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For each area shown below, determine: a) location of the centroid, (x, ỹ), with respect to point A, b) the moments of inertia, Ix and Iy, about the centroid axes.

For each area shown below, determine: a) location of the centroid, (x, ỹ), with respect to point A, b) the moments of inertia, Ix and Iy, about the centroid axes.

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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For each area shown below, determine: a) location of the centroid, \((\bar{x}, \bar{y})\), with respect to point \(A\), b) the moments of inertia, \(I_x\) and \(I_y\), about the centroid axes. **Problem 1:** - Shape: U-shaped cross section - Dimensions: - Total height: 60 mm - Bottom width: 50 mm - Side width: 10 mm each - Bottom thickness: 10 mm - Reference point \(A\) is located at the bottom left corner of the shape. - Axes: - \(x\)-axis is horizontal - \(y\)-axis is vertical **Problem 2:** - Shape: T-shaped cross section - Dimensions: - Vertical height: 3 in. for both top and bottom sections - Horizontal width of top section: 4 in. - Vertical section width: 2 in. - Reference point \(A\) is located at the bottom left corner of the vertical section. - Axes: - \(x\)-axis is horizontal - \(y\)-axis is vertical These problems require calculating the centroid location and the moments of inertia for the sections about the specified axes.
Transcribed Image Text:For each area shown below, determine: a) location of the centroid, \((\bar{x}, \bar{y})\), with respect to point \(A\), b) the moments of inertia, \(I_x\) and \(I_y\), about the centroid axes. **Problem 1:** - Shape: U-shaped cross section - Dimensions: - Total height: 60 mm - Bottom width: 50 mm - Side width: 10 mm each - Bottom thickness: 10 mm - Reference point \(A\) is located at the bottom left corner of the shape. - Axes: - \(x\)-axis is horizontal - \(y\)-axis is vertical **Problem 2:** - Shape: T-shaped cross section - Dimensions: - Vertical height: 3 in. for both top and bottom sections - Horizontal width of top section: 4 in. - Vertical section width: 2 in. - Reference point \(A\) is located at the bottom left corner of the vertical section. - Axes: - \(x\)-axis is horizontal - \(y\)-axis is vertical These problems require calculating the centroid location and the moments of inertia for the sections about the specified axes.
### Problem 3 Description: The image shows a composite shape with a yellow rectangular area from which a smaller rectangle has been removed. The larger rectangle has dimensions of 16 inches in height and 12 inches in width. An origin point, labeled as "A," is located at the bottom left corner of the shape. #### Dimensions: - The larger rectangle's dimensions are: - Height: 16 inches - Width: 12 inches - The cut-out rectangle's dimensions are: - Positioned 2 inches from the bottom. - Height: 6 inches - Width: 4 inches - Positioned 6 inches from the left edge and 6 inches from the right edge of the larger rectangle. ### Axes: - The coordinate axes, labeled \( x \) and \( y \), are shown with the origin located at point "A." ### Answers: 1. **Set 1** - \(\bar{x} = 25 \, \text{mm}, \, \bar{y} = 25 \, \text{mm}\) - \( I_x = 5.125 \times 10^5 \, \text{mm}^4 \) - \( I_y = 5.125 \times 10^5 \, \text{mm}^4 \) 2. **Set 2** - \(\bar{x} = 2 \, \text{in.}, \, \bar{y} = 4 \, \text{in.}\) - \( I_x = 88.0 \, \text{in.}^4 \) - \( I_y = 64.0 \, \text{in.}^4 \) 3. **Set 3** - \(\bar{x} = 5.71 \, \text{in.}, \, \bar{y} = 8.43 \, \text{in.}\) - \( I_x = 3777 \, \text{in.}^4 \) - \( I_y = 2162 \, \text{in.}^4 \)
Transcribed Image Text:### Problem 3 Description: The image shows a composite shape with a yellow rectangular area from which a smaller rectangle has been removed. The larger rectangle has dimensions of 16 inches in height and 12 inches in width. An origin point, labeled as "A," is located at the bottom left corner of the shape. #### Dimensions: - The larger rectangle's dimensions are: - Height: 16 inches - Width: 12 inches - The cut-out rectangle's dimensions are: - Positioned 2 inches from the bottom. - Height: 6 inches - Width: 4 inches - Positioned 6 inches from the left edge and 6 inches from the right edge of the larger rectangle. ### Axes: - The coordinate axes, labeled \( x \) and \( y \), are shown with the origin located at point "A." ### Answers: 1. **Set 1** - \(\bar{x} = 25 \, \text{mm}, \, \bar{y} = 25 \, \text{mm}\) - \( I_x = 5.125 \times 10^5 \, \text{mm}^4 \) - \( I_y = 5.125 \times 10^5 \, \text{mm}^4 \) 2. **Set 2** - \(\bar{x} = 2 \, \text{in.}, \, \bar{y} = 4 \, \text{in.}\) - \( I_x = 88.0 \, \text{in.}^4 \) - \( I_y = 64.0 \, \text{in.}^4 \) 3. **Set 3** - \(\bar{x} = 5.71 \, \text{in.}, \, \bar{y} = 8.43 \, \text{in.}\) - \( I_x = 3777 \, \text{in.}^4 \) - \( I_y = 2162 \, \text{in.}^4 \)
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