes for a trapezoidal area using the provided method.
### Diagram Explanation
Below is a diagram of a symmetric trapezoid with the following dimensions labeled:
- The top width of the trapezoid is 16 inches (8 inches from the center on both left and right sides).
- The bottom width of the trapezoid is 26 inches (13 inches from the center on both left and right sides).
- The height of the trapezoid is 11 inches.
- The origin, \( O \), is located at the center of the trapezoid at the intersection of the axes.
\(\begin{array}{c}
\quad \quad \quad \quad y \quad \quad \quad \quad \quad \quad y
\xrightarrow{ x}
\\ 8" \quad \quad \quad \quad \quad \quad 8"
\xrightarrow{\xrightarrow{}}
\\ \quad \quad \quad \quad ε \quad \quad \quad \quad \quad \quad \quad \quad 11"
\\ \quad \quad \quad \quad
\\ \quad \quad 13" \quad \quad \quad \quad 13"
\end{array}\\
### Calculations
The moments of inertia \( I_x \) and \( I_y \) are calculated using the integral calculus methods outlined in this section. For a composite shape like this trapezoid, the moments of inertia can be determined using predefined formulas or by applying the parallel and perpendicular axis theorems.
### Answers Section
Below, you will input the calculated values for the moments of inertia:
- \( I_x \) (moment of inertia about the \( x \)-axis):
\[ \text{I}_x =\_\_\_\_\_\_ \, \text{in}^4 \]
- \( I_y \) (moment of inertia about the \( y \)-axis):
\[ \text{I}_y =\_\_\_\_\_\_ \, \text{in}^4 \]
To find these values, break down the trapezoid into simpler shapes, calculate their individual moments of inertia, and then sum them using the parallel axis theorem
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