Center of gravity and Centroids

The Center of gravity, of any object is the point within that object from which the force of gravity appears to act. An object will remain at rest if it is balanced on any point along a vertical line passing through its center of gravity.

In terms of moments, the center of gravity of any object is the point around which the moments of the gravitational forces completely cancel one another. So Center of gravity is the point at which a object can be suspended Or withstand in perfect equilibrium. 

Centroid of a Section

The Centroid of  a two dimensional surface (such as the cross-sectional area of a structural shape) is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape. 

The planar surface area (or figure) may represent an actual area (like a the cross-section of a beam) or a figurative diagram (like a loading diagram). The centroid of the area is to be required often in the analysis structures. 

Difference between centroid and center of gravity: 

1. The term Center of gravity applies to the bodies with mass and weight, while the term centroid applies to plan areas. 

2. Center of gravity of a body is the point through which the resultant gravitational force (weight) of the body acts for any orientation of the body while centroid is the point in a plane area such that the moment of the area, about any axis, through that point is zero. 

Determination of Centroid

Take any arbitrary cross section of an area of A The centroid of this  plane area can be found by subdividing the area into differential elements dA as shown in figure.

Computing the 'moment ' of this element area about each of the coordinate axes (X and Y).

The first moment of area of the element are

qy = dAi.xi

qx = dAi.yi

The total first moment of area of the section,

The centroid is the location where concentrating the total area A  generates the same moments as the distributed area.

Qy = A.xc and Qx = A.yc 

 Total area A = ∑dA

Hence, the centroid (xc, yc) is given by 

 

Note: 

1. Symmetry can be very useful to help determine the location of the centroid of an area. If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry. 

2. If a body (or area or section) has two (or more) lines of symmetry, the centroid must lie somewhere along each of the lines. Thus, the centroid is at the point where the lines intersect. 

3. The centroid of a section is not always within the area or material of the section. Hollow pipes, L shaped and some irregular shaped sections all have their centroid located outside of the material of the section.

4. In composite sections we break up the composite shape into simple standard shapes with known areas 

(A = ∑dA' = A1 + A2 + ... + An) and known centroid locations (y'). 

The centroid will be calculated by the following equations. 

Centroids of Common Shapes 

1.Rectangular Section of bxh

The centroid of rectangular section 

 coordinates are

xc = b/2

yc = h/2

2. Triangular section bxh

 The centroid of triangular section 

 coordinates are

xc = b/3

yc = h/3

3. Semi circular section of radius r

The centroid of semi circular section 

coordinate is located at distance

 yc = 4r/3p

4. Quarter circular section of radius r

The Quarter circular section 

 centroid  coordinate are located at distance

 xc = 4r/3p

 yc = 4r/3p