Load and Balance Calculations: Arms, Moments and CG Shifts
Arms, moments and the CG formula
Distances are measured from the datum, positive aft. A station ahead of the datum has a negative arm. Moment = mass × arm (kg·m). CG = total moment / total mass. Loading tables, graphs and nomograms are simply pre-computed moments; index units are moments divided by a constant, and must be read with the correct units and sign.
The three standard operations
- Adding mass m at arm a: new CG = (M·cg + m·a)/(M + m), so ΔCG = m(a − cg)/(M + m).
- Removing mass: new CG = (M·cg − m·a)/(M − m).
- Shifting mass m by distance d: ΔCG = m·d/M, because total mass is unchanged; aft shift positive, forward negative.
Fuel burn is simply removal of mass at the tank arm. If the tank CG coincides with the aircraft CG the CG does not move; fuel in the wings generally lies close to the CG, so the shift is small but not zero.
Worked example
Mass 50 000 kg, CG 18.00 m. Load 2 000 kg at 26.00 m: moment = 50 000 × 18.00 + 2 000 × 26.00 = 900 000 + 52 000 = 952 000 kg·m; mass 52 000 kg; CG = 18.31 m. Then burn 5 000 kg from a tank at 17.00 m: moment = 952 000 − 85 000 = 867 000 kg·m; mass 47 000 kg; CG = 18.45 m. Burning fuel forward of the CG moved the CG aft.
Traps that cost marks
Forgetting to update the total mass when adding or removing; getting the sign of a shift backwards; using the CG instead of the datum as the reference for arms; mixing kg·m and kg·in; reading a nomogram without checking which scale applies (mass, index or %MAC); and assuming fuel burn never changes the CG.