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The object of the figure is comprised of three different materials, and should be divided into three sub- volumes or regions, using materials and material densities (p) as follows:

The object of the figure is comprised of three different materials, and should be divided into three sub- volumes or regions, using materials and material densities (p) as follows:

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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### Calculation of Mass and Centroid for Composite Bodies #### Table of Properties for Composite Bodies Below is a structured table used to calculate the mass and centroid of composite bodies. Each body has been divided into distinct regions, and their properties such as volume, mass, and centroids have been segmented. | Region | Volume (V_i) (mm³) | Mass (m_i) (kg) | \(\overline{x_i}\) (mm) | \(\overline{x_i}m_i\) (mm * kg) | |-----------------------------|-------------------:|----------------:|------------------------:|-------------------------------:| | I Solid Cone | | | | | | II Solid Cylinder | | | | | | III Solid Half-Sphere | | | | | | **Total** | \(\Sigma_i\) | \(\Sigma_i\) | | \(\Sigma_i\) #### Properties of Each Region - **Region I: Solid Cone**: - Material: Plastic - Density, \(\rho = 0.00000103 \text{ kg/mm}³\) - **Region II: Solid Cylinder**: - Material: Steel - Density, \(\rho = 0.00000786 \text{ kg/mm}³\) - **Region III: Solid Half-Sphere**: - Density, \(\rho = 0.00000847 \text{ kg/mm}³\) These regions are combined to form a composite body, and the following calculations are based on their individual properties. #### Calculation of Mass and Centroid - Use the properties given to calculate the mass \(m_i\) for each region using the formula: \(m_i = \rho \cdot V_i\). - Calculate the total mass by summing up the masses of each region. - Determine the centroids \(\overline{x_i}\) for each region. - Compute the weighted centroids \(\overline{x_i}m_i\) for each region. - Find the total weighted centroid by summing up the \(\overline{x_i}m_i\) values. #### Centroid Calculations \[ \overline{X} (\text{mm}) = \frac{\Sigma (\overline{x_i}m_i)}{\
Transcribed Image Text:### Calculation of Mass and Centroid for Composite Bodies #### Table of Properties for Composite Bodies Below is a structured table used to calculate the mass and centroid of composite bodies. Each body has been divided into distinct regions, and their properties such as volume, mass, and centroids have been segmented. | Region | Volume (V_i) (mm³) | Mass (m_i) (kg) | \(\overline{x_i}\) (mm) | \(\overline{x_i}m_i\) (mm * kg) | |-----------------------------|-------------------:|----------------:|------------------------:|-------------------------------:| | I Solid Cone | | | | | | II Solid Cylinder | | | | | | III Solid Half-Sphere | | | | | | **Total** | \(\Sigma_i\) | \(\Sigma_i\) | | \(\Sigma_i\) #### Properties of Each Region - **Region I: Solid Cone**: - Material: Plastic - Density, \(\rho = 0.00000103 \text{ kg/mm}³\) - **Region II: Solid Cylinder**: - Material: Steel - Density, \(\rho = 0.00000786 \text{ kg/mm}³\) - **Region III: Solid Half-Sphere**: - Density, \(\rho = 0.00000847 \text{ kg/mm}³\) These regions are combined to form a composite body, and the following calculations are based on their individual properties. #### Calculation of Mass and Centroid - Use the properties given to calculate the mass \(m_i\) for each region using the formula: \(m_i = \rho \cdot V_i\). - Calculate the total mass by summing up the masses of each region. - Determine the centroids \(\overline{x_i}\) for each region. - Compute the weighted centroids \(\overline{x_i}m_i\) for each region. - Find the total weighted centroid by summing up the \(\overline{x_i}m_i\) values. #### Centroid Calculations \[ \overline{X} (\text{mm}) = \frac{\Sigma (\overline{x_i}m_i)}{\
## Object Description and Instructions for Calculations ### Description The object shown in the figure is comprised of three different materials and will be divided into three sub-volumes or regions, each with distinct material densities (ρ). The breakdown is as follows: - **Region I: Solid Cone** - Material: Plastic - Density: \( \rho = 0.00000103 \, \text{kg} / \text{mm}^3 \) - **Region II: Solid Cylinder** - Material: Steel - Density: \( \rho = 0.00000786 \, \text{kg} / \text{mm}^3 \) - **Region III: Solid Half-Sphere** - Material: Brass - Density: \( \rho = 0.00000847 \, \text{kg} / \text{mm}^3 \) The dimensions of the figure are given in millimeters (mm). ### Tasks and Parts #### Part A: Complete all the entries of the attached table in the specified units. Enter the results in the spaces provided. If the sign (positive or negative) is necessary, ensure it is clearly included. #### Part B: Compute the "x̄" coordinate of the center-of-mass with respect to the coordinate system of the figure. Enter the results in the designated space on the attached sheet below the table. #### Part C: Compute the "ȳ" and "z̄" coordinates of the center-of-mass with respect to the coordinate system of the figure. Enter the results in the designated spaces on the attached sheet. #### Part D: Compute the total Weight, W, in Newtons and enter the result in the space provided on the attached sheet below the table. The acceleration of gravity is given as \( g = 9.81 \, \text{m/sec}^2 \). ### Diagram Explanation The provided diagram is a two-dimensional representation showing the cross-section of the object. It includes several key measurements and dimensions: - The object spans 170 mm along the x-axis. - It is segmented into three parts (indicated by the distinct regions) along the x-axis: - From -80 mm to 0 mm: Region I (Solid Cone) - From 0 mm to 60 mm: Region II (Solid Cylinder) - From 60 mm to 90 mm: Region III (
Transcribed Image Text:## Object Description and Instructions for Calculations ### Description The object shown in the figure is comprised of three different materials and will be divided into three sub-volumes or regions, each with distinct material densities (ρ). The breakdown is as follows: - **Region I: Solid Cone** - Material: Plastic - Density: \( \rho = 0.00000103 \, \text{kg} / \text{mm}^3 \) - **Region II: Solid Cylinder** - Material: Steel - Density: \( \rho = 0.00000786 \, \text{kg} / \text{mm}^3 \) - **Region III: Solid Half-Sphere** - Material: Brass - Density: \( \rho = 0.00000847 \, \text{kg} / \text{mm}^3 \) The dimensions of the figure are given in millimeters (mm). ### Tasks and Parts #### Part A: Complete all the entries of the attached table in the specified units. Enter the results in the spaces provided. If the sign (positive or negative) is necessary, ensure it is clearly included. #### Part B: Compute the "x̄" coordinate of the center-of-mass with respect to the coordinate system of the figure. Enter the results in the designated space on the attached sheet below the table. #### Part C: Compute the "ȳ" and "z̄" coordinates of the center-of-mass with respect to the coordinate system of the figure. Enter the results in the designated spaces on the attached sheet. #### Part D: Compute the total Weight, W, in Newtons and enter the result in the space provided on the attached sheet below the table. The acceleration of gravity is given as \( g = 9.81 \, \text{m/sec}^2 \). ### Diagram Explanation The provided diagram is a two-dimensional representation showing the cross-section of the object. It includes several key measurements and dimensions: - The object spans 170 mm along the x-axis. - It is segmented into three parts (indicated by the distinct regions) along the x-axis: - From -80 mm to 0 mm: Region I (Solid Cone) - From 0 mm to 60 mm: Region II (Solid Cylinder) - From 60 mm to 90 mm: Region III (
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