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12 in. r = 4 in.

12 in. r = 4 in.

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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**Locate the Centroid of the Plane Area** **Dimensions and Description:** - The main shape is a vertical rectangle of height 12 inches. - The rectangle is modified by removing a quarter-circle of radius \( r = 4 \) inches from its right side. - The base length of the rectangle is divided into two segments: \( a \) inches (the remaining width after the quarter-circle is removed) and the circular segment. **Given:** - \( a = 7 \) inches - Radius of circle segment \( r = 4 \) inches **Coordinate System:** - The origin of the coordinate system is placed at the bottom-left corner of the rectangle. - The x-axis runs horizontally, and the y-axis runs vertically. **Required:** - To find the centroid coordinates \(\bar{X}\) and \(\bar{Y}\) for the given plane area. **Solution Fields:** - The value of \(\bar{X}\) is _______ in. - The value of \(\bar{Y}\) is _______ in. Complete the calculations to determine the centroid coordinates based on these measurements and descriptions.
Transcribed Image Text:**Locate the Centroid of the Plane Area** **Dimensions and Description:** - The main shape is a vertical rectangle of height 12 inches. - The rectangle is modified by removing a quarter-circle of radius \( r = 4 \) inches from its right side. - The base length of the rectangle is divided into two segments: \( a \) inches (the remaining width after the quarter-circle is removed) and the circular segment. **Given:** - \( a = 7 \) inches - Radius of circle segment \( r = 4 \) inches **Coordinate System:** - The origin of the coordinate system is placed at the bottom-left corner of the rectangle. - The x-axis runs horizontally, and the y-axis runs vertically. **Required:** - To find the centroid coordinates \(\bar{X}\) and \(\bar{Y}\) for the given plane area. **Solution Fields:** - The value of \(\bar{X}\) is _______ in. - The value of \(\bar{Y}\) is _______ in. Complete the calculations to determine the centroid coordinates based on these measurements and descriptions.
Expert Solution
Step 1

The centroid of the rectangle part 1 is 

x1=b2=72=3.5 iny1=d2=122=6 ina1=b×d=12×7=84 in2Civil Engineering homework question answer, step 1, image 1

 

The centroid of the quarter circle part 2 is 

x1=y1=4r3π=4×43π=1.70 ina1=πr24=π(4)24=12.56 in2

 

The centroid of the quarter circle part 3 is 

x1=y1=4r3π=4×43π=1.70 ina1=πr24=π(4)24=12.56 in2

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