Three Adults, Full Baggage and an Aft CG in a Cessna 172
You are loading a Cessna 172 for a summer afternoon flight with three adults. Empty mass 760 kg at a moment arm of 1.02 m aft of datum; pilot and front passenger 160 kg at 0.95 m; rear passengers 150 kg at 1.45 m; baggage 40 kg at 1.75 m; fuel 100 kg at 1.15 m. The approved envelope at this mass runs from 0.91 m to 1.20 m aft of datum, and maximum take-off mass is 1,100 kg. The flight manual quotes a clean stall speed of 48 kt at maximum mass with the CG at the forward limit. The runway is 900 m of asphalt, temperature 30 °C, wind 210°/12 kt, and the runway in use is 21. You plan a short-field take-off with 10° of flap, rotating at 50 kt and climbing at Vx.
1. Compute the mass and CG position and state whether the loading is acceptable.
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Mass: 760 + 160 + 150 + 40 + 100 = 1,210 kg, which is 110 kg above the 1,100 kg maximum take-off mass. Moments (kg × m aft of datum): 760 × 1.02 = 775.2; 160 × 0.95 = 152.0; 150 × 1.45 = 217.5; 40 × 1.75 = 70.0; 100 × 1.15 = 115.0. Total moment 1,329.7 kg·m. CG = 1,329.7 ÷ 1,210 = 1.099 m.
That sits inside 0.91–1.20 m but only 0.10 m from the aft limit, and the envelope figures are only valid up to 1,100 kg - so the loading is not acceptable: the aircraft is overweight and aft-biased. A second point: as fuel burns from the 1.15 m tank, the CG moves further aft, so the landing margin is smaller than the take-off margin. The load sheet must be redone before anyone boards.
2. Explain the flight-mechanics consequences of that aft CG for stall, stability and the planned early rotation.
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Moving the CG aft reduces the nose-down moment the tailplane must produce, so the wing carries a slightly larger share of the weight and the clean stall speed falls by a knot or two; take-off distance shortens and cruise speed improves marginally. The cost is static margin: the pitch-down moment generated by a given angle-of-attack increase is smaller, so the aircraft is less stable in pitch, stick force per g falls, and a gust or a ham-fisted rotation pushes it to a much higher angle of attack for the same input. At the stall the nose-drop is weaker - the classic aft-CG stall is a stall that does not want to break - and if a wing drops, the likelihood of a spin and the time to recover both increase.
Rotating at 50 kt with the CG at 1.10 m combines a low speed with the smallest pitch-authority margin: the aircraft can leave the ground below flying speed, in ground effect, at high alpha. Rotate at the flight-manual speed, not before.
3. What load change brings the aircraft to 1,100 kg, and what CG results?
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Remove one rear passenger (75 kg at 1.45 m) and 35 kg of baggage (at 1.75 m), which takes out exactly 110 kg. New mass 1,100 kg. Moments removed: 75 × 1.45 = 108.75 and 35 × 1.75 = 61.25, total 170.0 kg·m. New total moment 1,329.7 − 170.0 = 1,159.7; CG = 1,159.7 ÷ 1,100 = 1.054 m.
That is comfortably inside 0.91–1.20 m and about 45 mm forward of the original figure, restoring a sensible pitch-stability margin. Alternatives - less fuel, or moving baggage from the aft hold into the cabin - change the CG differently for the same mass reduction, so the correct answer is whichever combination satisfies both limits with margin. Then recompute the take-off and climb performance for 1,100 kg at 30 °C, which is a separate limitation from the CG question.