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Centroid
The point at which the total area of a plane figure or lamina is assumed to be concentrated is called centroid
Centroid of a Line
Centroid of an Area
Characteristics of Centroid
- The centroid represents the geometric center of a body
- The centroid may be located at a point that does not lie on the line/area.
- The coordinates of centroid is calculated with reference to the chosen axis
- An area can have only one centroid for all positions of the figure.
- In case of symmetric figures, centroid is located along the axes of symmetry
Centroid Of Regular Figures
Centre of Gravity Centre of gravity is the point about which the resultant of the whole weight of the body may be considered to act. It is denoted by G
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A. CENTROID OF COMPOSITE FIGURES
Note : If the area has a hole or cut out portion, the first moment of inertia and area must be subtracted to yield the centroid
B. PAPPUS GULDINUS THEOREMS
Pappus Guldinus Theorems are two theorems describing a simple way to calculate volumes (solids) and surface areas (shells) of revolution.
First Theorem
The surface area A of a surface of revolution generated by rotating a plane curve about an axis external to it and on the same plane is equal to the product of the arc length of the curve and the distance y traveled by its geometric centroid.
Examples
Second Theorem
The second theorem states that the volume V of a solid of revolution generated by rotating a plane area about an external axis is equal to the product of the area A and the distance y traveled by its geometric centroid
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FAQs on Summary: Centroid - Engineering Mechanics - Civil Engineering (CE)
| 1. What is a centroid in mathematics? |  |
Ans. In mathematics, a centroid is the center point of a geometric shape, such as a triangle, where the medians intersect. It is often referred to as the "center of gravity" or the "center of mass" of the shape.
| 2. How is the centroid of a triangle calculated? | |
Ans. To calculate the centroid of a triangle, you can find the average of the coordinates of its three vertices. For example, if the coordinates of the vertices are (x1, y1), (x2, y2), and (x3, y3), the centroid can be calculated as ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3).
| 3. What are the properties of a centroid in a triangle? | |
Ans. The centroid of a triangle has several interesting properties. It divides each median into two segments, with the segment connecting the centroid to the vertex being twice as long as the segment connecting the centroid to the midpoint of the opposite side. Additionally, the centroid is the balancing point of a triangle, meaning that if you place the triangle on a pin at its centroid, it will balance perfectly.
| 4. Can the centroid of a triangle lie outside the triangle? | |
Ans. No, the centroid of a triangle always lies inside the triangle. This is because the centroid is the average of the coordinates of the vertices, so it cannot be outside the triangle formed by those vertices.
| 5. How is the centroid useful in real-life applications? | |
Ans. The centroid concept is widely used in various fields, such as physics, engineering, and computer graphics. It helps determine the center of mass or gravity of an object, which is crucial for balancing and stability. In architecture and urban planning, the centroid is employed to find the center of a building or a city, aiding in efficient design and distribution of resources.
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