Lecture on Centroids

Introduction

• Centroid is a location in space, similar to balance point.
• It can represent:
  • Center of gravity
  • Center of mass
  • Center of area
  • Center of volume
  • Center of pressure
• Centroid is a coordinate point (x, y, z in 3D).

Notation

• x̄ (x bar), ȳ (y bar): Represents the x and y coordinates of centroid.
• In 3D: z̄ (z bar) is also used.

Geometric Properties

• Reference table in the back of the textbook: Geometric properties of line and area elements.
• Contains formulas/locations of centroids for common shapes (e.g., circles, rectangles, triangles, parabolas).
• This table is provided during tests for reference.

Centroid Equations

• Basic equation for 2D centroids:
  • x̄ = (Σxᵢaᵢ) / (Σaᵢ)
  • ȳ = (Σyᵢaᵢ) / (Σaᵢ)
• xᵢ, yᵢ: Coordinates of elements, aᵢ: Area of elements.
• Σ: Indicates summation over parts of a shape.
• Different substitutions possible for various centers (e.g., volume, mass, pressure).
• For volume: x̄ = (Σxᵢvᵢ) / (Σvᵢ), ȳ = (Σyᵢvᵢ) / (Σvᵢ)
• z̄ equation: z̄ = (Σzᵢaᵢ) / (Σaᵢ)

Explanation via Moment Balance

• Centroid found by balancing moments (forces causing rotation).
• Demonstrated using a book:
  • Forces on either side of the centroid point balance each other.

Practical Example: State of Texas

• Centroid can be found by suspending shape and using gravity (plumb line).
• Demonstrated with a map of Texas, finding centroid by balance.

Derivation and Calculus

• Introduction to calculus in statics:
  • x̄ = ∫x dA / ∫dA
  • ȳ = ∫y dA / ∫dA
• Integration equals summation, for complex shapes not in table.
• Introduction to use of calculus for deriving centroids.

Conclusion

• Centroids help in understanding balance and equilibrium.
• Next lecture will cover using calculus to find centroids for more complex shapes.