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Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

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$### Centroidal Moment of Inertia (Iz0 and Iy0) Calculation for Built-Up Section #### Objective Given the detailed structural diagram below, our objective is to determine the centroidal moment of inertia (Iz0 and Iy0) of the built-up section. #### Diagram Description - The diagram shows a composite section composed of three distinct geometric shapes forming a symmetrical profile about the y-axis. #### Notations and Dimensions - The intersecting vertical and horizontal lines represent the y and x axes, respectively, indicating the centroidal axis of the section. - The bottom dimension, horizontal to the x-axis, is specified as 12" - 20.7/16". - The left and right vertical components have dimensions of 6" x 3/4". - The middle vertical component, positioned centrally along the y-axis, has dimensions of 5" x 3 x 1/2". #### Structural Components **1. Left Vertical Component** - Dimensions: 6 inches in height, 3/4 inches in thickness. **2. Central Vertical Component** - Positioned along the central y-axis. - Dimensions: 5 inches in height, 3 inches in width, with a thickness of 1/2 inches. **3. Right Vertical Component** - Similar to the left vertical component. - Dimensions: 6 inches in height, 3/4 inches in thickness. #### Calculations Needed To determine the centroidal moments of inertia $Iz_0$ and $Iy_0$, follow these steps: 1. **Calculate Individual Moments of Inertia:** - For each component (left, central, right), calculate the individual moments of inertia about their centroidal axes. 2. **Parallel Axis Theorem:** - Apply the parallel axis theorem to shift these moments of inertia to the centroidal axes (x and y) of the entire built-up section. 3. **Summation of Moments:** - Sum the individual moments of inertia to find the total $Iz_0$ and $Iy_0$. #### Conclusion The precise calculation of $Iz_0$ and $Iy_0$ requires detailed formula applications and parameter assessments. The provided dimensions and structural arrangement contribute to an accurate determination of these critical structural properties, ensuring the stability and performance of the built-up section in practical applications. --- This transcribed section provides clarity for students and professionals learning about centroidal moments of inertia$

Transcribed Image Text:### Centroidal Moment of Inertia (Iz0 and Iy0) Calculation for Built-Up Section #### Objective Given the detailed structural diagram below, our objective is to determine the centroidal moment of inertia (Iz0 and Iy0) of the built-up section. #### Diagram Description - The diagram shows a composite section composed of three distinct geometric shapes forming a symmetrical profile about the y-axis. #### Notations and Dimensions - The intersecting vertical and horizontal lines represent the y and x axes, respectively, indicating the centroidal axis of the section. - The bottom dimension, horizontal to the x-axis, is specified as 12" - 20.7/16". - The left and right vertical components have dimensions of 6" x 3/4". - The middle vertical component, positioned centrally along the y-axis, has dimensions of 5" x 3 x 1/2". #### Structural Components **1. Left Vertical Component** - Dimensions: 6 inches in height, 3/4 inches in thickness. **2. Central Vertical Component** - Positioned along the central y-axis. - Dimensions: 5 inches in height, 3 inches in width, with a thickness of 1/2 inches. **3. Right Vertical Component** - Similar to the left vertical component. - Dimensions: 6 inches in height, 3/4 inches in thickness. #### Calculations Needed To determine the centroidal moments of inertia $Iz_0$ and $Iy_0$, follow these steps: 1. **Calculate Individual Moments of Inertia:** - For each component (left, central, right), calculate the individual moments of inertia about their centroidal axes. 2. **Parallel Axis Theorem:** - Apply the parallel axis theorem to shift these moments of inertia to the centroidal axes (x and y) of the entire built-up section. 3. **Summation of Moments:** - Sum the individual moments of inertia to find the total $Iz_0$ and $Iy_0$. #### Conclusion The precise calculation of $Iz_0$ and $Iy_0$ requires detailed formula applications and parameter assessments. The provided dimensions and structural arrangement contribute to an accurate determination of these critical structural properties, ensuring the stability and performance of the built-up section in practical applications. --- This transcribed section provides clarity for students and professionals learning about centroidal moments of inertia

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om the given figure shown below, determine the centroidal moment of Inertia (Ixo and lyo) of the built-up section. 6x3/4 | 5x3x1/2" "12"-20.716 -X