Center of Mass Calculator

What Is Center of Mass?

The center of mass (COM) is the average position of all mass in a system, weighted by each mass value. It represents the point where the system would balance perfectly. In a uniform gravitational field, the center of mass equals the center of gravity.

Center of Mass Formula:
x_cm = Σ(mᵢxᵢ) / Σ(mᵢ)
y_cm = Σ(mᵢyᵢ) / Σ(mᵢ)

Where mᵢ is each mass and (xᵢ, yᵢ) is its position

Applications

Physics: Projectile motion, collision analysis, orbital mechanics
Engineering: Structural design, vehicle stability, robotics
Aerospace: Aircraft balance, satellite orientation
Architecture: Load distribution, structural support

How to Use

Enter the mass and position (x, y) for up to 3 point masses. The calculator computes the center of mass coordinates and total mass.

Properties

The center of mass always lies within the convex hull of the point masses. For symmetric mass distributions, the center of mass is at the geometric center.

Center of Mass vs Centroid

The centroid is the geometric center (assumes uniform density). The center of mass accounts for different masses at different positions. They coincide only when mass is uniformly distributed.

Extension to 3D

The same formula extends to 3D by adding a z-component: z_cm = Σ(mᵢzᵢ) / Σ(mᵢ).

Frequently Asked Questions

The average position of mass in a system, weighted by each mass. It is the balance point of the system.

x_cm = (m₁x₁ + m₂x₂ + ...) / (m₁ + m₂ + ...). Sum of mass-position products divided by total mass.

In a uniform gravitational field (small scale), yes. They differ only in non-uniform fields (very large scales).

Yes, for objects with hollow centers (like a donut), the center of mass can be in empty space.

The center of mass moves as if all external forces act on it. It simplifies analysis of complex systems.

As fuel burns, the center of mass shifts, affecting stability. Rocket design ensures the center of mass stays ahead of the center of pressure.