Center of Mass Calculator — 2D System of Particles
Enter the mass and (x, y) coordinates of up to three point particles to find their center of mass — the unique point where the system balances under uniform gravity.
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ȳ = 3 m • Total mass = 10 kg
- 1
Total mass M
2 + 3 + 5 = 10 - 2
Weighted x sum Σ(mᵢ × xᵢ)
2 × 1 + 3 × 4 + 5 × 2 = 24 - 3
Center of mass x̄ = Σ(mᵢxᵢ) ÷ M
24 ÷ 10 = 2.400000
How does this calculator work?
Center of mass coordinates: x̄ = Σ(mᵢ xᵢ) / Σmᵢ, ȳ = Σ(mᵢ yᵢ) / Σmᵢ. It is the mass-weighted average position — heavier particles pull the CoM closer to them. For two equal masses the CoM is the midpoint; for unequal masses it shifts toward the heavier one. The CoM can lie outside the object for non-convex shapes.
Formula
How this is calculated
The center of mass (CoM) of a system is the mass-weighted average position of all particles. If you balanced the system on a pin placed at the CoM, it would be in equilibrium under a uniform gravitational field — which is why it is sometimes called the center of gravity (the two concepts coincide in a uniform field, but differ in non-uniform fields, such as tidal effects).
For a discrete system of N point masses, x̄ = Σ(mᵢ xᵢ) / Σmᵢ and ȳ = Σ(mᵢ yᵢ) / Σmᵢ. These are simply weighted averages: positions of heavier masses count more. Equivalently, x̄ minimizes Σmᵢ(x − xᵢ)², the mass-weighted sum of squared distances from x to each particle.
This calculator treats particles as point masses. For extended rigid bodies with a uniform density, the center of mass coincides with the geometric centroid and can be computed using integration or composite-body techniques. For non-uniform or continuous distributions, the sums become integrals: x̄ = ∫x dm / ∫dm. Units are length units of the coordinate system you choose — the calculator uses metres and kilograms by convention, but any consistent unit system works.
Frequently asked questions
The centroid is the geometric center of a shape (pure geometry, no masses). The center of mass weights positions by mass. For a uniform-density object, they coincide; for non-uniform distributions, they differ.
Yes — for non-convex or hollow objects, the CoM can be in empty space. A classic example is a horseshoe or a donut ring, where the CoM sits in the hole.
The CoM position relative to the body does not change from an external force alone (internal forces cancel). However, Newton's second law applied to the CoM gives F_net = M × a_CoM, so the CoM accelerates as if all external forces act on a single point particle of total mass M.
Also known as
TG we-Calculate Editorial Team. (2026). Center of Mass Calculator — 2D System of Particles [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/center-of-mass-calculator
TG we-Calculate Editorial Team. "Center of Mass Calculator — 2D System of Particles." TG we-Calculate. 2026. https://we-calculate.com/calculator/center-of-mass-calculator.
TG we-Calculate Editorial Team, "Center of Mass Calculator — 2D System of Particles," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/center-of-mass-calculator
@misc{wecalculate_center_of_mass_calculator, title = {Center of Mass Calculator — 2D System of Particles}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/center-of-mass-calculator}}, year = {2026}, note = {TG we-Calculate} }
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