Lesson 2 of 13 · 8 min
Centre of mass
NCERT §6.2
The tea stall at the mela hangs its name on an L-shaped wooden signboard. The owner wants to hang it from a single hook so that it does not tip. Where should the hook go? The answer is the board's centre of mass.
The lesson in notes
In short
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.