**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

Dimensions are given for the figures embedded in diagram .

Answered: Determine the coordinates of the…

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

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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Concept explainers

Length and Length related Variables

Length can de defined as the measurement or extent of physical quantities or objects from one terminal to another terminal. Stone age humans used finger length, step length, and so on to measure the size or displacement of an object. Nowadays, depending on how precise the measurement requires to be and the object’s size is measured, we use measuring devices for measurements like rulers and steel tape. Most people have used a ruler or tape measure to measure a particular distance or the size of an object. For more precise measurements of tiny objects, the development of measurement tools such as the Vernier caliper, which permits us to measure dimensions within 1/1000 of an inch. In the last few decades, electronic distance measuring instruments (EDMI) have been implemented that permit us to measure distances from a few feet to many miles with reasonable accuracy.

Question

**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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