Center of Mass Calculator (point masses in 1-D, 2-D or 3-D)

Understanding the centre of mass calculator — concept and principle

The centre of mass is the average position of a system's mass. This calculator finds it for a set of point masses in one, two or three dimensions, with total mass and the distance of the centre from the origin.

How it is calculated

• x̄ = Σmᵢxᵢ ÷ M; ȳ = Σmᵢyᵢ ÷ M; z̄ = Σmᵢzᵢ ÷ M.
• Total mass M = Σmᵢ.
• Each coordinate is computed independently; unused coordinates stay 0.
• Distance from the origin = √(x̄² + ȳ² + z̄²).

The principle

The centre of mass is a weighted average: heavier masses pull it towards themselves. Under uniform gravity it is also the centre of gravity, the point where the whole weight can be considered to act, so an object supported there balances. For an extended body, split it into simple parts, find each part's own centre and mass, and combine them like point masses.

Worked example

Default inputs (x, y, z in m; mass in kg): (0, 0, 0) 2 kg; (4, 0, 0) 1 kg; (0, 3, 0) 1 kg; (0, 0, 2) 4 kg.

1. Total mass M = 2 + 1 + 1 + 4 = 8 kg; 4 masses.
2. x̄ = (1 × 4) ÷ 8 = 0.5 m.
3. ȳ = (1 × 3) ÷ 8 = 0.375 m.
4. z̄ = (4 × 2) ÷ 8 = 1 m; centre = (0.5, 0.375, 1).
5. Distance from origin = √(0.25 + 0.1406 + 1) ≈ 1.17925 m.

How to read the result

The centre lies closest to the heaviest mass, here the 4 kg mass at z = 2 m. It need not lie on any of the masses, or even inside an object (think of a ring). Engineers use it to check stability, lifting points and balance.

Common mistakes

• Forgetting to divide by total mass.
• Mixing units between coordinates.
• Using the geometric centre of a non-uniform object.
• Entering a part's corner position instead of its own centre.

Frequently asked questions

How do you find the centre of mass?

Multiply each mass by its position, add them up and divide by the total mass, for each coordinate.

Is the centre of mass the same as the centre of gravity?

Yes, in a uniform gravitational field.

Can the centre of mass be outside an object?

Yes, for shapes such as rings, horseshoes or a bent rod.

Sources: Halliday, Resnick & Walker, Fundamentals of Physics (centre of mass) · OpenStax, University Physics Volume 1, Section 9.6 Center of Mass