05 The centre of gravity
The centre of gravity (C.G.) is the point through which the resultant of all the gravitational forces on the body passes, whatever the body's position.
- Uniform rod → its midpoint. Uniform rectangular lamina → intersection of the diagonals. Uniform circle → the centre. Uniform triangle → the intersection of the medians (1/3 of the way up any median).
- For parallel weights W₁, W₂, … at positions x₁, x₂, …: x̄ = Σ(Wᵢxᵢ)/ΣWᵢ, and the same in y.
- Composite laminae: divide into standard shapes, tabulate area and centroid, then take weighted averages. A removed part (hole) enters with a negative area.
- Suspended-body method: when a body hangs freely from a point, its C.G. lies on the vertical line through the point of suspension; repeating from a second point fixes the C.G. at the intersection.
Worked example — a lamina with a cut-out
A circular disc of radius 6 cm has a circular hole of radius 2 cm cut from it, the hole's centre 3 cm from the disc's centre. Taking moments of area about the disc centre: x̄ = [π(36)(0) − π(4)(3)]/[π(36) − π(4)] = −12π/32π = −0.375 cm, i.e. the C.G. of the remaining part is 0.375 cm from the centre on the side opposite the hole.
Worked example — two masses on a rod
A light rod 1.2 m long carries 4 kg at one end and 8 kg at the other. The C.G. is at x = (4×0 + 8×1.2)/12 = 0.8 m from the 4 kg end.