Centre-of-mass calculations

In Moments, couples and centre of gravity the weight of a body acts at one point. For a symmetrical uniform body you can find that point by eye. For anything else, such as a set of particles or a flat shape with a piece cut out, it has to be calculated.

A position weighted by mass

The centre of mass is the mean position of the mass, with each piece weighted by how much mass it has. For particles of mass mᵢ at positions xᵢ, measured from one origin,

x_CM = (∑ mᵢxᵢ)/(∑ mᵢ)

In two dimensions, find each coordinate separately with the same total mass:

y_CM = (∑ mᵢyᵢ)/(∑ mᵢ)

The answer always lies between the smallest and largest coordinates, and closer to the larger masses. Moving the origin changes the numbers but not the point itself. In a uniform gravitational field the centre of mass is also the centre of gravity.

A lamina with a piece cut out

For a flat sheet (lamina) of uniform thickness and density, mass is proportional to area. Split the shape into rectangles or other simple parts whose centres you know, and use their areas Aᵢ in place of the masses.

A hole is a part with negative area at the centre of the piece removed. Subtract its area from the total, and subtract its area × coordinate from each sum. A table with columns Aᵢ, xᵢ, yᵢ, Aᵢxᵢ and Aᵢyᵢ keeps the signs straight.

Common mistakes

• Averaging positions without weighting them by mass.
• Measuring positions from different origins for different parts.
• Adding the area of a hole instead of subtracting it, or subtracting its area but not its area × coordinate.
• Dividing by the area of the original shape instead of the area that remains.