Center of Gravity Calculator

Compute the centroid (center of mass) for a system of point masses in 2D. Essential for statics, physics, and engineering.

Centroid formula: For masses mᵢ at (xᵢ, yᵢ):

X_cg = (Σ mᵢ·xᵢ) / (Σ mᵢ)    Y_cg = (Σ mᵢ·yᵢ) / (Σ mᵢ)

Two equal masses (0,0) & (4,0)
Triangle corners (0,0),(2,4),(4,0) m=1 each
Heavy mass at (0,0) m=5, two m=1 at (3,3)
Three masses along line y=x
Calculating...

Understanding Center of Gravity

The center of gravity (or centroid) of a system is the average position of its mass. In a uniform gravitational field, it's the point where the entire weight can be considered to act. This calculator handles discrete point masses, but the concept extends to continuous objects.

Mathematical Definition (discrete masses):

X̄ = (∑ mᵢ xᵢ) / (∑ mᵢ) , Ȳ = (∑ mᵢ yᵢ) / (∑ mᵢ)

For continuous objects, integrals replace the sums: x̄ = (1/M) ∫ x dm , ȳ = (1/M) ∫ y dm.

Key Properties

1

Balance point: If you support the object at the centroid, it will balance perfectly (in uniform gravity).

2

Additivity: For composite objects, the overall centroid is the mass-weighted average of individual centroids.

3

Symmetry: If a system has an axis of symmetry, the centroid lies on that axis.

Experimental Determination

?

Plumb-line method (suspension): Suspend the object from two different points; the intersection of vertical lines (using a plumb bob) gives the centroid.

⚖️

Balance method: Support the object on a knife-edge or pivot; adjust until it balances. The centroid lies directly above the support.

Center of Gravity & Stability

Stability condition: An object is stable if its centroid lies above its base of support. The lower the centroid and the wider the base, the more stable the object.

Examples: Racing cars have very low CG to avoid tipping; the leaning tower of Pisa stays upright because its CG is still above its foundation.

Common Centroids for Continuous Bodies (Uniform Density)

Shape Location
Rectangle (width w, height h) (w/2, h/2)
Triangle (vertices at (0,0), (b,0), (0,h)) (b/3, h/3) from right-angle corner
Circle (center at (xc,yc)) (xc, yc)
Semicircular arc (radius R) (0, 2R/π) from center of full circle
Uniform rod (length L) midpoint (L/2)
Rectangular plate intersection of diagonals

For irregular shapes, you can approximate by breaking them into small elements (like this calculator does with point masses).

Applications in Real World

  • Engineering: Stability analysis of buildings, bridges, vehicles. Determining support reactions in statics.
  • Physics: Orbital mechanics (center of mass of binary stars), pendulum center of oscillation.
  • Design: Balancing products (e.g., ergonomic tools, furniture), sports equipment (sweet spot).
  • Aerospace: Aircraft CG limits for safe flight; rocket stability.
  • Biomechanics: Human body centroid changes with posture; used in gait analysis.

Frequently Asked Questions

In a uniform gravitational field, they coincide. Center of mass depends only on mass distribution; center of gravity depends on gravitational field. For most everyday situations, they are the same.

Yes! For example, with two equal masses separated by a distance, the centroid is at the midpoint between them — which is empty space. For a ring, the centroid is at the center (no mass there). This calculator will show that.

It uses standard double-precision floating point. Results are shown to two decimal places, but internal precision is higher. Great for student problems and quick checks.

Approximate the shape by a set of point masses: divide it into small pieces, treat each piece's mass as concentrated at its own centroid. More pieces give better accuracy. This is the fundamental idea behind numerical integration.