Lecture Notes: Determining the Centroid of an Area

Introduction to Centroid

• Centroid is the geometric center of an area.
• Equivalent to the center of gravity and center of mass if the density is uniform.
• Previous knowledge includes determining the center of gravity for a rigid body.

Key Concepts

• Center of Gravity: Location where weight equals mass times gravitational acceleration (9.81 m/s² near Earth).
• Center of Mass: Coordinates can be determined using relevant equations if density is uniform.
• Centroid of Volume: If volume has uniform density, center of gravity, mass, and centroid are the same.

Reducing Dimensions

• If volume has uniform thickness, simplify problem from 3D to 2D to find centroid of an area.
• Further reduction possible by considering constant width, leading to centroid of a line.

Symmetry Consideration

• Centroid lies on any axis of symmetry (e.g., cube, circle, rectangle, straight line).
• Use symmetry to find centroid of complex or unsymmetrical shapes.

Example: Finding Centroid of a Right Triangle

• Example of a right triangle with base 'B' and height 'H'.
• Two approaches demonstrated:

First Approach: Using Double Integration

1. Place triangle in an XY coordinate system.
2. Define differential element with sides DX and Dy.
3. Line equation: y = (H/B) * X.
4. Integrate to find ( \bar{x} ) and ( \bar{y} ):
   • ( \bar{x} ) calculated to be ( \frac{2}{3}B ).
   • ( \bar{y} ) calculated to be ( \frac{1}{3}H ).
5. Denominator in formulas is the total area of the triangle.*

Second Approach: Using Single Variable Integration

1. Use a vertical strip of width DX as differential element.
2. The height of the strip is determined by the line equation.
3. Replace X and Y with coordinates of the centroid of the element (rectangle).
4. Result is the same as the first approach but using simpler calculations.

Conclusion

• Centroid location on the triangle is at 1/3 location from the base and height.
• Choosing the differential element wisely simplifies calculations.
• Centroid locations for common shapes are documented and can be found online or in textbooks.

Additional Resources

• Screenshot of a Wikipedia page on centroids. Useful for finding centroid locations of various shapes.