Centre of Mass (Centre of Gravity) Calculator

What is the centre of mass?

The centre of mass is the point where the whole mass of a body or a group of parts acts as if it were concentrated. Hang the body from that point and it stays balanced. With gravity taken as uniform, the centre of gravity is the same point.

How is it calculated?

Multiply the mass of each part by the coordinate of that part's own centre of mass, add the products and divide by the total mass: x̄ = Σ (m × x) / Σ m. Do the same separately for y and z. It is a weighted average, so heavy parts pull the centre towards themselves.

Example

Three parts sit on a base frame: a 120 kg frame at (400, 250) mm, a 45 kg motor at (150, 200) mm and a 30 kg gearbox at (650, 300) mm. The total mass is 195 kg. In x, 120 × 400 + 45 × 150 + 30 × 650 = 74 250, divided by 195 gives 380.8 mm. In y, 120 × 250 + 45 × 200 + 30 × 300 = 48 000, divided by 195 gives 246.2 mm.

Holes and voids

If a hole is cut in a plate or a piece is removed from a body, add the removed portion to the list with a negative mass. Enter the solid plate as positive and the hole as negative, and the calculation gives the right answer on its own.

What can be entered instead of mass?

Because the result depends only on ratios, you can enter weight instead of mass. For pieces of a plate of the same thickness and material, area works as well, and for a bar of constant section, length does.

Where is it used?

Choosing sling points when lifting a machine, axle load distribution on vehicles, unbalance of rotating parts and tipping checks for equipment on legs all rest on the position of the centre of mass.

Limits

The tool treats parts as point masses and does the arithmetic on the numbers you enter. Finding each part's own centre of mass correctly is up to you. For complex geometry the mass properties of a CAD program are more reliable. For lifting work, confirm the calculation by weighing or a trial lift.