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Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE) PDF Download

What is the center of gravity?

The center of gravity G is a point that locates the resultant weight of a system of particles.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)


What is the composite body?

A composite body consists of a series of connected simpler-shaped bodies, which may be rectangular, triangular, semicircular, etc. 

What is centroid?

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

The line of action of the resultant force passes through the geometric center or centroid of the volume under the distributed–loading diagram.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Find the centroid of a rectangular area by integration

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Find the centroid of a quarter circle by double integration in rectangular coordinates

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Procedure for locating the center of gravity of a body or the centroid of a composite geometrical object

  • Using a sketch, divide the body or object into a finite number of composite parts that have simpler shapes.
  • If a composite part has a hole, or a geometric region having no material, then consider the composite part without the hole and consider the hole as an additional composite part having negative. 
  • Establish the coordinate axes on the sketch and determine the coordinates Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE) of the center of gravity or centroid of each part.
  • Determine Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)  by applying the center of gravity equations
  • If an object is symmetrical about an axis, the centroid of the object lies on this axis. 

First Moments of Areas and Lines 

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

  • An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’.

 

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)
  • The first moment of an area with respect to a line of symmetry is zero.
  • If an area possesses a line of symmetry, its centroid lies on that axis
  • If an area possesses two lines of symmetry, its centroid lies at their intersection.

 

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

 

  • An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y).  
  • The centroid of the area coincides with the center of symmetry.

 


Center of Mass – locates the point in a system where the resultant mass can be concentrated so that the moment of the concentrated mass with respect to any axis equals the moment of the distributed mass with respect to the same axis.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)


Center of gravity 

The center of gravity (G) is a point that locates the resultant weight of a system of particles.

Particles with weights W1, W2, …, Wn can be replaced by a resultant force of weight W located at the center of gravity G.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

To find the location of the center of gravity G(x,y,z):

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

(We can obtain z by imagining the coordinate system, with the particles fixed in it, as being rotated 90 degrees about the x (or the y) axis).

Knowing W=mg, if we assume that the acceleration due to gravity (g) for every particle is constant (g will be cancelled out)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

A rigid body is composed of an infinite number of particles, hence it is necessary to use integration instead of summation.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

But dm=ρdV, with ρ being the density and dv the volume of each particle. Therefore, the centre of mass has the coordinates of

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid:

The centroid (C) is a point that defines the geometric center of an object.  If the material composing a body is uniform or homogeneous, the density of the material is constant (ρ = constant). Hence, the resulting formulas that define the centroid of a body depend only on the geometry of the body {Volume (V), Area (A), or Length (L)}.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid (volume):

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid (area):

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid (line):

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

The centroids of some shapes may be partially or completely specified by using conditions of symmetry.  In cases where the shape has an axis of symmetry, the centroid of the shape will lie along that axis.

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE)

The document Centroid & Center of Mass of the Composite Bodies | Engineering Mechanics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mechanics.
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FAQs on Centroid & Center of Mass of the Composite Bodies - Engineering Mechanics - Civil Engineering (CE)

1. What is the difference between centroid and center of mass?
Ans. The centroid of a body refers to the geometric center or the average position of all the points in the body. On the other hand, the center of mass refers to the point where the entire mass of the body is considered to be concentrated. While the centroid is a purely geometric property, the center of mass takes into account the distribution of mass within the body.
2. How is the centroid of a composite body calculated?
Ans. To find the centroid of a composite body, the centroid of each individual component needs to be determined first. Then, the weighted average of the respective centroids, based on their masses or areas, is calculated. This gives the overall centroid of the composite body. The formula for calculating the centroid of a composite body depends on the specific shape and arrangement of its components.
3. What is the significance of the centroid in civil engineering?
Ans. The centroid is an important concept in civil engineering as it helps in analyzing the stability and equilibrium of structures. It is used to determine the location of the resultant force acting on a structure, which is crucial for designing beams, columns, and other load-bearing elements. The centroid also plays a role in calculating moments of inertia and determining the behavior of structures under different loading conditions.
4. Can the centroid of a composite body lie outside its boundaries?
Ans. No, the centroid of a composite body will always lie within its boundaries. This is because the centroid is a property that depends on the distribution of mass or area within the body. Since all the components are included in the calculation of the centroid, it cannot be located outside the overall boundaries of the composite body.
5. How is the center of mass of a composite body different from the centroid?
Ans. While the centroid and center of mass are often used interchangeably, they may not always coincide in certain cases. The center of mass takes into account the distribution of mass within the body, whereas the centroid only considers the geometric properties. In a composite body, if there are variations in the mass distribution, the center of mass may not align with the centroid. However, in most cases, the centroid and center of mass are close enough to be treated as the same point.
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