Centres of mass

Objective. Locate and use centres of mass for simple laminas.

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Learn: the key idea

The centre of mass of a lamina or system of particles is the single point at which its total weight can be considered to act for the purposes of moments and equilibrium. For a system of point masses, the centre of mass coordinates are found by taking the weighted average of positions, x̄ = Σ(mx)/Σm, applied separately for x and y. For uniform symmetric laminas, such as rectangles or circles, the centre of mass lies at the geometric centre by symmetry, while composite shapes are handled by summing moments of each part about an axis. For example, two masses of 2 kg and 3 kg at x = 0 and x = 5 have centre of mass at x = 15/5 = 3.

Worked example

Particles of mass 4 kg and 6 kg lie at x = 1 and x = 6 respectively. Find the x-coordinate of the centre of mass of this system.

1 $x̄ = Σ(mx) \div Σm$
2 $Σ(mx) = 4(1) + 6(6) = 4 + 36 = 40$
3 $Σm = 4 + 6 = 10$
4 $x̄ = 40 \div 10 = 4$

Answer: x = 4

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Common mistake. Students average positions directly, ignoring the mass weighting, e.g. taking (0+5)/2 = 2.5 instead of weighting by mass. Always weight each position by its own mass before dividing by the total mass.

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