The centre of mass is the mass-weighted average position of a system. It is the single point that moves as though the entire mass were concentrated there and every external force acted on it, which is what makes it the most useful simplification in mechanics. The calculator above locates it either from a set of point masses with coordinates, or from a vehicle's axle weights and wheelbase.
Arb Digital publishes free physics calculators that own one job and say plainly what they are not. This page returns a position. It is not a moment of inertia, and the distinction between the two is the single most common confusion in this corner of mechanics, so there is a section below devoted entirely to it.
What This Center of Mass Calculator Does
In point-mass mode the tool takes up to four masses with x and y coordinates and returns the coordinates of the centre of mass, the total mass, and the distance from your chosen origin. Setting a mass to zero removes that row from the sum without you having to clear its coordinates, so you can work with two or three bodies without rearranging anything.
In vehicle mode the tool takes the weight carried by each axle and the wheelbase, and returns how far the centre of mass sits behind the front axle. This is the same weighted-average sum viewed from the other side: instead of telling the calculator where the mass is, you let a pair of scales measure the weighting for you and solve backwards for the position.
Both modes are unit-agnostic. Because the mass appears in both numerator and denominator, any consistent mass unit works — kilograms, pounds, or newtons of weight — and the position comes out in whatever length unit you used for the coordinates or the wheelbase.
How to Use It
- Choose your origin deliberately. The answer is a position relative to whatever point you called zero. Putting the origin at a physical feature — a corner, a front axle, a datum face — makes the result immediately usable.
- Keep units consistent within each column. Mixing millimetres and metres across coordinates is the fastest way to a wrong answer, and nothing in the arithmetic will warn you.
- Zero out unused rows. A mass of zero contributes nothing to either sum, so a three-body problem needs no editing beyond leaving the fourth mass at zero.
- For a vehicle, weigh on the level. Any slope transfers weight between axles and shifts the apparent longitudinal position of the centre of mass.
- Read the grid, not just the hero. The total mass is a useful check: if it does not match what you expect the system to weigh, an input is wrong before the position is even worth reading.
The Formula: How Centre of Mass Is Calculated
For a system of point masses the definition is xcm = Σ(mixi) ÷ Σmi, and identically for y. Each coordinate is weighted by its own mass, the weighted values are summed, and the sum is divided by the total mass. OpenStax University Physics Volume 1, section 9.6 on centre of mass, gives this definition for a discrete system and shows that the centre of mass obeys Newton's second law for the system as a whole.
Work the defaults. Masses of 2, 3 and 5 units sit at (0, 0), (4, 0) and (2, 6). The total mass is 10. The weighted x sum is (2×0) + (3×4) + (5×2) = 22, so xcm = 2.2. The weighted y sum is (2×0) + (3×0) + (5×6) = 30, so ycm = 3.0. The centre of mass is at (2.2, 3.0), which is inside the triangle the three masses form but pulled toward the heaviest one.
For the vehicle the balance is a moment sum about an axle. If Wf is the front axle weight, Wr the rear, and L the wheelbase, then the distance behind the front axle is a = WrL ÷ (Wf + Wr). With 900 on the front, 600 on the rear and a 2.6 m wheelbase: a = 600 × 2.6 ÷ 1,500 = 1.04 m, and the front carries 60 per cent of the weight. NASA Glenn Research Center's page on centre of gravity describes the same weighted-average construction for a body of distributed weight.
Centre of Mass Is Not a Moment of Inertia
Several tools on this site have the word "moment" in the name, and none of them do what this page does. A centre of mass is a position: three coordinates, measured in metres. A moment of inertia is a resistance to angular acceleration: a single number for a chosen axis, measured in kilogram metres squared for mass moments or metres to the fourth power for area moments.
The moment of inertia calculator works with cross-sections and returns a second moment of area for beam bending. The mass moment of inertia calculator works with solid bodies and returns rotational inertia about an axis. Both of them assume you already know where the centroid or the centre of mass is, and both ask you for an offset distance if you want to shift the axis. This page is where that offset comes from.
The link between them is the parallel-axis theorem, which says the moment of inertia about any axis equals the moment about a parallel axis through the centre of mass plus the mass times the square of the separation. That theorem is unusable without a centre of mass, which is exactly why the two calculations are separate steps in the same workflow rather than the same calculation. The polar moment of inertia calculator applies the same logic to torsion about a longitudinal axis.
Centre of Mass Versus Centre of Gravity Versus Centroid
Three terms that get used interchangeably and are not identical. The centre of mass is the mass-weighted average position. The centre of gravity is the weight-weighted average position, and they coincide exactly whenever the gravitational field is uniform, which it is for anything smaller than a small mountain. They diverge only for objects large enough that gravity is measurably stronger at one end than the other — which matters for orbiting spacecraft and for tidal effects, and for nothing you are likely to weigh on a driveway.
The centroid is a purely geometric quantity: the average position of the volume or the area, with no reference to mass at all. It coincides with the centre of mass when density is uniform, and departs from it when it is not. A steel-and-foam sandwich panel has its centroid at the geometric middle and its centre of mass much closer to the steel face.
The practical consequence is that you cannot infer a centre of mass from a drawing unless the density is uniform, and most assemblies are not. That is why a vehicle's longitudinal balance is measured on scales rather than computed from a CAD model, and why the axle-weight mode on this page exists.
What Weight Distribution Actually Tells You
A longitudinal balance figure is the easy half of a vehicle's mass properties. It tells you how the static load is shared between the axles, which is the starting point for tyre loading, braking balance and the relative grip available at each end. What it does not tell you is anything about the height of the centre of mass, and height is what governs load transfer under braking, acceleration and cornering.
Finding that height requires a second measurement with one axle raised through a known height, so the horizontal distance between the axles changes and the weight redistributes. The mathematics is straightforward, but the measurement is unforgiving: small errors in the raise height, in suspension movement, or in tyre deflection produce large errors in the result. That is why this page deliberately stops at the longitudinal position rather than pretending a two-scale measurement can produce a three-dimensional answer.
For the same reason, a static distribution figure is not a handling verdict. It is one input among many, alongside roll centres, spring and damper rates, tyre construction and aerodynamic load. The lever calculator and the torque calculator handle the moment arithmetic that follows once you know the position.
Where This Sits Among the Other Mechanics Tools
This page produces a position, and several other tools consume it. The momentum calculator and the angular momentum calculator both describe motion that is properly referenced to the centre of mass, since a rigid body's motion splits cleanly into translation of the centre of mass plus rotation about it. The weighted average calculator performs the identical arithmetic in a non-physical context, and the 3D distance calculator measures separations once positions are known. Where density rather than mass is the given, start with the density calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Averaging the coordinates without weighting — the plain geometric mean of the positions is the centroid of the points, not the centre of mass, and the two agree only when every mass is equal.
- Mixing length units between rows — nothing in the arithmetic detects it, and the answer will look perfectly plausible while being badly wrong.
- Assuming the centre of mass lies inside the object — rings, arches and L-shapes routinely place it in empty space, and that is not an error.
- Weighing a vehicle on a slope — any gradient transfers weight between axles and moves the computed longitudinal position.
- Confusing this with a moment of inertia — one is a location and the other is a resistance to angular acceleration. They are different quantities with different units.
Related Free Tools From Arb Digital
Take the position into the moment of inertia calculator or the mass moment of inertia calculator when you need rotational resistance rather than location, and into the polar moment of inertia calculator for torsion. Use the momentum calculator and the angular momentum calculator for motion, the lever calculator and the torque calculator for moment arithmetic, and the weighted average calculator when the same sum is not about mass at all. The 3D distance calculator and the density calculator cover the geometry and material sides. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the mass-weighted average position: multiply each mass by its coordinate, add those products together, and divide by the total mass. The same sum is done independently for each axis, so a two-dimensional problem is simply two one-dimensional problems sharing a denominator.
They coincide in any uniform gravitational field, which covers essentially every terrestrial situation. They differ only when the field varies measurably across the object, so the distinction matters for very large bodies and for orbital mechanics, and not for a car, a beam or a machine part.
A centre of mass is a position measured in units of length. A moment of inertia is a resistance to angular acceleration about a chosen axis, measured in kilogram metres squared. They are related through the parallel-axis theorem, which needs the centre of mass as an input, but they are not the same quantity.
Yes, and frequently is. A ring, a horseshoe, a boomerang and a hollow box all place it in empty space. Nothing in the physics requires material to exist at that point; it is a weighted average, not a physical location on the body.
Weigh each axle on a level surface, then divide each axle weight by the total. The centre of mass sits behind the front axle by the rear axle weight times the wheelbase, divided by the total weight, which is exactly what the vehicle mode on this page computes.
No. Height requires a second weighing with one axle raised by a known amount, and the result is very sensitive to measurement error in the raise height, suspension travel and tyre deflection. This page deliberately reports only the longitudinal position that two axle weights can honestly support.
A centroid is the average position of geometry, with no reference to mass. A centre of mass weights by mass. They coincide when density is uniform and separate when it is not, which is why a composite panel with a dense face has its centre of mass offset from its geometric middle.
This tool is provided for educational and preliminary engineering use. It models idealised point masses and a two-axle static weight balance, and does not account for suspension movement, dynamic load transfer or vertical centre of mass height, so treat its output as a teaching result rather than a validated measurement.
