Zerdly

Centre of Gravity Calculator

The centre of gravity (CoG) of a set of point masses is the mass-weighted average of their positions.

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How to use this calculator

Enter Values

Enter your values and press Calculate to see results.

Before you rely on this: First-pass guide only. Verify safety-critical or regulated work against the relevant standards, your project requirements and a qualified professional.

How to use this calculator

  1. Enter Mass 1 and its x position (and y position if you want a 2-D result). All positions are measured from the same origin you choose — e.g. the left end or a datum edge.
  2. Add Mass 2 and Mass 3 with their positions if you have more components; leave them blank for a single- or two-mass problem.
  3. Read the centre-of-gravity coordinate(s) — the point where the combined weight effectively acts, i.e. the balance point for lifting, slinging or tipping checks.

How it works

The centre of gravity (CoG) of a set of point masses is the mass-weighted average of their positions. For each axis the formula is x̄ = Σ(mᵢ·xᵢ) / Σmᵢ, where mᵢ is each mass and xᵢ its coordinate. The numerator is the total first moment of mass about the origin; dividing by the total mass gives the single point at which that whole mass could be concentrated without changing the moment. The y-axis is handled identically with ȳ = Σ(mᵢ·yᵢ) / Σmᵢ.

This tool treats each component as a point mass located at its own centre. Because the weight (m·g) shares the constant g, the CoG and the centre of mass coincide, so masses can be used directly. If your components are areas or volumes of the same material, you can enter their areas or volumes as the weights instead and the result is the centroid. The calculator sums up to three masses; for more complex bodies, break them into simple parts, find each part's own centre, and enter those. Results are a first-pass estimate — for rigging, lifting-plan or stability sign-off, verify against measured weights and the relevant standard (AS 4991 / AS 1418, Eurocode or AISC) and a competent structural or mechanical engineer.

Worked example

Balance point of three components on a frame. A fabricated frame carries three masses along its length: 40 kg at x = 0 m, 25 kg at x = 1.2 m and 15 kg at x = 2.0 m. Total mass = 40 + 25 + 15 = 80 kg. Σ(M·x) = 40×0 + 25×1.2 + 15×2.0 = 0 + 30 + 30 = 60 kg·m. Centre of gravity x̄ = 60 / 80 = 0.75 m from the origin. So the frame balances 0.75 m from the left end — that is where you would sling or support it to keep it level.

Common mistakes

  • Measuring each mass's position from a different origin. Every x (and y) must be measured from the same datum, or the weighted average is meaningless.
  • Forgetting that positions can be negative. A mass to the left of, or below, the origin has a negative coordinate — use the ± toggle so its moment subtracts correctly.
  • Treating the geometric mid-point as the CoG. The balance point shifts toward the heavier masses; only a mass-weighted average (not a simple average of positions) gives the true centre of gravity.

Frequently asked questions

What is the difference between centre of gravity, centre of mass and centroid?

For a body in a uniform gravity field they all coincide. Centre of mass is the mass-weighted average position; centre of gravity is the point where the total weight acts (identical to centre of mass because g is constant); centroid is the same idea but weighted by area or volume for a body of uniform density. Enter masses for the CoG, or areas/volumes for the centroid.

Why does the y-axis result only sometimes appear?

The y result is shown only when you enter at least one y position. Many CoG problems are one-dimensional (e.g. finding the balance point along a beam or trailer), so if you leave every y blank the tool reports just the x-axis centre of gravity to keep the answer clean.

Can I use this to plan a lift or find a sling point?

It gives a good first estimate of where a load balances, which is where you would centre a sling or spreader. But it assumes accurate masses and positions and ignores flexibility, dynamic effects and load shift. Any actual lift plan, tipping or stability check must use verified weights and be signed off against the relevant standard by a competent engineer or dogger/rigger.

What if two masses sit at the same position?

That is fine — their moments simply add. The formula sums every mass's moment about the origin regardless of whether positions repeat, so overlapping or coincident components are handled correctly.

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