A load sheet that is legal but will not get airborne
Your four-seat single-engine aeroplane has a basic empty mass of 700 kg at an arm of 2.40 m aft of the datum and a maximum take-off mass of 1,150 kg. CG limits are 2.35 m to 2.55 m aft of datum at all masses. Planned load: pilot 85 kg and front passenger 85 kg at 2.30 m; rear passengers 80 kg and 85 kg at 3.10 m; 40 kg of baggage, the maximum permitted, at 4.00 m; and 140 L of usable fuel at 2.50 m, density 0.72 kg/L. The departure aerodrome is at 1,200 ft elevation, QNH 1013, OAT 24 °C, runway 09, TODA 900 m, dry grass with a 1% upslope, wind calm. Under these conditions the POH gives a take-off distance to 50 ft of 620 m at 1,150 kg, sea level, ISA, with corrections of +10% per 1,000 ft of density altitude, +15% for dry grass and +10% per 1% of upslope.
1. Check the load sheet against MTOW and the CG envelope, and state the maximum fuel you may carry with this payload.
Read the worked answer
Mass: 700 + 85 + 85 + 80 + 85 + 40 + (140 × 0.72 = 100.8) = 1,175.8 kg. That is 25.8 kg over MTOW - the aeroplane cannot legally or safely depart as loaded.
Moment: (700 × 2.40) 1,680 + (170 × 2.30) 391 + (165 × 3.10) 511.5 + (40 × 4.00) 160 + (100.8 × 2.50) 252 = 2,994.5 kg·m. CG = 2,994.5 / 1,175.8 = 2.547 m, inside the 2.55 m aft limit - but only just.
Maximum fuel with this payload is 100.8 − 25.8 = 75 kg, that is 104 L. Redo the calculation at 104 L: mass 1,150 kg, moment 2,994.5 − (25.8 × 2.50) = 2,930.0 kg·m, CG = 2.548 m. The aeroplane is now at MTOW and the CG has moved aft, because the fuel tank arm (2.50 m) is forward of the CG you are trimming to. Offloading fuel does not move the CG forward here; it moves it back.
So: 104 L maximum, and the CG is 2 mm inside the aft limit. There is effectively no loading freedom left, which matters for how the aeroplane will handle in the climb and at the stall.
2. Compute the take-off distance required and decide whether the departure can be made.
Read the worked answer
Density altitude first. ISA temperature at 1,200 ft is 15 − (2 × 1.2) = 12.8 °C; the actual 24 °C is an ISA deviation of about +11 °C. Density altitude ≈ 1,200 + (120 × 11) = about 2,520 ft.
Apply the corrections to the book figure of 620 m:
- Density altitude: 2,520 / 1,000 × 10% = +25.2% → 620 × 1.252 = 776 m
- Dry grass: +15% → 776 × 1.15 = 893 m
- 1% upslope: +10% → 893 × 1.10 = 982 m
Required 982 m against a TODA of 900 m: short by about 82 m, roughly 9%. The departure cannot be made as loaded.
Note the order of magnitude. The grass and slope corrections together add 26%, but density altitude alone adds 25% - a hot day at a 1,200 ft strip in a normally aspirated aeroplane is the dominant penalty. The unaided eye reads 900 m of dry grass as generous; the numbers say otherwise. Reject the departure and re-plan rather than trying it and hoping the aeroplane climbs through the gap.
3. The passenger suggests simply offloading the 40 kg of baggage. Evaluate that, and identify what would actually fix the problem.
Read the worked answer
Offloading the baggage fixes the mass and does nothing for the performance.
The runway requires the total correction factor to be no more than 900 / 620 = 1.45. Grass and slope already account for 1.15 × 1.10 = 1.265, which leaves only 1.15 for density altitude: about 1,500 ft DA, meaning an OAT at or below roughly 15 °C.
Removing 40 kg from 1,150 kg is a 3.5% mass reduction. Take-off distance does not fall one-for-one with mass - in this class of aeroplane a 3.5% mass reduction buys a few per cent of distance, well short of the 82 m (9%) shortfall, and you have also shed the payload you wanted to carry while still being over the limiting correction. Mass reduction alone will not solve a density-altitude problem of this size.
What actually works: depart in the cool of the early morning when the OAT is near 15 °C or below; or use a hard-surface, level runway, which removes both the 15% grass penalty and the 10% slope penalty; or split the load. If none of those is available, the flight does not go.
Before any of these, re-check the CG after every change. With this payload the aft limit is only 2 mm away, and a lighter aeroplane with the same rear-seat load will rotate more easily and stall at a higher speed for the same attitude.