Definitions and Calculations — Page 50, Lesson 66

Let’s work through this example together, because it pulls together everything we’ve been doing with mass, arm, and moment into one complete take-off and landing calculation. We have a fictitious aircraft, and we’re given a set of data. Our job is to find the centre of gravity position for take-off as loaded, and then for landing after a flight of 1 hour 30 minutes. First, the limiting data. Maximum take-off mass is 2245 lb. Maximum landing mass is 2100 lb. The centre of gravity limits are 2 inches forward to 6 inches aft of the datum. Fuel consumption is 7.0 US gallons per hour. Oil consumption is 1.0 US quart per hour. Now the load items. Basic mass is 1275 lb at an arm of minus 5 inches. Seats 1 and 2, 340 lb at minus 2. Seats 3 and 4, 170 lb at plus 30. Fuel, 35 US gallons, specific gravity 0.72, at arm plus 2. Oil, 8 US quarts, specific gravity 0.9, at arm minus 48. Baggage, 45 lb at arm plus 70. Before we build the table, we must convert fuel and oil volumes into mass, using the densities given. For fuel: 35 US gallons divided by 1.2, times 0.72, times 10, gives 210 lb. That factor of 1.2 and 10 is the standard conversion from US gallons to pounds for a liquid of given specific gravity. For oil: 8 US quarts divided by 4, divided by 1.2, times 0.9, times 10, gives 15 lb. Notice the extra division by 4 — that converts quarts into gallons first, because there are 4 quarts in a US gallon. Now we lay out the load sheet. For each item, mass times arm gives moment. Basic mass: 1275 times minus 5 is minus 6375. Seats 1 and 2: 340 times minus 2 is minus 680. Seats 3 and 4: 170 times plus 30 is plus 5100. Fuel: 210 times plus 2 is plus 420. Oil: 15 times minus 48 is minus 720. Baggage: 45 times plus 70 is plus 3150. Sum the masses: 1275 plus 340 plus 170 plus 210 plus 15 plus 45 gives a take-off mass of 2055 lb. That is below the maximum take-off mass of 2245 lb, so we’re fine there. Sum the moments: minus 6375, minus 680, plus 5100, plus 420, minus 720, plus 3150. That totals plus 895. So take-off moment is plus 895 lb-inches. Take-off CG is total moment divided by total mass: 895 divided by 2055, which is 0.435 inches aft of the datum. That is within the limits of 2 inches forward to 6 inches aft, so the take-off loading is acceptable. Now for landing. We need to know how much fuel and oil were consumed during the 1 hour 30 minute flight. Fuel used: 1.5 hours times 7 gallons per hour, divided by 1.2, times 0.72, times 10, gives 63 lb. Oil used: 1.5 hours times 0.25 quarts per hour — because 1.0 quart per hour divided by 4 gives 0.25 gallons per hour — divided by 1.2, times 0.9, times 10, gives 2.8 lb. Landing mass is take-off mass minus fuel used minus oil used: 2055 minus 63 minus 2.8 equals 1989.2 lb. That is below the maximum landing mass of 2100 lb, so we’re fine there too. Landing moment is take-off moment minus the moment of the fuel used minus the moment of the oil used. Fuel used moment is 63 lb times its arm of plus 2, which is plus 126. Oil used moment is 2.8 lb times its arm of minus 48, which is minus 134.4 — the excerpt rounds it to minus 134. So landing moment equals plus 895 minus 126 minus (minus 134). And here’s the key arithmetic point: minus and minus give plus. So that becomes plus 895 minus 126 plus 134, which equals plus 903. Finally, landing CG is landing moment divided by landing mass: plus 903 divided by 1989.2, which gives approximately 0.454 inches aft of the datum. That is still within the 2 inches forward to 6 inches aft limits, so the landing loading is acceptable. Notice the pattern: we always compute mass and moment for take-off, check the CG against limits, then subtract the fuel and oil consumed — with their own moments — to find the landing condition. The sign convention matters throughout: arms forward of the datum are negative, aft are positive, and when you subtract a negative moment, it adds. That’s the complete worked example.