Center of Mass Calculator
Enter the mass and position of each object to find the center of mass of the system. Choose between a 1D line, a 2D plane, or full 3D space, then add up to 6 point masses. The calculator shows the weighted-average formula worked out with your exact numbers, so you can follow every step.
Formula
Worked example
Three masses: 3 kg at (0, 0), 5 kg at (4, 6), and 2 kg at (2, 3). Total mass M = 10 kg. x_cm = (3*0 + 5*4 + 2*2) / 10 = 24/10 = 2.4 m. y_cm = (3*0 + 5*6 + 2*3) / 10 = 36/10 = 3.6 m. Center of mass: (2.4, 3.6).
What is the center of mass?
The center of mass (COM) of a system of particles is the unique point where the total mass of the system can be considered to be concentrated for the purpose of analyzing translational motion. If you apply a net external force anywhere on the system, the center of mass accelerates exactly as Newton's second law predicts for a single particle of the same total mass. For a two-mass seesaw, the center of mass lies along the beam between them, closer to the heavier side. For an irregularly shaped object, it may not even lie on the object itself - think of a boomerang or a hollow ring, whose center of mass sits in empty space at the geometric center.
The weighted-average formula
For a system of n discrete point masses, the center of mass coordinate along each axis is the mass-weighted average of the positions along that axis: x_cm = (m1*x1 + m2*x2 + ... + mn*xn) / (m1 + m2 + ... + mn). The same formula applies independently to each axis, so a 3D calculation is just three 1D calculations run in parallel. The weighting ensures that a mass of 10 kg at position 5 m contributes ten times as much to the numerator as a 1 kg mass at the same position. If every mass is identical, all the weights cancel and you recover the simple geometric centroid: the ordinary arithmetic average of all positions.
Center of mass vs. center of gravity
In a uniform gravitational field, the center of mass and center of gravity coincide exactly, which is why the two terms are interchangeable in everyday settings. The distinction only matters in a non-uniform field: a very large object in orbit, for example, experiences stronger gravity at its near end than its far end. In that case the center of gravity shifts slightly toward the stronger-field side, while the center of mass stays at the geometric average. For all terrestrial and most engineering applications you can treat them as the same point.
Practical applications
Engineers use the center of mass to predict how structures, vehicles, and mechanisms will respond to forces. Aircraft designers locate the center of mass relative to the center of lift to ensure stable flight. Vehicle designers keep the center of mass low and within the wheelbase to prevent tipping. Roboticists compute the center of mass of each link in a kinematic chain to control balance and joint torques. In sports science, the whole-body center of mass trajectory tells coaches how efficiently an athlete generates and transfers momentum. Even in space mission design, the center of mass of a satellite dictates where thrusters must fire to produce pure translation rather than unwanted rotation.
Center of mass formulas by configuration
| Object type | Formula or location | Notes |
|---|---|---|
| Two point masses | x = m1*x1 + m2*x2) / (m1+m2) | Falls on the line connecting them |
| n point masses (1D) | x_cm = sum(mi*xi) / sum(mi) | Weighted average of positions |
| n point masses (2D/3D) | (x_cm, y_cm, z_cm) each axis independently | Apply 1D formula per axis |
| Uniform rod (length L) | x_cm = L/2 from one end | Geometric midpoint |
| Uniform triangle (2D) | Centroid: ((x1+x2+x3)/3, (y1+y2+y3)/3) | Intersection of medians |
| Uniform rectangle | Center of the rectangle | Geometric center |
| Uniform sphere / cube | Geometric center | Symmetry implies COM = centroid |
| Hollow ring (1D) | Geometric center | COM may lie in empty space |
Quick-reference formulas for common discrete and uniform-density shapes.
Frequently asked questions
Can the center of mass lie outside the physical object?
Yes. For any non-convex or hollow object the center of mass can fall in empty space. A classic example is a horseshoe magnet: the center of mass is in the gap between the poles. Similarly, a ring or a boomerang has its center of mass at the geometric center, not on the material itself. This is perfectly valid physically; it means you can balance the object by supporting it at that point if you attach a rigid platform there.
How is center of mass different from centroid?
The centroid is a purely geometric concept: the arithmetic mean of all points in a shape, independent of any physical properties. The center of mass takes mass distribution into account. For a uniform-density object the two coincide because every volume element contributes the same mass. For a non-uniform object, the center of mass shifts toward the denser regions while the centroid stays at the geometric average.
What happens to the center of mass when no external forces act?
The center of mass moves at constant velocity (or stays still) whenever the net external force on the system is zero. This is a direct consequence of Newton's second law and the conservation of momentum. Individual masses in the system can move and collide freely; their center of mass still glides on a straight line at the same speed. Rocket engineers and orbital mechanics use this principle constantly.
Does the center of mass depend on the coordinate system?
Its coordinates change when you shift or rotate the coordinate system, but the physical point it describes does not. If you move the origin 2 m to the right, every x-coordinate shifts by 2 m and so does x_cm. To compare center-of-mass positions across problems, always specify the reference point or origin you are measuring from.
How do I find the center of mass of a composite object?
Treat each part as a point mass located at its own center of mass, then apply the weighted-average formula to those reduced masses and positions. For example, an L-shaped bracket is two rectangles; find the center of mass of each rectangle, assign each one its total mass, and use the formula. This calculator handles exactly this workflow for up to 6 sub-masses.
Is center of mass the same in all dimensions?
The concept is the same: a mass-weighted average of positions. In 1D you compute one number. In 2D you compute x_cm and y_cm independently using the same formula on each axis. In 3D you add z_cm the same way. The three calculations are completely independent of each other; you can solve each axis without knowing the others.