Find the centre of mass.

Place point masses (or square tiles of an L-plate, ring segments) on a grid; the centre of mass marker moves and may fall outside the material.

Place a 2 kg mass so the system's centre of mass lands on the marked point (1, 1) m; then build an L-plate whose COM lies off the plate.

The centre of mass must lie inside the body.

• For two particles on the x-axis, the centre of mass is at X = (m₁x₁ + m₂x₂)/(m₁ + m₂), the mass-weighted mean of their positions. For equal masses it lies midway.
• In three dimensions, with total mass M = Σmᵢ: X = Σmᵢxᵢ/M, Y = Σmᵢyᵢ/M and Z = Σmᵢzᵢ/M; compactly, R = Σmᵢrᵢ/M.
• Three equal masses: the centre of mass is at the centroid of their triangle. If the centre of mass is taken as the origin, Σmᵢrᵢ = 0.
• For a continuous body the sums become integrals: R = (1/M)∫r dm. By reflection symmetry, uniform rods, rings, discs, spheres and cubes have their centre of mass at their geometric centre.
• Masses 100 g, 150 g and 200 g at the corners (0, 0), (0.5, 0) and (0.25, 0.25√3) m of an equilateral triangle of side 0.5 m have their centre of mass at (5/18 m, 1/(3√3) m), not at the centroid, because the masses are unequal.
• A uniform triangular plate can be cut into thin strips parallel to one side; each strip balances at its midpoint, so the centre of mass lies on every median, at the centroid.
• A uniform 3 kg L-shaped plate made of three 1 m squares: treat each square as 1 kg at its centre, (½, ½), (3/2, ½), (½, 3/2) m. The centre of mass is at (5/6 m, 5/6 m), on the plate's line of symmetry.
• The centre of mass need not lie inside the material of the body; for a ring it is at the empty centre.

Teaching this idea? Put the simulation on your own page or LMS with this code.