Introduction to Center of Gravity, Center of Mass, and Centroid

The concept of center of gravity (COG), center of mass (COM), and centroid are fundamental in engineering mechanics and play a crucial role in understanding the stability, equilibrium and motion of rigid bodies.

Center of Gravity (COG)

• An object is made up of countless tiny particles, each incredibly small. These particles experience the force of gravity, which pulls them downward vertically, as illustrated in Figure.
• COG is an imaginary point where the weight of the body can be assumed to be concentrated. In simple words, "it is that point within or outside a body through which the resultant of the distributed parallel gravity force passes, no matter how the object is positioned in space . "
• The symbol 'G' is commonly used to represent the center of gravity for three-dimensional rigid objects.

Location the Center of Gravity

• Imagine a flat plate with a consistent thickness (t), positioned on the xy plane as shown in the figure below. This plate can be broken down into small components, each carrying a weight like w ₁ , w ₂ , and so on. These components are positioned at specific coordinates from origin such as (x ₁ , y ₁ ), (x ₂ , y ₂ ), and so forth.
• To figure out the location of the centre of gravity on any object, we use a rule called the principle of moments (Varignon’s theorem).

Moment due to resultant weight about Y axis = net moment due to weight components about Y axis

• The coordinates of the center of gravity relative to the origin may be expressed as;

Mathematically, we can also represent the above expressions in the following manner:;

Center of Mass (COM)

• “The center of mass (COM) of an object is the point at which the entire mass of the object is considered to be concentrated.”
• The center of mass is crucial for understanding acceleration's impact on a body. The resulting force aligns with acceleration and acts through the center of mass.
• By using principle of moments, the coordinates of the center of mass relative to the origin may be expressed as; Mathematically, we can also represent the above expressions in the following manner:
• Centre of mass lies at centre of gravity in a uniform gravitational field.

Centroid

• Centroid is a concept often encountered in engineering design and analysis, particularly in the study of structures and materials. It represents the central point of a 2D, or 3D shape, influenced by the location and size of each tiny area or volume part.
• “A centroid is a point either inside or outside a shape where the total area of the shape can be considered to be concentrated.”
• The coordinates of the centroid of a plane area relative to the origin may be expressed as;

Mathematically, we can also represent the above expressions in the following manner:

Note: When finding the centroid of a line or a volume, we use the same formula as before, but instead of A, we use L for lines or V for volumes in the expression mentioned above.

• If the density of the body is uniform throughout, then the centroid coincides with the center of mass.
• If an area or volume exhibits symmetry about any axis, the centroid will always lie on that axis of symmetry.

Centroid of Some Common Geometrical Shapes

Rectangle	Right – Angled Triangle
Isosceles/Equilateral Triangle	Circle
Semi Circle	Quarter Circle
Sector of a Circle	Parabolic Spandrel

Centroid Calculation of Composite Area

• Step 1: Divide the composite area into simpler known geometric areas.
• Step 2: Determine the area and the centroid coordinates for each basic geometric area from the origin.
• Step 3: Substitute the values into the equation and compute the centroid coordinates of the composite area.