Mass and Balance: CG, Moments and %MAC
Mass and balance questions test the arithmetic and operational limits that keep the aircraft within certified CG and mass limits. You must handle datum arms, moments, fuel burn and %MAC without losing signs or units.
Core rules
- Moment = mass × arm from the datum. Total mass is the sum of masses; CG arm = total moment / total mass.
- Operating empty mass, payload and fuel make up take-off mass. Zero-fuel mass excludes usable fuel. Landing mass is take-off mass minus trip fuel, but must respect maximum landing mass.
- Forward CG limits affect rotation, stall speed and trim; aft limits reduce static stability and can worsen spin recovery. Floor loading and maximum zero-fuel mass are separate limits.
- %MAC = 100 × (CG arm − LE MAC arm) / MAC length. Moving a mass shifts CG by (mass moved × distance moved) / total mass.
Worked example: Empty mass 30,000 kg at 12 m; payload 8,000 kg at 16 m; fuel 6,000 kg at 14 m. Total moment = 360,000 + 128,000 + 84,000 = 572,000 kg·m. Total mass = 44,000 kg. CG = 572,000 / 44,000 = 13.00 m. If LE MAC is 10.0 m and MAC is 8.0 m, %MAC = 100 × (13.0 − 10.0) / 8.0 = 37.5%. Moving 500 kg aft by 4 m shifts CG by (500 × 4) / 44,000 = 0.045 m aft, so the new CG is 13.045 m.
Traps
Do not include unusable fuel in zero-fuel mass unless the definition specifies it. Keep arms signed relative to the datum. Do not mix kilograms and pounds or metres and inches. Check that fuel burn moves CG as fuel leaves; a forward limit can be breached at landing even when take-off was legal.