Calculators
Center of Mass Calculator
Find a system's center of mass from masses and positions.
Find the one-dimensional center of mass using x̄ = Σ(mᵢxᵢ) / Σmᵢ. Positions may be positive or negative relative to your chosen origin.
Center of mass
6 m
Total mass: 5 kg · x̄ = Σ(mᵢxᵢ) / Σmᵢ
This calculator treats each entry as a point mass on one axis. For extended objects, use each object's own center-of-mass position, and use a consistent origin and axis direction.
About This Tool
The center of mass is the mass-weighted average position of a system. This calculator handles two or more point masses on one axis using x̄ = Σ(mᵢxᵢ) / Σmᵢ. It is useful for mechanics exercises, balance-point estimates, and combining objects whose individual center positions are already known.
How To Use It
- Choose common units for the masses and positions.
- Enter at least two mass points. Positions are coordinates measured from the same origin and may be negative.
- Add more points when the system contains additional masses.
- Read the center-of-mass coordinate in your preferred position unit. Interpret it using the same origin and positive direction used for the inputs.
Examples
Two unequal masses
A 2 kg mass at 0 m and a 3 kg mass at 10 m have a center of mass at 6 m.
Equal masses
Equal masses at 0 m and 10 m place the center of mass halfway between them at 5 m.
Negative coordinate
A 1 kg mass at −2 m and a 3 kg mass at 2 m give a center of mass at 1 m.
Centimeter inputs
500 g at 0 cm and 1500 g at 40 cm give a center of mass at 30 cm.
Useful Notes
Center of mass formula
For discrete masses on one axis, x̄ = Σ(mᵢxᵢ) / Σmᵢ. Each coordinate is multiplied by its mass, the products are summed, and the result is divided by total mass.
Choose an origin first
A center-of-mass result is a coordinate, so every position must use the same origin and positive direction. Negative coordinates are valid when a mass lies on the negative side of that origin.
Point masses and extended objects
The formula can combine point masses directly. An extended rigid object can also be represented by its total mass located at that object's own center of mass when that center is already known.
Center of mass versus center of gravity
In a uniform gravitational field, center of gravity and center of mass coincide for ordinary mechanics problems. In a significantly nonuniform gravitational field they are different concepts, so this calculator should be interpreted as center of mass.
One-dimensional scope
This tool solves one coordinate at a time. For a 2D or 3D system, apply the same weighted-average formula separately to x, y, and z coordinates.
FAQ
Can positions be negative?
Yes. Negative coordinates are meaningful and often necessary. Keep one consistent origin and axis direction for every mass point.
Do all masses need the same unit?
The interface uses one selected mass unit for all rows so inputs remain easy to compare. Convert values first if your source data uses mixed units.
Can I add more than two masses?
Yes. Add as many mass points as needed; the calculator includes every row in the weighted average.
What happens with a zero-mass point?
A zero-mass point contributes nothing to the weighted position. The total mass of the complete system must still be greater than zero.
Is this the same as a geometric centroid?
Only when mass is distributed uniformly in the relevant geometry. Center of mass weights position by mass, while a purely geometric centroid depends on shape rather than mass distribution.
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