# Center of Mass Calculator

Calculate the center of mass for up to 5 point masses in 2D. Enter mass and (x, y) position for each object. Shows weighted average position of the system.

## What this calculates

The center of mass is the weighted average position of all the mass in a system. For two objects, a 5 kg mass at x=0 and a 3 kg mass at x=4 m have their center of mass at x = (5*0 + 3*4)/(5+3) = 1.5 m, closer to the heavier object. This calculator handles up to 5 point masses in two dimensions.

## Inputs

- **Mass 1** (kg) — min 0 — Mass of object 1.
- **X₁ Position** (m) — X-coordinate of object 1.
- **Y₁ Position** (m) — Y-coordinate of object 1.
- **Mass 2** (kg) — min 0 — Mass of object 2 (set to 0 to ignore).
- **X₂ Position** (m) — X-coordinate of object 2.
- **Y₂ Position** (m) — Y-coordinate of object 2.
- **Mass 3** (kg) — min 0 — Mass of object 3 (set to 0 to ignore).
- **X₃ Position** (m) — X-coordinate of object 3.
- **Y₃ Position** (m) — Y-coordinate of object 3.
- **Mass 4** (kg) — min 0 — Mass of object 4 (set to 0 to ignore).
- **X₄ Position** (m) — X-coordinate of object 4.
- **Y₄ Position** (m) — Y-coordinate of object 4.
- **Mass 5** (kg) — min 0 — Mass of object 5 (set to 0 to ignore).
- **X₅ Position** (m) — X-coordinate of object 5.
- **Y₅ Position** (m) — Y-coordinate of object 5.

## Outputs

- **Center of Mass X** (m) — X-coordinate of the center of mass.
- **Center of Mass Y** (m) — Y-coordinate of the center of mass.
- **Total Mass** (kg) — Sum of all masses in the system.
- **Objects Used** — formatted as text — Number of objects with non-zero mass.

## Details

The center of mass formulas for a system of point masses are:

Xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)

Ycm = (m1y1 + m2y2 + ... + mnyn) / (m1 + m2 + ... + mn)

Or more compactly: Rcm = Sum(mi * ri) / Sum(mi)

**Why Center of Mass Matters:**
- **Stability:** An object tips over when its center of mass is no longer above the base of support (this is why loaded trucks and SUVs are more prone to rollover)
- **Orbital mechanics:** Two orbiting bodies rotate around their common center of mass (the barycenter). The Earth-Moon barycenter is about 4,670 km from Earth's center
- **Sports:** A high jumper arches their back so their center of mass passes under the bar while their body clears it (the Fosbury Flop technique)
- **Engineering:** Crane load calculations, vehicle weight distribution, and satellite balancing all rely on center of mass

To find the center of mass of an irregular object experimentally, hang it from two different points and draw plumb lines. The center of mass is where the lines intersect.

## Frequently Asked Questions

**Q: What is the difference between center of mass and center of gravity?**

A: For most everyday situations they are identical. Center of mass is the weighted average position of all mass. Center of gravity is the point where gravitational torque sums to zero. They differ only when the gravitational field is non-uniform (e.g., very tall structures or objects in orbit).

**Q: Can the center of mass be outside the object?**

A: Yes. A donut's center of mass is in the hole. A boomerang's center of mass is in the empty space between its arms. An L-shaped bracket's center of mass may be in the empty corner. The center of mass is the weighted average position of all the mass, which can land outside the physical material.

**Q: How is center of mass used in orbital mechanics?**

A: Two orbiting bodies both orbit their common center of mass (barycenter). For the Earth-Sun system, the barycenter is inside the Sun. For Pluto and Charon, the barycenter is in empty space between them. The wobble of a star around a barycenter is how astronomers detect exoplanets.

**Q: How do I use this for more than 5 objects?**

A: You can break a larger system into groups. Find the center of mass of the first 5 objects, then treat that result as a single point mass equal to the sum of those 5 masses. Then calculate again with that combined object and the remaining masses.

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Source: https://vastcalc.com/calculators/physics/center-of-mass
Category: Physics
Last updated: 2026-04-08