Weight and balance shift after a late load change
Your aeroplane has a maximum take-off mass of 1150 kg and an empty mass of 700 kg with an empty moment of 1,050 kg·m. The forward and aft centre-of-gravity limits are 0.80 m and 1.05 m from the datum. You plan two occupants at the front seats, 170 kg total at 0.60 m, baggage of 40 kg at 1.80 m, and 160 litres of fuel at 0.90 m (fuel density 0.72 kg per litre). At the last moment a passenger arrives with 30 kg of luggage, and you must decide where to put it. The baggage compartment limit is 50 kg.
1. Compute the current mass and centre of gravity before the late addition, and check the limits.
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Fuel mass is 160 x 0.72 = 115.2 kg. Total mass is 700 + 170 + 40 + 115.2 = 1025.2 kg, which is 124.8 kg below the maximum.
Moments: empty 1050; occupants 170 x 0.60 = 102; baggage 40 x 1.80 = 72; fuel 115.2 x 0.90 = 103.68. Total moment = 1050 + 102 + 72 + 103.68 = 1327.68 kg·m.
Centre of gravity = 1327.68 / 1025.2 = 1.295 m from the datum. That is well aft of the 1.05 m aft limit, so the aircraft is already outside the envelope before the late addition. The loading as planned is not legal.
2. What does this result tell you about the loading plan, and how would you correct it?
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The centre of gravity at 1.295 m is about 0.245 m aft of the limit, which is a serious out-of-balance condition, not a marginal one. The likely cause is that the baggage and fuel arms are far aft relative to the occupants, and the empty moment may already place the basic aircraft toward the rear of the envelope.
To correct it, move mass forward: reduce baggage, add ballast in a forward station if the handbook permits, or redistribute occupants. Reducing the 40 kg of baggage to zero removes 72 kg·m and brings the centre of gravity to (1327.68 - 72)/(1025.2 - 40) = 1255.68/985.2 = 1.275 m, still aft of the limit. The dominant term is the empty moment, so the aircraft must be checked against its actual weighing data; if the empty centre of gravity is 1050/700 = 1.50 m, the datum or the figures are inconsistent and the calculation must be redone with the correct arms.
3. Assuming corrected arms place the aircraft within limits, how would you decide whether to accept the extra 30 kg of luggage?
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Add the 30 kg at the baggage arm of 1.80 m: new mass 1055.2 kg, new moment 1327.68 + 54 = 1381.68, giving a centre of gravity of 1.309 m if the original figures are used. That moves the centre of gravity further aft, which is the wrong direction, and it also uses up 30 of the 50 kg baggage allowance.
If the corrected loading is within limits, accept the luggage only if the resulting centre of gravity remains inside the aft limit with a margin, the mass remains below 1150 kg, and the take-off and landing performance at the expected temperature and elevation is still satisfactory. Otherwise, refuse the extra luggage or offload fuel. The decision must be based on the recalculated envelope, not on the fact that the compartment limit alone is respected.