Short Answer:

The centroid is the geometric center or the average position of all the points in a given plane figure or body. It is the point through which the total area (for plane figures) or total volume (for solid bodies) may be assumed to act. The centroid represents the center of area or shape, just as the center of gravity represents the center of mass. For uniform density materials, the centroid and the center of gravity coincide. The location of a centroid depends on the shape and dimensions of the body.

Detailed Explanation :

Centroid

In Engineering Mechanics, the centroid is an important concept used to locate the geometric center of an object. It is a point that indicates the mean position of all the points that make up the area or volume of a body. When an area is uniform and its thickness and material are consistent, the centroid represents the balance point of the shape.

For example, if a uniform plate of metal or cardboard is cut into a particular shape and supported exactly at its centroid, it will remain in balance in all directions. In structural and mechanical engineering, the concept of centroid is essential in analyzing beams, trusses, and other structures to determine stresses, moments, and load distributions.

Definition

The centroid is defined as:
“The point at which the entire area or volume of a plane figure or solid body may be assumed to be concentrated.”

Alternatively,
“The centroid is the geometric center or mean position of all the points in a plane or volume.”

For a plane figure, the centroid represents the center of area.
For a solid body, the centroid represents the center of volume.

Mathematically, for a plane area, the coordinates of the centroid are given by:

where,

• and = coordinates of the elemental area ,
• = total area of the figure.

For simple geometric shapes, the centroid can be located using symmetry or standard formulas.

Physical Meaning of Centroid

The centroid can be thought of as the point at which the area of a figure is evenly distributed in all directions. If a shape is made of a uniform sheet of material, and you try to balance it on a pin, the point where it balances perfectly is the centroid.

The concept of the centroid is purely geometric and does not depend on the weight or material of the body (unlike the center of gravity, which depends on mass).

Centroid of Simple Plane Figures

1. Rectangle:
   The centroid lies at the intersection of its diagonals, i.e.,

where = base and = height.

1. Triangle:
   The centroid is located at the intersection of its medians (each median divides the opposite side in a 2:1 ratio).

measured from the base and height.

1. Circle:
   The centroid lies at its geometric center.
2. Semicircle:
   The centroid lies on the axis of symmetry at a distance of from the base.
3. Trapezium:
   The centroid is located by dividing the shape into simpler components and using the formula for the center of gravity of composite areas.

Centroid of Composite Areas

For complex shapes, the centroid can be found by dividing the figure into a number of simple areas such as rectangles, triangles, and circles. Then, the centroid of each simple area is found and combined using the formula:

where,

• = individual area of the i-th shape,
• = coordinates of the centroid of each individual shape.

This process is known as the method of moments or composite area method.

Centroid of Solid Bodies

In three-dimensional bodies, the centroid represents the center of volume. It can be determined using the following relations:

where is the small element of volume.

Examples:

1. Sphere: The centroid lies at its geometric center.
2. Cylinder: The centroid is at the midpoint of its axis.
3. Cone: The centroid lies along the axis at a height of from the base.

Relation Between Centroid and Center of Gravity

The centroid and the center of gravity (C.G.) are closely related but not always the same.

• Centroid depends only on the geometry of the body (area or volume).
• Center of gravity depends on both geometry and mass distribution.

For uniform density materials, both points coincide because the weight is distributed evenly.

Example:
A uniform metal plate and its geometric shape have the same centroid and center of gravity. However, if the plate has varying thickness or density, the center of gravity will not coincide with the centroid.

Importance of Centroid in Engineering

The concept of the centroid is widely used in mechanical and civil engineering for design, analysis, and balance of structures and machines.

1. Structural Analysis:
   • Used to locate the neutral axis in beams and bending calculations.
2. Machine Design:
   • Helps in balancing rotating components and determining points of support.
3. Civil Engineering:
   • Used in determining the center of pressure, load distribution, and stability of structures.
4. Moment Calculations:
   • Required in calculating moments of inertia and product of inertia.
5. Manufacturing Applications:
   • Used in the design of symmetrical components for uniform stress distribution.

Practical Example

Consider a steel plate in the shape of a triangle with a base of 120 mm and a height of 60 mm. The centroid from the base can be calculated using the formula:

Thus, the centroid lies 20 mm above the base along the axis of symmetry.

If this triangular plate were suspended from the centroid, it would remain perfectly balanced.

Graphical Determination of Centroid

For irregular or complex shapes, the centroid can be found graphically using the planimeter or coordinate method. By dividing the area into smaller segments and finding their centroids, an accurate average position can be obtained.

Conclusion

In conclusion, the centroid is the geometric center or average position of an area or volume, representing the balance point of a shape. It is a purely geometric property, independent of weight or material. The centroid plays a vital role in the design and analysis of mechanical and structural systems, helping engineers calculate stresses, moments, and equilibrium conditions. For uniform materials, the centroid coincides with the center of gravity, making it one of the most fundamental concepts in mechanics and engineering design.