Centre of Mass Calculator
The balance point of several masses along a line or across a plane, with each one's share of the total.
Results
What this tool does
The centre of mass is the single point where you could hold an object and it would not tip — the place where all the mass balances out. For a collection of parts it is a weighted average: each position counted in proportion to its mass, so the heavy things pull the answer towards them and the light things barely move it. List the parts with their masses and positions and this page finds the point, in one dimension or two, along with how much each part is really deciding the answer.
Formula
centre x = Σ(m x) ÷ Σm · centre y = Σ(m y) ÷ Σm
Variables
| Symbol | Meaning | Unit |
|---|---|---|
bodies | The bodies: name = mass, position | — |
CX | Balance point along the first axis | — |
CY | Balance point along the second axis | — |
TB | Name · mass · position · share of the mass | — |
TM | Total mass | — |
NB | Bodies | — |
TD | Was a second coordinate given? | — |
HV | Heaviest body | — |
HS | Its share of the mass | % |
IN | Inertia about that balance point | — |
GY | Radius of gyration | — |
Worked example
- The bodies: name = mass, positionEngine = 180, 0.8 Gearbox = 70, 1.9 Battery = 25, 2.6 Fuel tank = 45, 3.4
- Balance point along the first axis1.546875
- Balance point along the second axis0.000000
- Name · mass · position · share of the mass Engine 180 0.8000 56.25 Gearbox 70 1.9000 21.88 Battery 25 2.6000 7.81 Fuel tank 45 3.4000 14.06
- Total mass320.000000
- Bodies4
- Was a second coordinate given?No
- Heaviest bodyEngine
- Its share of the mass56.25 %
- Inertia about that balance point291.396875
- Radius of gyration0.954262
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
Frequently asked questions
How do I write the list?
One body per line: a name, an equals sign, then its mass and its position along the axis you care about. Add a third number and it is read as a second coordinate, so you get the balance point in two dimensions instead of one. The positions must all be measured from the same origin, and the origin can be anywhere — put it at the front of the car, at one end of the beam, wherever makes the numbers easiest. The answer comes back in the same frame you put in.
Is the centre of mass the same as the centre of gravity?
For anything you are likely to be weighing, yes. They only separate when gravity is noticeably stronger at one end of the object than the other, which needs something the size of a mountain or a spacecraft in low orbit. For a car, a shelf or a boat the two points are the same to far more decimal places than you can measure. The distinction matters in orbital mechanics, where the difference is what slowly turns a satellite to face the planet, and essentially nowhere else.