The balance point of a system
Every object, or group of objects, has a point that behaves as if all of its mass were concentrated there: the center of mass. Push on it and the system moves without turning; hang the system from it and it balances. In a uniform gravitational field, which is the case for anything on Earth’s surface, it is the same as the center of gravity. Mathematicians call the same idea a weighted average, or barycenter.
The formula
For point masses m₁, m₂, … at positions (x₁, y₁), (x₂, y₂), …:
- x = (m₁x₁ + m₂x₂ + …) ÷ (m₁ + m₂ + …)
- y = (m₁y₁ + m₂y₂ + …) ÷ (m₁ + m₂ + …)
Each coordinate is the average of the positions, weighted by the masses. A heavier mass pulls the center of mass toward itself.
Worked example
Three masses lie in a plane:
| Mass | Position |
|---|---|
| 2 | (0, 0) |
| 3 | (4, 0) |
| 5 | (2, 3) |
- Total mass: 2 + 3 + 5 = 10.
- x = (2 × 0 + 3 × 4 + 5 × 2) ÷ 10 = 22 ÷ 10 = 2.2.
- y = (2 × 0 + 3 × 0 + 5 × 3) ÷ 10 = 15 ÷ 10 = 1.5.
- Distance from the origin: √(2.2² + 1.5²) = 2.66.
The center of mass (2.2, 1.5) is pulled toward the heaviest mass at (2, 3), but stays inside the triangle formed by the three masses.
Units
Use any mass unit and any length unit, as long as you use the same ones throughout. Pounds and kilograms give the same result, because only the ratios between the masses matter. You can even use counts: three identical boxes stacked in one spot count as a mass of 3.
Where it is used
- Balancing loads: the center of mass of a loaded truck, trailer or boat must stay within limits for stability. Aircraft are weighed and balanced before every flight.
- Seesaws and mobiles: the system balances when the pivot is under the center of mass.
- Robotics and sports: a walking robot or a gymnast stays upright when the center of mass is above the base of support.
- Astronomy: the Earth and the Moon both orbit their common center of mass, the barycenter, about 2,900 miles from the Earth’s center, inside the Earth.
Extended objects
For a real object, imagine it cut into small pieces, each with its own mass at its own center, and apply the same formula. For uniform symmetric shapes, the center of mass is at the geometric center: the middle of a rod, the center of a disc or sphere. For a uniform flat polygon, it is the centroid, which the polygon area calculator gives directly. To combine several shapes, enter each shape’s mass at its own centroid here.
In three dimensions
Add a z coordinate and apply the same weighted average: z = Σ(m × z) ÷ Σm. The calculation works the same way along each axis independently.