Centroids and Centers of Gravity - Theory & Concepts

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.

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.

\bar{x}=\frac{\sum A_ix_i}{\sum A_i} \qquad \bar{y}=\frac{\sum A_iy_i}{\sum A_i}

Variables

Symbol	Description	Unit
$A_i$	Signed area of component i; negative for a hole	mm²
$x_i$	Component-centroid x-coordinate	mm
$y_i$	Component-centroid y-coordinate	mm
$\bar{x}$	Composite centroid x-coordinate	mm
$\bar{y}$	Composite centroid y-coordinate	mm

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 $A_1=6000\ \text{mm}^2$ at $y_1=150\ \text{mm}$ and $A_2=4000\ \text{mm}^2$ at $y_2=50\ \text{mm}$ , the centroid is $\bar{y}=(6000\cdot150+4000\cdot50)/(10000)=110\ \text{mm}$ .

Simulation 1 Instructions

Resize the flange and web of the composite section and inspect the component table for $A_i$ , $x_i$ , $y_i$ , $A_ix_i$ , and $A_iy_i$ .

Advanced engineering statics simulation

Centroids and Centers of Gravity Suite

Validated signed area, density weighting, polygon geometry, and length-weighted wire centroids.

Simulation

Combine a flange and web without overlap and inspect their first-moment contributions.

Primary width160 mm

60–300

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Web height120 mm

40–260

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Flange thickness40 mm

10–100

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

valid geometry

Signed area

8800.00

mm²

x centroid

80.000 mm

y centroid

118.182 mm

Component	measure	xi	yi	measure·xi	measure·yi
Flange	6400.00	80.00	140.00	512000.00	896000.00
Web	2400.00	80.00	60.00	192000.00	144000.00

\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

For uniform areas use mᵢ=Aᵢ; use negative area for a fully contained hole, density-weighted area for nonuniform materials, and length for wires.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

Advanced engineering statics simulation

Centroids and Centers of Gravity Suite

Validated signed area, density weighting, polygon geometry, and length-weighted wire centroids.

Simulation

Treat a fully contained opening as negative area; invalid openings are rejected instead of silently subtracting a full circle.

Primary width160 mm

60–300

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Primary height120 mm

40–260

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Opening diameter40 mm

10–180

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Opening offset from plate center25 mm

-140–140

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

valid geometry

Signed area

17943.36

mm²

x centroid

78.249 mm

y centroid

60.000 mm

Component	measure	xi	yi	measure·xi	measure·yi
Plate	19200.00	80.00	60.00	1536000.00	1152000.00
Circular opening	-1256.64	105.00	60.00	-131946.89	-75398.22

\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

For uniform areas use mᵢ=Aᵢ; use negative area for a fully contained hole, density-weighted area for nonuniform materials, and length for wires.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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=\frac{1}{2}\sum_i(x_iy_{i+1}-x_{i+1}y_i)

\bar{x}=\frac{1}{6A}\sum_i(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i)

\bar{y}=\frac{1}{6A}\sum_i(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)

Variables

Symbol	Description	Unit
$A$	Signed polygon area	mm²
$x_i,y_i$	Ordered polygon-vertex coordinates	mm

Simulation 3 Instructions

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

Advanced engineering statics simulation

Centroids and Centers of Gravity Suite

Validated signed area, density weighting, polygon geometry, and length-weighted wire centroids.

Simulation

Move the two upper vertices through dimensional controls and apply the shoelace centroid equations.

Primary width160 mm

60–300

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Primary height120 mm

40–260

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Upper-left vertex x40 mm

0–180

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Upper-right inset25 mm

0–180

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

valid geometry

Signed area

16650.00

mm²

x centroid

81.396 mm

y centroid

59.760 mm

\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

For uniform areas use mᵢ=Aᵢ; use negative area for a fully contained hole, density-weighted area for nonuniform materials, and length for wires.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

\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

Symbol	Description	Unit
$\rho_i$	Material density or relative weight density	kg/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.

Advanced engineering statics simulation

Centroids and Centers of Gravity Suite

Validated signed area, density weighting, polygon geometry, and length-weighted wire centroids.

Simulation

Use equal geometric halves with different densities to distinguish centroid from center of gravity.

Primary width160 mm

60–300

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Primary height120 mm

40–260

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Right-to-left density ratio2.4

0.2–8.0

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

valid geometry

Density-weighted area

32640.00

relative mass measure

x centroid

96.471 mm

y centroid

60.000 mm

Component	measure	xi	yi	measure·xi	measure·yi
Material A	9600.00	40.00	60.00	384000.00	576000.00
Material B	23040.00	120.00	60.00	2764800.00	1382400.00

\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

For uniform areas use mᵢ=Aᵢ; use negative area for a fully contained hole, density-weighted area for nonuniform materials, and length for wires.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

Advanced engineering statics simulation

Centroids and Centers of Gravity Suite

Validated signed area, density weighting, polygon geometry, and length-weighted wire centroids.

Simulation

Use length-weighted centroids for vertical, horizontal, and semicircular wire elements.

Horizontal segment length160 mm

60–300

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Vertical segment length120 mm

40–260

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Semicircular arc diameter40 mm

10–180

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

valid geometry

Total length

342.83

x centroid

66.660 mm

y centroid

23.335 mm

Component	measure	xi	yi	measure·xi	measure·yi
Vertical segment	120.00	0.00	60.00	0.00	7200.00
Horizontal segment	160.00	80.00	0.00	12800.00	0.00
Upper semicircular arc	62.83	160.00	12.73	10053.10	800.00

\bar{x}=\frac{\sum m_i x_i}{\sum m_i},\qquad \bar{y}=\frac{\sum m_i y_i}{\sum m_i}

For uniform areas use mᵢ=Aᵢ; use negative area for a fully contained hole, density-weighted area for nonuniform materials, and length for wires.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

Simulation 5 Concept Question

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

Composite Centroid Procedure

STEP-BY-STEP

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

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.

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