Short Answer:

The centroid is the geometric center of a body or figure, representing the mean position of all its points in a given shape or area. It depends only on the geometry of the body.
The center of gravity (C.G.), on the other hand, is the point through which the total weight of a body acts, considering the effect of gravity. It depends on both the geometry and mass distribution of the body. For uniform bodies made of homogeneous material, the centroid and center of gravity coincide, but they differ for non-uniform materials.

Detailed Explanation :

Difference between Centroid and Center of Gravity

In Engineering Mechanics, both centroid and center of gravity are fundamental concepts used to describe the position of a point that represents the body’s geometric or weight center. Though they seem similar, they are conceptually different in their definitions, meaning, and application.

The centroid is a purely geometric property that applies to plane figures or volumes irrespective of mass, while the center of gravity is a physical property that depends on the distribution of weight and the effect of gravitational force. Both are used extensively in mechanical, civil, and structural engineering to ensure stability, balance, and accurate design of mechanical and structural systems.

Definition of Centroid

The centroid is the point that represents the geometric center of a figure or body. It is the mean position of all the points making up the area or volume. The centroid is independent of the material or its weight; it is determined solely by the geometry of the object.

Mathematically, for a plane figure,

where and represent the coordinates of the centroid, and represents an infinitesimal area element.

For a uniform plate or object, the centroid lies at its geometric center — for example, the intersection of the diagonals of a rectangle or the point where the medians of a triangle meet.

Definition of Center of Gravity

The center of gravity (C.G.) is the point through which the total weight of a body acts, irrespective of its orientation. It is the point where the resultant of all gravitational forces on the body’s particles passes. The C.G. depends on both the shape and the mass distribution of the body.

Mathematically, for a system of particles,

where is the weight of each particle, and are their coordinates.

If a body is suspended at its C.G., it will remain in stable equilibrium because the gravitational force acts vertically downward through this point.

Basic Difference

The key difference between centroid and center of gravity lies in their dependency on physical properties:

• The centroid depends on geometry only.
• The center of gravity depends on both geometry and mass distribution.

Let us now explain their differences in detail under various aspects.

1. Nature and Meaning

• The centroid is the geometric center of an area or volume, representing the mean position of its points.
• The center of gravity is the weight center of a body, representing the point through which the resultant gravitational force acts.

Example: The centroid of a uniform circular plate lies at its center, and its center of gravity also lies at the same point. However, if the plate has varying thickness, the centroid remains at the geometric center, but the center of gravity shifts toward the thicker part.

2. Dependency

• Centroid: Depends only on the shape and dimensions of the object.
• Center of Gravity: Depends on the weight distribution and material density of the body.

For homogeneous materials (same density throughout), both coincide. For heterogeneous materials (varying density), they differ.

3. Type of Bodies

• The centroid is mainly defined for plane figures (like rectangles, triangles, and circles) and volumes (like cylinders or cones).
• The center of gravity applies to bodies with weight or mass, whether two-dimensional or three-dimensional.

Example:
A flat metal plate has a centroid determined by its geometry, while its center of gravity depends on its weight distribution.

4. Effect of Gravity

• The centroid does not depend on gravity or external forces.
• The center of gravity always depends on the gravitational field and changes direction if gravity changes.

Example: In zero gravity (space), a body still has a centroid but no center of gravity.

5. Location

• The centroid is always located within the geometric boundaries of the object (for simple shapes).
• The center of gravity may lie inside or outside the object, depending on the shape and mass distribution.

Example: The centroid of a semicircular plate lies within the plate, but the C.G. of a ring lies at its center, which is outside the material boundary.

6. Coincidence Condition

The centroid and center of gravity coincide only when the body is of uniform density and the gravitational field is uniform.

For example:

• A uniform rectangular plate → C.G. and centroid coincide.
• A non-uniform plate with different materials → C.G. shifts toward the heavier material.

Mathematical Relationship

If the material is uniform, the total weight is proportional to its volume, so:

where is density and is acceleration due to gravity.

Thus, the formulas for centroid and C.G. become equivalent:

Therefore, for uniform bodies, centroid = center of gravity.

Practical Examples

1. Uniform Beam:
   The centroid and center of gravity lie at the midpoint of the beam.
2. Irregular Plate:
   The centroid is determined geometrically, but if one side is heavier, the center of gravity lies closer to that side.
3. Hollow Sphere or Ring:
   The C.G. is at the center, even though the material is distributed around it, lying outside the material body.
4. Composite Bodies:
   In structures made of different materials (steel and concrete), the C.G. is calculated by taking into account the densities of each material.

Importance in Engineering

1. Structural Design:
   The centroid helps determine bending stresses and deflection in beams.
   The C.G. helps ensure stability of structures.
2. Vehicle Design:
   The C.G. must be kept low for better stability and safety.
3. Aerospace and Ship Design:
   The correct location of C.G. ensures proper balance and control.
4. Machine Design:
   In rotating machines, balancing depends on accurately locating the C.G.

Summary of Key Differences

Aspect	Centroid	Center of Gravity (C.G.)
Definition	Geometric center of shape	Point where total weight acts
Depends on	Shape and geometry	Shape and mass distribution
Gravity	Independent of gravity	Depends on gravity
Applicable to	Plane figures or solids	Bodies with mass and weight
Coincides when	Material is uniform	Material is uniform
Example	Rectangle → midpoint	Triangle plate → weight center

Conclusion

In conclusion, both centroid and center of gravity describe a body’s central point but differ in meaning. The centroid is the geometric center of an area or volume, while the center of gravity is the point where the total weight acts. They coincide only when the material is uniform and gravity is constant. The distinction between these two concepts is fundamental in engineering design and analysis, as it affects balance, stability, and load distribution in machines and structures.