Centroids and Centers of Gravity

Learning Objectives

  • Compute centroids from component first moments.
  • Treat holes and cutouts as negative areas.
  • Apply polygon centroid equations to editable vertices.
  • Distinguish geometric centroid from density-weighted center of gravity.
  • Determine centroids of straight and curved line elements.
Composite-area centroid with an openingComposite centroids use first moments of signed areas. Material contributes positive area, while an opening is treated as negative area about the same reference axes.C( x̄, ȳ )negative areaΣAᵢxᵢ / ΣAᵢ · holes use negative area
Composite-area centroid with an opening
Composite centroids use first moments of signed areas. Material contributes positive area, while an opening is treated as negative area about the same reference axes.

Centroid

The centroid is the geometric point at which the first moments of an area, line, or volume can be represented as concentrated.

Composite-Area Centroid

Centroid coordinates obtained from signed component areas and their centroid locations.

xˉ=∑Aixi∑Aiyˉ=∑Aiyi∑Ai\bar{x}=\frac{\sum A_ix_i}{\sum A_i} \qquad \bar{y}=\frac{\sum A_iy_i}{\sum A_i}

Variables

SymbolDescriptionUnit
AiA_iSigned area of component i; negative for a holemm²
xix_iComponent-centroid x-coordinatemm
yiy_iComponent-centroid y-coordinatemm
xˉ\bar{x}Composite centroid x-coordinatemm
yˉ\bar{y}Composite centroid y-coordinatemm

Invalid Signed Area

If the signed areas sum to zero, the composite-area centroid formula is undefined. This is an invalid area model rather than a centroid at infinity.

Worked Example Summary

For two rectangles with A1=6000 mm2A_1=6000\ \text{mm}^2 at y1=150 mmy_1=150\ \text{mm} and A2=4000 mm2A_2=4000\ \text{mm}^2 at y2=50 mmy_2=50\ \text{mm}, the centroid is yˉ=(6000⋅150+4000⋅50)/(10000)=110 mm\bar{y}=(6000\cdot150+4000\cdot50)/(10000)=110\ \text{mm}.

Simulation 1 Instructions

Resize the flange and web of the composite section and inspect the component table for AiA_i, xix_i, yiy_i, AixiA_ix_i, and AiyiA_iy_i.

Centroids and Centers of Gravity Suite

Concept and model scope

Combine non-overlapping flange and web areas and inspect A, Ax, and Ay first-moment contributions.

Simulation purpose: First-moment, signed-area, density, polygon, and wire-centroid models tied directly to the displayed geometry.

Model scope: Positive composite pieces are non-overlapping; the composite web width is explicitly parameterized as 0.30B and shown on the drawing. Openings are subtracted only after full-containment validation. Polygon centroid uses a simple-boundary shoelace model, density uses mass weighting, and wire centroid uses actual segment/arc length.

Verification: Check minimum/default/maximum geometry. Reverse polygon vertex order analytically to verify the physical centroid is unchanged; test zero-area/self-intersecting polygons; move the opening until edge clearance becomes negative; compare geometric centroid with density-weighted CG; verify the semicircular arc centroid at 2r/π from its diameter.

Controls
Primary width

Primary width

Overall flange width of the non-overlapping composite section. The flange and web share the same horizontal centroid line.

160 mm
Web height

Web height

Clear web height below the flange. The web is a separate positive area so flange and web are not double-counted.

120 mm
Flange thickness

Flange thickness

Thickness of the top flange measured normal to its width. It changes flange area and its centroid without overlapping the web area.

40 mm
B = 160 mmweb b = 48 mmweb h = 120 mmt = 40 mmC
valid geometry
Composite area
12160.00 mm²
x centroid
80.000 mm
y centroid
102.105 mm
First-moment contributions
Flange
Aᵢ
6400.00
xᵢ
80.00
yᵢ
140.00
Aᵢxᵢ
512000.00
Aᵢyᵢ
896000.00
Web
Aᵢ
5760.00
xᵢ
80.00
yᵢ
60.00
Aᵢxᵢ
460800.00
Aᵢyᵢ
345600.00
xˉ=∑mixi∑mi,yˉ=∑miyi∑mi\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

Equation concept

For uniform areas use mᵢ=Aᵢ, with Aᵢ<0 only for fully contained openings. For nonuniform materials use mᵢ=ρᵢAᵢ. For wires use mᵢ=Lᵢ; a semicircular wire has arc length πr and centroid 2r/π from its diameter.

Simulation 1 Concept Question

Which component moves the centroid more: a large area close to the origin or a smaller area far from it?

Simulation 2 Instructions

Move and resize the circular cutout. Confirm that its negative area shifts the centroid away from the opening.

Centroids and Centers of Gravity Suite

Concept and model scope

Treat only a fully contained circular opening as negative area and reject partial/out-of-bounds holes.

Simulation purpose: First-moment, signed-area, density, polygon, and wire-centroid models tied directly to the displayed geometry.

Model scope: Positive composite pieces are non-overlapping; the composite web width is explicitly parameterized as 0.30B and shown on the drawing. Openings are subtracted only after full-containment validation. Polygon centroid uses a simple-boundary shoelace model, density uses mass weighting, and wire centroid uses actual segment/arc length.

Verification: Check minimum/default/maximum geometry. Reverse polygon vertex order analytically to verify the physical centroid is unchanged; test zero-area/self-intersecting polygons; move the opening until edge clearance becomes negative; compare geometric centroid with density-weighted CG; verify the semicircular arc centroid at 2r/π from its diameter.

Controls
Primary width

Primary width

Physical plate width used both in the area calculation and the rendered containment boundary.

160 mm
Primary height

Primary height

Physical plate height used for area, opening containment, dimensions, and centroid location.

120 mm
Opening diameter

Opening diameter

Physical diameter of the circular opening. Its full area is subtracted only while the complete circle remains inside the plate.

40 mm
Opening offset from plate center

Opening offset from plate center

Signed horizontal eccentricity of the opening center from the plate center; positive values move the opening right and shift the net centroid away from it.

25 mm
r = 20 mm160 mme = 25 mm120 mmC
valid geometry
Net area
17943.36 mm²
x centroid
78.249 mm
y centroid
60.000 mm
Minimum edge clearance
35.00 mm
First-moment contributions
Plate
Aᵢ
19200.00
xᵢ
80.00
yᵢ
60.00
Aᵢxᵢ
1536000.00
Aᵢyᵢ
1152000.00
Circular opening
Aᵢ
-1256.64
xᵢ
105.00
yᵢ
60.00
Aᵢxᵢ
-131946.89
Aᵢyᵢ
-75398.22
xˉ=∑mixi∑mi,yˉ=∑miyi∑mi\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

Equation concept

For uniform areas use mᵢ=Aᵢ, with Aᵢ<0 only for fully contained openings. For nonuniform materials use mᵢ=ρᵢAᵢ. For wires use mᵢ=Lᵢ; a semicircular wire has arc length πr and centroid 2r/π from its diameter.

Simulation 2 Concept Question

Why must both the area and its first-moment contribution be negative for a hole?

Polygon Centroid by the Shoelace Method

Centroid of a non-self-intersecting polygon with ordered vertices.

A=12∑i(xiyi+1−xi+1yi)A=\frac{1}{2}\sum_i(x_iy_{i+1}-x_{i+1}y_i)xˉ=16A∑i(xi+xi+1)(xiyi+1−xi+1yi)\bar{x}=\frac{1}{6A}\sum_i(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i)yˉ=16A∑i(yi+yi+1)(xiyi+1−xi+1yi)\bar{y}=\frac{1}{6A}\sum_i(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)

Variables

SymbolDescriptionUnit
AASigned polygon areamm²
xi,yix_i,y_iOrdered polygon-vertex coordinatesmm

Simulation 3 Instructions

Edit the polygon dimensions and vertex offset. Degenerate geometry is reported as invalid instead of producing a misleading centroid.

Centroids and Centers of Gravity Suite

Concept and model scope

Apply the shoelace equations to a simple polygon; self-intersecting and zero-area states are rejected.

Simulation purpose: First-moment, signed-area, density, polygon, and wire-centroid models tied directly to the displayed geometry.

Model scope: Positive composite pieces are non-overlapping; the composite web width is explicitly parameterized as 0.30B and shown on the drawing. Openings are subtracted only after full-containment validation. Polygon centroid uses a simple-boundary shoelace model, density uses mass weighting, and wire centroid uses actual segment/arc length.

Verification: Check minimum/default/maximum geometry. Reverse polygon vertex order analytically to verify the physical centroid is unchanged; test zero-area/self-intersecting polygons; move the opening until edge clearance becomes negative; compare geometric centroid with density-weighted CG; verify the semicircular arc centroid at 2r/π from its diameter.

Controls
Primary width

Primary width

Reference base width used to position the editable polygon vertices and shown dimensions.

160 mm
Primary height

Primary height

Reference polygon height used by the editable upper vertices and the shoelace centroid model.

120 mm
Upper-left vertex x

Upper-left vertex x

Horizontal coordinate of the polygon upper-left vertex in the displayed model coordinate system. Invalid self-intersecting or degenerate boundaries are rejected.

40 mm
Upper-right inset

Upper-right inset

Horizontal inset of the upper-right polygon vertex from the primary width. It changes the ordered boundary used by the shoelace area and centroid equations.

25 mm
P1P2P3P4base = 160 mmh = 120 mmC
valid geometry
Polygon area
15300.00 mm²
x centroid
83.431 mm
y centroid
54.902 mm
Vertex orientation
counterclockwise
xˉ=∑mixi∑mi,yˉ=∑miyi∑mi\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

Equation concept

For uniform areas use mᵢ=Aᵢ, with Aᵢ<0 only for fully contained openings. For nonuniform materials use mᵢ=ρᵢAᵢ. For wires use mᵢ=Lᵢ; a semicircular wire has arc length πr and centroid 2r/π from its diameter.

Simulation 3 Concept Question

Why does reversing the vertex order change the signed area but not the physical centroid?

Density-Weighted Center of Gravity

Center of gravity for components of different uniform densities.

xˉG=∑ρiAixi∑ρiAiyˉG=∑ρiAiyi∑ρiAi\bar{x}_G=\frac{\sum \rho_iA_ix_i}{\sum \rho_iA_i} \qquad \bar{y}_G=\frac{\sum \rho_iA_iy_i}{\sum \rho_iA_i}

Variables

SymbolDescriptionUnit
ρi\rho_iMaterial density or relative weight densitykg/m³ or ratio

Simulation 4 Instructions

Keep the geometry symmetric while changing the density ratio. Observe the center of gravity move even though the geometric centroid does not.

Centroids and Centers of Gravity Suite

Concept and model scope

Compare geometric centroid with density-weighted center of gravity for equal geometric halves.

Simulation purpose: First-moment, signed-area, density, polygon, and wire-centroid models tied directly to the displayed geometry.

Model scope: Positive composite pieces are non-overlapping; the composite web width is explicitly parameterized as 0.30B and shown on the drawing. Openings are subtracted only after full-containment validation. Polygon centroid uses a simple-boundary shoelace model, density uses mass weighting, and wire centroid uses actual segment/arc length.

Verification: Check minimum/default/maximum geometry. Reverse polygon vertex order analytically to verify the physical centroid is unchanged; test zero-area/self-intersecting polygons; move the opening until edge clearance becomes negative; compare geometric centroid with density-weighted CG; verify the semicircular arc centroid at 2r/π from its diameter.

Controls
Primary width

Primary width

Overall width of the two equal geometric halves. Changing width moves each half-centroid while preserving the density weighting model.

160 mm
Primary height

Primary height

Common height of both material halves. Their equal geometry keeps the geometric centroid fixed while density changes the center of gravity.

120 mm
Right-to-left density ratio

Right-to-left density ratio

Relative density of the right half divided by the left-half density. Geometry is unchanged, so only mass weighting moves the center of gravity.

2.4
geometric Cρ2.4ρ160 mm120 mmCG
valid geometry
Relative mass measure
32640.00 (relative)
CG x-coordinate
96.471 mm
CG y-coordinate
60.000 mm
Geometric centroid x
80.000 mm
First-moment contributions
Material A
ρᵢAᵢ
9600.00
xᵢ
40.00
yᵢ
60.00
ρᵢAᵢxᵢ
384000.00
ρᵢAᵢyᵢ
576000.00
Material B
ρᵢAᵢ
23040.00
xᵢ
120.00
yᵢ
60.00
ρᵢAᵢxᵢ
2764800.00
ρᵢAᵢyᵢ
1382400.00
xˉ=∑mixi∑mi,yˉ=∑miyi∑mi\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

Equation concept

For uniform areas use mᵢ=Aᵢ, with Aᵢ<0 only for fully contained openings. For nonuniform materials use mᵢ=ρᵢAᵢ. For wires use mᵢ=Lᵢ; a semicircular wire has arc length πr and centroid 2r/π from its diameter.

Simulation 4 Concept Question

Can the center of gravity lie outside the denser component? Explain using weighted first moments.

Simulation 5 Instructions

Combine straight line segments and a semicircular wire. The table uses segment length in place of area.

Centroids and Centers of Gravity Suite

Concept and model scope

Use length weighting for connected straight segments and the exact centroid of a semicircular wire arc.

Simulation purpose: First-moment, signed-area, density, polygon, and wire-centroid models tied directly to the displayed geometry.

Model scope: Positive composite pieces are non-overlapping; the composite web width is explicitly parameterized as 0.30B and shown on the drawing. Openings are subtracted only after full-containment validation. Polygon centroid uses a simple-boundary shoelace model, density uses mass weighting, and wire centroid uses actual segment/arc length.

Verification: Check minimum/default/maximum geometry. Reverse polygon vertex order analytically to verify the physical centroid is unchanged; test zero-area/self-intersecting polygons; move the opening until edge clearance becomes negative; compare geometric centroid with density-weighted CG; verify the semicircular arc centroid at 2r/π from its diameter.

Controls
Horizontal segment length

Horizontal segment length

Length of the straight horizontal wire segment measured from the vertical-segment junction to the semicircular arc. It contributes by length, not enclosed area.

160 mm
Vertical segment length

Vertical segment length

Length of the straight vertical wire segment from the shared junction upward. Its centroid is at half this length.

120 mm
Semicircular arc diameter

Semicircular arc diameter

Physical diameter of the connected semicircular wire arc. The arc contributes length πr and has its wire centroid 2r/π from the diameter.

40 mm
arc centroidstraight = 160 mm120 mmsemicircle Ø40 mmC
valid geometry
Total wire length
342.83 mm
x centroid
70.325 mm
y centroid
23.335 mm
First-moment contributions
Vertical segment
Lᵢ
120.00
xᵢ
0.00
yᵢ
60.00
Lᵢxᵢ
0.00
Lᵢyᵢ
7200.00
Horizontal segment
Lᵢ
160.00
xᵢ
80.00
yᵢ
0.00
Lᵢxᵢ
12800.00
Lᵢyᵢ
0.00
Upper semicircular arc
Lᵢ
62.83
xᵢ
180.00
yᵢ
12.73
Lᵢxᵢ
11309.73
Lᵢyᵢ
800.00
xˉ=∑mixi∑mi,yˉ=∑miyi∑mi\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

Equation concept

For uniform areas use mᵢ=Aᵢ, with Aᵢ<0 only for fully contained openings. For nonuniform materials use mᵢ=ρᵢAᵢ. For wires use mᵢ=Lᵢ; a semicircular wire has arc length πr and centroid 2r/π from its diameter.

Simulation 5 Concept Question

Why is a curved-wire centroid weighted by arc length rather than enclosed area?

Composite Centroid Procedure

  1. Select an origin and consistent axes.
  2. Divide the geometry into standard components.
  3. Assign positive measures to material and negative measures to holes.
  4. List each component measure and centroid coordinates.
  5. Sum first moments and divide by the signed total measure.
  6. Check that the result is physically plausible and that the signed total is nonzero.
Composite Centroid Workflow

Compute area or mass centroids by signed component summation or direct integration, then verify the result with first-moment and symmetry checks.

Composite Centroid WorkflowCompute area or mass centroids by signed component summation or direct integration, then verify the result with first-moment and symmetry checks.. Choose reference axes → Record any symmetry constraints before calculating; Record any symmetry constraints before calculating → Can the body be represented by simple components?; Can the body be represented by simple components? — Yes → Decompose into signed areas or masses; openings are negative; Can the body be represented by simple components? — No → Use direct integration with the correct area, mass, or density measure; Decompose into signed areas or masses; openings are negative → Compute total A or m and the corresponding first moments; Use direct integration with the correct area, mass, or density measure → Compute total A or m and the corresponding first moments; Compute total A or m and the corresponding first moments → Net physical area or mass is positive?; Net physical area or mass is positive? — No → Physical centroid model is invalid for a nonpositive total; Net physical area or mass is positive? — Yes → Compute the centroid coordinates; Compute the centroid coordinates → First moments about the centroid vanish and symmetry checks pass?; First moments about the centroid vanish and symmetry checks pass? — Yes → Centroid verified; First moments about the centroid vanish and symmetry checks pass? — No → Correct component signs, centroids, density, or integration limits; Correct component signs, centroids, density, or integration limits → Can the body be represented by simple components?

Choose reference axes → Record any symmetry constraints before calculating; Record any symmetry constraints before calculating → Can the body be represented by simple components?; Can the body be represented by simple components? — Yes → Decompose into signed areas or masses; openings are negative; Can the body be represented by simple components? — No → Use direct integration with the correct area, mass, or density measure; Decompose into signed areas or masses; openings are negative → Compute total A or m and the corresponding first moments; Use direct integration with the correct area, mass, or density measure → Compute total A or m and the corresponding first moments; Compute total A or m and the corresponding first moments → Net physical area or mass is positive?; Net physical area or mass is positive? — No → Physical centroid model is invalid for a nonpositive total; Net physical area or mass is positive? — Yes → Compute the centroid coordinates; Compute the centroid coordinates → First moments about the centroid vanish and symmetry checks pass?; First moments about the centroid vanish and symmetry checks pass? — Yes → Centroid verified; First moments about the centroid vanish and symmetry checks pass? — No → Correct component signs, centroids, density, or integration limits; Correct component signs, centroids, density, or integration limits → Can the body be represented by simple components?

  • Choose reference axes: terminator
  • Record any symmetry constraints before calculating: process
  • Can the body be represented by simple components?: decision
  • Decompose into signed areas or masses; openings are negative: process
  • Use direct integration with the correct area, mass, or density measure: process
  • Compute total A or m and the corresponding first moments: process
  • Net physical area or mass is positive?: decision
  • Physical centroid model is invalid for a nonpositive total: terminator
  • Compute the centroid coordinates: process
  • First moments about the centroid vanish and symmetry checks pass?: decision
  • Correct component signs, centroids, density, or integration limits: process
  • Centroid verified: terminator
Key Takeaways
  • Composite centroids are obtained from first moments.
  • Holes must subtract both area and first moments.
  • Polygon vertex ordering controls signed area but not the physical centroid.
  • Center of gravity requires density or weight weighting when materials differ.
  • Line and curved-wire centroids use length rather than area.