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Centroids and centers of gravity2D

Density-weighted center of gravity

Calculate signed-area, polygon, density-weighted, and line centroids with explicit invalid-geometry checks.

Open the complete lesson

Advanced engineering statics simulation

Centroids and Centers of Gravity Suite

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

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

Primary width
160 mm
mm
60–300

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

Primary height
120 mm
mm
40–260

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

Right-to-left density ratio
2.4
0.2–8.0

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

valid geometry
ρ2.4ρ
Density-weighted area
32640.00
relative mass measure
x centroid
96.471 mm
y centroid
60.000 mm
Componentmeasurexiyimeasure·ximeasure·yi
Material A9600.0040.0060.00384000.00576000.00
Material B23040.00120.0060.002764800.001382400.00
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}

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

Concept question: Predict the center-of-gravity x-coordinate after changing the density ratio.

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.

Related simulations

Composite-area centroid builderPlate with circular openingParameter-edited polygon centroidStraight and curved wire centroid