Weight and Balance Plus Density Altitude
You are loading a Cessna 182 for a commercial sightseeing flight. Empty weight is 1,500 lb with an empty moment of 127,500 lb-in. The centre-of-gravity envelope is 82.0 to 90.0 in aft of datum. Pilot and front passenger weigh 170 lb and 160 lb at arm 85 in. Two rear passengers weigh 150 lb each at arm 118 in. Baggage is 80 lb at arm 140 in. Fuel is 40 US gal at 6 lb/gal and arm 90 in. Maximum gross weight is 2,550 lb. The airport elevation is 3,000 ft, altimeter 29.92, temperature 30°C. The POH takeoff chart gives 1,500 ft at sea level ISA and 2,700 ft at 5,500 ft density altitude. Runway available is 3,000 ft.
1. Compute total weight and centre of gravity. Is the loading within limits? If not, how much baggage must be removed to reach the aft limit?
Read the worked answer
Weight = 1,500 + 170 + 160 + 300 + 80 + 240 = 2,450 lb. Moment = 127,500 + (170×85) + (160×85) + (300×118) + (80×140) + (240×90) = 127,500 + 14,450 + 13,600 + 35,400 + 11,200 + 21,600 = 223,750 lb-in. CG = 223,750 / 2,450 = 91.3 in, which is 1.3 in aft of the 90.0 in limit. The loading is not within the envelope.
To reach 90.0 in by removing baggage at arm 140 in: (223,750 − 140x)/(2,450 − x) = 90; 223,750 − 140x = 220,500 − 90x; 3,250 = 50x; x = 65 lb. Remove at least 65 lb of baggage, or move load forward.
2. Compute density altitude and evaluate the takeoff distance against the runway available.
Read the worked answer
Pressure altitude = 3,000 ft (29.92). ISA temperature at 3,000 ft = 15 − 2×3 = 9°C. Difference = 30 − 9 = 21°C. Density altitude ≈ 3,000 + 120×21 = 5,520 ft. The chart at 5,500 ft gives 2,700 ft; with a 3,000 ft runway the margin is 300 ft (about 10%).
That is not adequate for a commercial operation with variable wind, runway slope, or a late liftoff. Removing 65 lb of baggage improves the CG and slightly reduces weight, but the density altitude still dominates. Reduce fuel/payload, wait for a cooler time, or use a longer runway; do not accept the flight on the basis of the 300 ft margin.