Centre of Mass Calculator
The centre of mass is the balance point of a system, the single spot where all of its mass can be imagined to act, and this calculator finds it for a set of point masses arranged along a line. Enter the masses and their positions as two matching lists, and it returns the position of the centre of mass along with the total mass, updating as you type. The idea is that of a weighted average: every mass pulls the balance point towards itself in proportion to how heavy it is, so the centre of mass always lies nearer the heavier objects. To find it you multiply each mass by its position, add up all those products, and divide by the total mass, which is exactly the calculation the tool performs. This deceptively simple quantity is one of the great labour-saving ideas in physics, because a complicated object made of many parts moves, under gravity or any external force, exactly as if its entire mass were concentrated at the centre of mass, letting you replace a swarm of particles with a single point. It also explains balance and stability in the everyday world: a stack, a vehicle or a person stays upright only while the centre of mass remains above the base of support, which is why a tall load tips so easily and why we lean into a turn. The same principle, extended to two and three dimensions, governs everything from gymnastics to spacecraft. That makes the tool genuinely useful for physics students learning about mass, balance and motion and checking homework, and for anyone working out a balance point or load distribution. Use any consistent units for position. The formula and a worked example are explained clearly below.
How it works
The centre of mass is the sum of each mass times its position, divided by the total mass. This is a weighted average of the positions, with the masses as the weights, so it sits closer to the heavier objects. The same idea extends to two and three dimensions by doing this separately for each coordinate.
Worked example
For masses 2, 3 and 5 at positions 0, 4 and 10: the weighted sum is 2 times 0, plus 3 times 4, plus 5 times 10, which is 0 plus 12 plus 50, or 62. The total mass is 10, so the centre of mass is 62 over 10, which is 6.2.