Aircraft Weight and Balance (CG Envelope)
The in-gross-weight-but-out-of-CG load weight-and-balance exists to catch: sum the station moments (empty, occupants, fuel, baggage), CG = total moment / total weight, legal only if weight <= max gross AND fwd_limit <= CG <= aft_limit. A 1,500 lb (39 in) aircraft with 340 lb front, 180 lb fuel, 200 lb baggage is 2,220 lb, CG 44.47 in -- legal. Fly lighter but pack 300 lb of baggage aft and it is 2,100 lb (under gross) yet CG 47.24 in -- BEHIND the 47 in aft limit and dangerously unstable. Fuel burn moves the CG, so both takeoff and landing CG must be in the envelope. A loading aid; the aircraft flight manual and the pilot in command govern.
Formula and source
total_weight = sum(w); total_moment = sum(w x arm); CG = total_moment / total_weight; in_envelope = weight <= max_gross AND fwd_limit <= CG <= aft_limit.
The station-moment weight-and-balance check of the FAA Weight & Balance Handbook (FAA-H-8083-1; AC 91-23), by name; the specific aircraft flight manual and the pilot in command govern.
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