Last updated: 2026-07-01

Definition

The center of mass of a system of point masses is the mass-weighted average of their positions — the single point that behaves, for the purposes of motion, as if all the mass were concentrated there. For objects arranged along a straight line, enter each mass and its position and this calculator returns x_cm = Σ(mᵢxᵢ) ÷ Σmᵢ, supporting up to four point masses.

The formula

Each mass mᵢ is multiplied by its position xᵢ, the products are summed, and the result is divided by the total mass. A heavier object contributes a larger product and therefore pulls the center of mass toward its own position. When every mass is equal, the formula reduces to the plain average of the positions, (x₁ + x₂ + … + xₙ) ÷ n — the geometric midpoint.

Positions are measured along a single axis, so they can be negative. The choice of origin does not affect the physics: shifting all positions by the same amount shifts the center of mass by exactly that amount.

Worked example — two objects

A 2 kg ball sits at x = 0 m and a 3 kg ball sits at x = 5 m.

Total mass: 2 + 3 = 5 kg

Weighted sum: 2 × 0 + 3 × 5 = 0 + 15 = 15 kg·m

Center of mass: 15 ÷ 5 = 3 m

The center of mass lands at 3 m — past the midpoint of 2.5 m and closer to the heavier 3 kg ball, exactly as intuition suggests.

Worked example — three objects

Three masses lie along a track:

Total mass: 2 + 3 + 5 = 10 kg

Weighted sum: 0 + 3 + 10 = 13 kg·m

Center of mass: 13 ÷ 10 = 1.3 m

Because the 5 kg mass at x = 2 m dominates, the center of mass is pulled toward the right end of the track, well past the geometric midpoint of the three positions (1.0 m).

Worked example — using negative positions

Place a 4 kg block at x = −3 m and a 1 kg block at x = 2 m, with the origin between them.

Total mass: 4 + 1 = 5 kg

Weighted sum: 4 × (−3) + 1 × 2 = −12 + 2 = −10 kg·m

Center of mass: −10 ÷ 5 = −2 m

The result is negative, sitting on the same side as the heavier block. This shows that the center of mass need not coincide with any object and need not be positive — it simply tracks where the mass is concentrated.

Center of mass versus centroid

The centroid is the purely geometric average of positions, treating every point equally. The center of mass weights each position by its mass. The two agree only when the masses are identical; once they differ, the center of mass moves toward the heavier objects while the centroid stays at the geometric center.

Using unused slots

This calculator supports up to four mass–position pairs. To model fewer objects, leave the extra masses at 0. A pair with mass 0 contributes nothing to either the total mass or the weighted sum, so its position is ignored. The total mass must be greater than 0 — with no mass present the center of mass is undefined.