The center of mass is where an object or system balances. Engineers need it for vehicle stability, aircraft loading and cranes; physicists use it to describe the motion of a system as if all the mass were in one point.
How to use the Center of Mass Calculator
- Dimension
- Number of point masses
- Mass m₁
- x₁
- y₁
- z₁
- Mass m₂
- x₂
- y₂
- z₂
- Mass m₃
- x₃
- y₃
- z₃
- Mass m₄
- x₄
- y₄
- z₄
- Mass m₅
- x₅
- y₅
- z₅
- Mass m₆
- x₆
- y₆
- z₆
Results update as you type, and everything is calculated in your browser.
How it works
Choose the dimension and number of masses, then enter each mass and its coordinates (any consistent units). Heavier masses pull the centre toward themselves.
Inputs explained
| Field | What to enter |
|---|---|
| Dimension | Choose one: 1D (on a line), 2D (in a plane), 3D (in space) |
| Number of point masses | Choose one: 2, 3, 4, 5 … |
| Mass m₁ | A number |
| x₁ | A number |
| y₁ | A number |
| z₁ | A number |
| Mass m₂ | A number |
| x₂ | A number |
| y₂ | A number |
| z₂ | A number |
| Mass m₃ | A number |
| x₃ | A number |
| y₃ | A number |
| z₃ | A number |
| Mass m₄ | A number |
| x₄ | A number |
| y₄ | A number |
| z₄ | A number |
| Mass m₅ | A number |
| x₅ | A number |
| y₅ | A number |
| z₅ | A number |
| Mass m₆ | A number |
| x₆ | A number |
| y₆ | A number |
| z₆ | A number |
Example values
How the main result changes when Mass m₁ changes and everything else stays at the example values above:
| Mass m₁ | x of the center of mass |
|---|---|
| 1 | 0.3333 |
| 1.5 | 0.3684 |
| 2 | 0.4 |
| 2.5 | 0.4286 |
| 3 | 0.4545 |
| 4 | 0.5 |
Formula
x_cm = Σ mᵢxᵢ / Σ mᵢ, y_cm = Σ mᵢyᵢ / Σ mᵢ, z_cm = Σ mᵢzᵢ / Σ mᵢ
Worked example
Inputs — Dimension: 2D (in a plane); Number of point masses: 3; Mass m₁: 2; x₁: 1; y₁: 1; Mass m₂: 3; x₂: 4; y₂: 2; Mass m₃: 5; x₃: -2; y₃: 3.
Results — x of the center of mass = 0.4; y of the center of mass = 2.3; Total mass = 10.
Key terms
- Center of mass
- The mass-weighted average position.
Frequently asked questions
Is the center of mass the same as the centre of gravity?
In a uniform gravitational field, yes.
Can the center of mass be outside the object?
Yes – a ring or a boomerang balances at an empty point.
What units should I use?
Any, as long as all masses share one unit and all coordinates another.
What if one mass is zero?
It simply does not affect the result.