Centroid and Centre of Gravity

Definition

The centroid of a plane figure is the geometric center of the shape, where the whole area may be assumed to be concentrated for calculating moments of area. The centre of gravity is the point in a body through which the resultant gravitational force acts, and the entire weight of the body is considered to act at that point.

For uniform, homogeneous bodies in a uniform gravitational field, the centroid and centre of gravity coincide. However, in non-uniform bodies or in varying gravitational fields, they may differ.

Main Content

1. Centroid of Plane Areas

• The centroid is the point at which a plane lamina’s area is considered to be concentrated for geometric analysis.
• It is denoted by coordinates $(\bar{x}, \bar{y})$, which are calculated using first moments of area: $$\bar{x} = \frac{\sum A_i x_i}{\sum A_i}, \qquad \bar{y} = \frac{\sum A_i y_i}{\sum A_i}$$ where $A_i$ are component areas and $x_i, y_i$ are the distances of their centroids from the reference axes.

The centroid depends only on the shape, not on mass or material density, provided the figure is uniformly thick and homogeneous. For simple shapes such as rectangles, triangles, circles, and semicircles, the centroid is found using standard formulas. For example, the centroid of a rectangle lies at the intersection of its diagonals, while the centroid of a triangle lies at the intersection of its medians, one-third of the distance from the base to the opposite vertex.

For composite areas, the shape is divided into simpler geometric parts, and the centroid of the whole is obtained by algebraic addition of areas and their moments. If a portion is removed, its area is taken as negative in the calculation.

2. Centre of Gravity of Bodies

• The centre of gravity is the point through which the resultant weight of a body acts vertically downward.
• It is the point where the whole mass of the body may be assumed to be concentrated for studying gravitational effects.

The centre of gravity is influenced by the mass distribution of the body. In a uniform gravitational field and for bodies of uniform density, the centre of gravity coincides with the centroid. If density varies from one part of the body to another, the centre of gravity shifts toward the region of greater mass. For irregular or non-homogeneous bodies, finding the centre of gravity is essential for ensuring balance and stability.

The concept is important in practical systems such as vehicles, cranes, bridges, and machinery. A low centre of gravity increases stability, while a high centre of gravity may cause tipping or overturning. In everyday life, we observe this when balancing an object on a point or trying to lift a loaded body.

3. Relationship Between Centroid and Centre of Gravity

• The centroid refers to a geometric property, whereas the centre of gravity refers to a physical property related to weight.
• They coincide for homogeneous bodies in uniform gravitational conditions.

This relationship is one of the most important ideas in Unit V. For a thin, uniform lamina placed in a uniform gravitational field, the weight is proportional to the area, so the centroid and centre of gravity are the same point. However, when the density is non-uniform, the two points can be different. This distinction is especially useful in advanced applications such as composite sections, aerospace structures, and irregular bodies.

Understanding this relationship helps in simplifying many engineering problems. In most basic problems involving flat plates of uniform thickness and density, one can use centroid calculations directly to locate the centre of gravity.

Working / Process

1. Identify the body or plane figure

• Determine whether the object is a simple shape or a composite shape.
• Note whether it is uniform in density and thickness, since this decides whether centroid and centre of gravity are the same.

2. Select a suitable reference axis or datum

• Choose axes from which distances of component parts will be measured.
• Break the shape into simple standard figures such as rectangles, triangles, circles, semicircles, or trapeziums.

3. Calculate the centroid or centre of gravity

• For each part, find its area or mass and the location of its centroid or mass center.

• Apply the moment formula: $$\bar{x} = \frac{\sum A_i x_i}{\sum A_i}, \qquad \bar{y} = \frac{\sum A_i y_i}{\sum A_i}$$ for areas, or $$\bar{x} = \frac{\sum m_i x_i}{\sum m_i}, \qquad \bar{y} = \frac{\sum m_i y_i}{\sum m_i}$$ for masses.

• Interpret the result as the balance point of the body or figure.

Advantages / Applications

Structural design and analysis

• Used to find the balance and load-carrying behavior of beams, columns, arches, and composite sections.

Stability of objects and vehicles

• Helps in designing vehicles, aircraft, ships, and machines with proper stability by controlling the centre of gravity.

Engineering and mechanical calculations

• Essential in determining moments, equilibrium, and safe positioning of loads in cranes, bridges, and rotating bodies.

Summary

• The centroid is the geometric center of a plane figure or area.
• The centre of gravity is the point through which the resultant weight of a body acts.
• For uniform bodies in a uniform gravitational field, both points coincide.
• Important terms to remember: centroid, centre of gravity, composite area, first moment of area, equilibrium.