Center of Mass Calculator
Enter up to three masses and their positions along a line to find the center of mass.
The center of mass of objects along a line is the mass-weighted average of their positions: multiply each mass by its position, add these up, and divide by the total mass. Two masses of 2 kg at 0 and 3 kg at 4 have a center of mass at (0 + 12) รท 5 = 2.4.
How to Use This Calculator
Enter each object's mass and its position along a straight line โ a shared axis you measure all positions from. You can use one, two or three masses; leave any you don't need with a mass of zero. The calculator returns the center of mass, the single point where the system would balance.
Positions can be negative if some masses sit to one side of your zero point. The result is in the same distance units you use for the positions.
How the Calculation Works
The center of mass is a weighted average, where each position is weighted by its mass. Heavier objects pull the balance point toward themselves, which is why the center of mass sits closer to the larger mass.
Each mass is multiplied by its position, the products are summed, and the total is divided by the combined mass. If all masses were equal, the result would be the plain average of the positions.
x_cm = ฮฃ(mแตขxแตข) รท ฮฃmแตข
Worked Example
Take a 2 kg mass at position 0 and a 3 kg mass at position 4. The weighted sum is 2 ร 0 + 3 ร 4 = 12, and the total mass is 5.
The center of mass is 12 รท 5 = 2.4 โ closer to the heavier 3 kg mass, as expected.
Assumptions and Limitations
- Handles point masses arranged along a single line (1D)
- Positions can be negative; use a consistent reference point
- For 2D or 3D, compute each coordinate separately the same way
- Total mass must be greater than zero
Who Uses a Center of Mass Calculator
Physics and engineering students use this to find the balance point of up to three masses along a line. Run it for statics and mechanics problems or to build intuition for how mass distribution sets the balance point. Enter each mass and its position to get the center of mass.
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Open calculator โFrequently Asked Questions
How do you find the center of mass?
Multiply each mass by its position, add the results, and divide by the total mass. It is a mass-weighted average, so heavier objects pull the balance point toward them.
What is the difference between center of mass and center of gravity?
They coincide in a uniform gravitational field, which covers almost all everyday situations. Center of mass depends only on mass distribution; center of gravity also depends on how gravity varies across the object.
Can positions be negative?
Yes. Choose any point as zero and measure positions from it; masses on one side are negative and on the other positive. The formula handles the signs automatically.
How do I find center of mass in 2D?
Do the same mass-weighted average separately for the x-coordinates and the y-coordinates. Each axis is computed independently using the same formula.
Where is the center of mass of two equal masses?
Exactly halfway between them. When masses are equal, the center of mass is just the midpoint, because the weighted average reduces to a plain average.
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