Determine the coordinates of the centroid of the shaded area. 10 40 y 1 Answers: X = y = 15 MI F 10 15 ! mm 30 55 Dimensions in millimeters ! mm 20 -x
Determine the coordinates of the centroid of the shaded area. 10 40 y 1 Answers: X = y = 15 MI F 10 15 ! mm 30 55 Dimensions in millimeters ! mm 20 -x
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![**Title: Determining the Coordinates of the Centroid of a Composite Area**
**Objective:**
Determine the coordinates of the centroid of the shaded area in the given diagram.
**Diagram Description:**
The diagram shows a composite shape that consists of a rectangle, a triangle, and a semicircle with a cutout of a rectangle in the middle. The dimensions are provided in millimeters:
- The overall height is 40 mm and the total length is 55 mm.
- The rectangle has a height of 10 mm and lengths of 15 mm, 10 mm, and 15 mm.
- The semicircle on the right has a diameter of 30 mm, with an inner cutout 10 mm smaller than the outer diameter.
- The cutout rectangle in the middle has dimensions of 20 mm by 30 mm.
**Detailed Explanation of the Diagram:**
- **Triangle Dimensions:**
- Base: 55 mm (overall length)
- Height: 30 mm (height excluding the lower rectangle)
- **Rectangle Cutout Dimensions:**
- Width: 20 mm
- Height: 30 mm
- **Semicircle Dimensions:**
- Outer Radius: 15 mm (half of the total diameter)
- Inner Radius: 10 mm
- **Coordinate Axes:**
- Origin (\(0, 0\)) is positioned at the bottom-left corner of the entire shape.
**Process:**
Calculate the centroids of each basic shape component and then apply the principle of composite areas to find the centroid of the entire composite shape.
**Formulas:**
- **Centroid of a Rectangle:**
\[ \bar{x} = \frac{b}{2}, \quad \bar{y} = \frac{h}{2} \]
- **Centroid of a Triangle:**
\[ \bar{x} = \frac{b}{3}, \quad \bar{y} = \frac{h}{3} \]
- **Centroid of a Semicircle:**
\[ \bar{y} = \frac{4r}{3\pi} \text{ from flat side}, \quad \bar{x} \text{ is at center} \]
**Application:**
1. Divide the composite area into simpler shapes: the left triangle, the right semicircle, and subtract the cutout rectangle.
2. Find the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9eb052b-0c42-425f-80d7-ea5c1f6fa7cb%2Fbf4fa69a-86c4-4778-9a25-d398bbdb74b7%2Fxe6wq8p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determining the Coordinates of the Centroid of a Composite Area**
**Objective:**
Determine the coordinates of the centroid of the shaded area in the given diagram.
**Diagram Description:**
The diagram shows a composite shape that consists of a rectangle, a triangle, and a semicircle with a cutout of a rectangle in the middle. The dimensions are provided in millimeters:
- The overall height is 40 mm and the total length is 55 mm.
- The rectangle has a height of 10 mm and lengths of 15 mm, 10 mm, and 15 mm.
- The semicircle on the right has a diameter of 30 mm, with an inner cutout 10 mm smaller than the outer diameter.
- The cutout rectangle in the middle has dimensions of 20 mm by 30 mm.
**Detailed Explanation of the Diagram:**
- **Triangle Dimensions:**
- Base: 55 mm (overall length)
- Height: 30 mm (height excluding the lower rectangle)
- **Rectangle Cutout Dimensions:**
- Width: 20 mm
- Height: 30 mm
- **Semicircle Dimensions:**
- Outer Radius: 15 mm (half of the total diameter)
- Inner Radius: 10 mm
- **Coordinate Axes:**
- Origin (\(0, 0\)) is positioned at the bottom-left corner of the entire shape.
**Process:**
Calculate the centroids of each basic shape component and then apply the principle of composite areas to find the centroid of the entire composite shape.
**Formulas:**
- **Centroid of a Rectangle:**
\[ \bar{x} = \frac{b}{2}, \quad \bar{y} = \frac{h}{2} \]
- **Centroid of a Triangle:**
\[ \bar{x} = \frac{b}{3}, \quad \bar{y} = \frac{h}{3} \]
- **Centroid of a Semicircle:**
\[ \bar{y} = \frac{4r}{3\pi} \text{ from flat side}, \quad \bar{x} \text{ is at center} \]
**Application:**
1. Divide the composite area into simpler shapes: the left triangle, the right semicircle, and subtract the cutout rectangle.
2. Find the
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