Centroid vs. Center of Mass: Key Differences Explained—
Introduction
The terms “centroid” and “center of mass” are often used interchangeably in casual conversation, but they have distinct meanings and applications in mathematics, physics, and engineering. Understanding their differences clarifies how to analyze shapes, materials, and physical systems — from pure geometry problems to real-world structures and dynamics.
Basic definitions
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Centroid: In geometry, the centroid is the arithmetic mean position of all the points in a shape. For a two-dimensional lamina of uniform density (a flat shape of constant thickness and uniform material), the centroid coincides with the geometric center — the point obtained by averaging coordinates of all the points in the region.
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Center of mass (COM): In physics, the center of mass is the point at which the total mass of a system can be considered to be concentrated for the purposes of translational motion. It depends on the distribution of mass, which may vary across the object or system.
When they coincide
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Uniform density: For a homogeneous body (constant density throughout), the centroid and center of mass are the same point. That makes the centroid a convenient geometric tool for many engineering tasks where material is uniform.
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Simple geometric bodies: Rectangles, circles, regular polygons, and uniform triangles have centroids that also serve as their centers of mass.
When they differ
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Non-uniform density: If density varies spatially, the center of mass shifts toward heavier regions and no longer coincides with the centroid. Example: a plate whose density increases from left to right will have its COM shifted right of the centroid.
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Discrete mass systems: For a system of point masses (e.g., masses connected by rods, or a collection of particles), the center of mass is computed from the weighted average of positions by mass; there is no “centroid” in the geometric sense unless you treat masses as uniform points.
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Hollow versus solid shapes: The centroid of an area differs from the center of mass of a structure if thickness or density distribution differs (for instance, a thin hollow ring vs. a filled disk).
Mathematical formulas
For simple cases, both concepts use averaging, but with different weights.
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Centroid (2D region R of uniform density): x_centroid = (1/Area) ∫_R x dA
y_centroid = (1/Area) ∫_R y dA -
Center of mass (system of point masses or a body with density ρ(x,y,z)): x_COM = (1/M) ∫ x ρ(x) dV
y_COM = (1/M) ∫ y ρ(x) dV
z_COM = (1/M) ∫ z ρ(x) dV
where M = ∫ ρ(x) dV is the total mass.
For discrete point masses mi at positions (xi, yi, zi): x_COM = (Σ mi xi) / (Σ mi)
y_COM = (Σ mi yi) / (Σ mi)
z_COM = (Σ mi zi) / (Σ mi)
Calculation examples
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Uniform triangle (vertices at coordinates): the centroid is at the average of vertex coordinates:
x_centroid = (x1 + x2 + x3)/3, y_centroid = (y1 + y2 + y3)/3. -
Two masses m1 at x=0 and m2 at x=L: center of mass is at x = (m2 L) / (m1 + m2).
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Non-uniform lamina where density ρ(x) varies: use the weighted integral formulas above to find COM; centroid formula fails unless ρ is constant.
Physical significance and applications
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Engineering design: Centroid location is key for structural bending, area moments of inertia, and locating neutral axes in beams. COM is critical for dynamics, balance, stability, and motion predictions.
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Robotics and aerospace: COM determines balance, stability, and control responses. For payloads with uneven mass distribution, COM calculations guide design and control.
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Graphics and computational geometry: Centroids are used for mesh processing, object alignment, and geometric simplification.
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Statics vs. dynamics: Centroid-related methods often serve static structural analyses; COM is essential for dynamic behavior and interaction with forces (gravity, inertia).
Practical tips
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Always check density assumptions: If density is not uniform, compute center of mass with density as a weight.
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Use symmetry: Symmetry can pinpoint centroid/COM without integrals (e.g., symmetric objects place centroid/COM on symmetry axes).
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Choose the right model: For thin plates of uniform thickness, area centroid = COM. For solids, use volume integrals.
Quick reference
- Nature: Centroid is a geometric property of a region; COM is a physical property depending on mass distribution.
- When equal: They coincide only for uniform density.
- Use centroid for geometry and structural area problems; use COM for mass, balance, and dynamics.
If you want, I can add worked examples with full integrals for a non-uniform density lamina, or produce diagrams and step-by-step calculations for common shapes.