Center of Mass Calculator

The formula behind it

For a system of point masses, the center of mass is the mass-weighted average position: X = Σ(mᵢ × xᵢ) / Σmᵢ, and Y is computed the same way using the y-coordinates. Each point pulls the balance point toward itself in proportion to its mass — a heavier point has more influence than a light one at the same distance.

Worked example

Picture two weights on a see-saw laid along the x-axis: a 4 kg mass sitting at x = 0, and a 6 kg mass sitting at x = 10 (both at y = 0). Total mass is 4 + 6 = 10 kg. The weighted sum of x-positions is (0 × 4) + (10 × 6) = 60, so the center of mass is at X = 60 / 10 = 6, Y = 0. Notice the balance point (6) sits closer to the heavier 6 kg mass than to the lighter one — that's expected, since the heavier object contributes more to the weighted average.

Adding more points

"Add Another Point" appends a fresh row with its own X, Y, and mass fields; the calculator sums across however many rows are present when you click Calculate, so a three- or ten-point system uses the exact same Σ(mᵢxᵢ)/Σmᵢ formula, just with more terms in the sum. "Remove" on a row deletes it and renumbers the remaining points; "Clear All" resets back to a single blank row.

Where this applies

This is the standard discrete center-of-mass calculation used in introductory mechanics for systems of separate point masses — not for a single continuous solid object, which would need an integral over its mass distribution instead of a finite sum. It's the same math used to check whether a loaded structure, a set of counterweights, or a cluster of point loads balances where you expect.