Centroids, Centre of Gravity, and Moment of Inertia Study Notes

Fundamentals of Centroids and Center of Gravity (COG)

Centroid (Geometric Center)

  • Defined as the point at which the whole area of a plane figure (e.g., rectangle, circle, triangle) acts.
  • Applicable only to 2D figures with area but no volume.

Center of Gravity (COG)

  • The point at which the whole mass or weight of a body is assumed to be concentrated.
  • For plane figures where weight is negligible or proportional to area, the centroid and COG coincide.

Key Differences

  • Centroid: Relates to the concentration of area.
  • COG: Relates to the concentration of weight/mass.

Locating the Centroid

Centroids of Standard Plane Figures

  • Rectangle (b×hb \times h): x=b2x = \frac{b}{2}, y=h2y = \frac{h}{2}; Area=bhArea = bh
  • Triangle: x=b3x = \frac{b}{3}, y=h3y = \frac{h}{3}; Area=bh2Area = \frac{bh}{2}
  • Circle: x=rx = r, y=ry = r; Area=πr2Area = \pi r^2
  • Semicircle: y=4r3πy = \frac{4r}{3\pi} (≈0.424r\approx 0.424r); Area=πr22Area = \frac{\pi r^2}{2}
  • Trapezium: y=h3×(2a+ba+b)y = \frac{h}{3} \times \left(\frac{2a + b}{a + b}\right); Area=(a+b2)hArea = \left(\frac{a + b}{2}\right)h

Centroids of Regular Solids (C.G.)

  • Cylinder: Centroid is at h2\frac{h}{2}.
  • Hemisphere: Centroid is at 3r8\frac{3r}{8}.
  • Right Circular Cone: Centroid is at h4\frac{h}{4}.

Centroid location of Composite Figures To find the coordinates (xc,yc)(x_c, y_c) of a figure made of multiple parts:

  1. Place the figure in the first quadrant (lowest point on x-axis, left edge on y-axis).
  2. Split the composite figure into simple plane figures.
  3. Apply the formulas: xc=A1x1+A2x2+⋯+AnxnA1+A2+⋯+Anx_c = \frac{A_1x_1 + A_2x_2 + \dots + A_nx_n}{A_1 + A_2 + \dots + A_n}

yc=A1y1+A2y2+⋯+AnynA1+A2+⋯+Any_c = \frac{A_1y_1 + A_2y_2 + \dots + A_ny_n}{A_1 + A_2 + \dots + A_n}

Symmetry in Centroids

  • One Axis of Symmetry: The centroid always lies on that axis.
  • Two Axes of Symmetry: The centroid lies at the intersection of the two axes.
  • No Axis of Symmetry: The centroid is located by inspection or calculation; such figures may have a "center of symmetry" where any line through it contacts the area symmetrically.

Moment of Inertia (MOI)

Definitions

  • 1st Moment of Area: Area×perpendicular distanceArea \times \text{perpendicular distance}. Units are (length)3(\text{length})^3.
  • 2nd Moment of Area (MOI): Area×(perpendicular distance)2Area \times (\text{perpendicular distance})^2. Units are m4m^4 or mm4mm^4. It is always a positive quantity.

Physical Significance

  • Mass MOI: Measures a body's resistance to rotation.
  • Area MOI: Measures a member's resistance to bending.
  • Polar MOI: Measures a member's resistance to torsion.

Standard MOI Formulas (Centroidal Axis)

  • Rectangle: Ixx=bd312I_{xx} = \frac{bd^3}{12}, Iyy=db312I_{yy} = \frac{db^3}{12}
  • Triangle: Ixx=bh336I_{xx} = \frac{bh^3}{36}
  • Circle: Ixx=Iyy=πd464=πr44I_{xx} = I_{yy} = \frac{\pi d^4}{64} = \frac{\pi r^4}{4}
  • Hollow Rectangle: Ixx=bd3−b1d1312I_{xx} = \frac{bd^3 - b_1d_1^3}{12}
  • Hollow Circle: Ixx=π(D4−d4)64I_{xx} = \frac{\pi(D^4 - d^4)}{64}

Parallel Axis Theorem

The moment of area of an object about any axis parallel to its centroidal axis is the sum of its centroidal MOI and the product of its area with the square of the distance between the two axes.

Mathematical Formulas:

  • Ixx=Icx+A(y)2I_{xx} = I_{cx} + A(y)^2
  • Iyy=Icy+A(x)2I_{yy} = I_{cy} + A(x)^2

Note: In these formulas, one of the two parallel axes must be the centroidal axis. The MOI about the centroidal axis is always the minimum MOI.