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 for take-off as loaded, and then the CG for landing after a flight of 1 hour 30 minutes. First, let's note the limiting values. The Maximum Take-off Mass is 2245 lb, and the Maximum Landing Mass is 2100 lb. The Centre of Gravity limits are 2 inches forward to 6 inches aft of datum. Fuel consumption is 7.0 US Gallons per hour, and oil consumption is 1.0 US quart per hour. Now, the table gives us the items. We have the Basic Mass at 1275 lb with an arm of -5 inches. Seats 1 and 2 at 340 lb, arm -2. Seats 3 and 4 at 170 lb, arm 30. Fuel, 35 US Gallons, with a specific gravity of 0.72, and the arm is given as 2. Oil, 8 US quarts, specific gravity 0.9, arm -48. And Baggage at 45 lb, arm 70. Now, the first thing we must do is convert fuel and oil from volume to mass, because the load sheet works in pounds. The book gives us the conversion factors. For fuel: 35 divided by 1.2, times 0.72, times 10, equals 210 lb. Let me explain that. The 1.2 is the conversion from US gallons to imperial gallons, and the 10 is the density of water in pounds per imperial gallon. So we're converting US gallons to imperial gallons, then multiplying by the specific gravity to get the density of the fuel, then by 10 to get pounds. For oil: 8 divided by 4, divided by 1.2, times 0.9, times 10, equals 15 lb. Here the extra division by 4 converts US quarts to US gallons first, because there are 4 quarts in a gallon. So now we can build our moment table. Moment is mass times arm. For the Basic Mass: 1275 times -5 gives -6375. Seats 1 and 2: 340 times -2 gives -680. Seats 3 and 4: 170 times 30 gives 5100. Fuel: 210 times 2 gives 420. Oil: 15 times -48 gives -720. Baggage: 45 times 70 gives 3150. Now we sum the masses. 1275 plus 340 plus 170 plus 210 plus 15 plus 45 gives a Take-off Mass of 2055 lb. That's within the 2245 limit, good. Summing the moments: -6375 plus -680 plus 5100 plus 420 plus -720 plus 3150 gives a Take-off Moment of +895. Now, the CG position is the total moment divided by the total mass. So 895 divided by 2055 gives 0.435 inches aft of datum. And that's within our limits of 2 forward to 6 aft. So take-off is fine. Now for landing. The flight lasts 1 hour 30 minutes, which is 1.5 hours. Fuel used in that time is 1.5 times 7 gallons per hour, which is 10.5 gallons. Converting that to pounds: 1.5 times 7, divided by 1.2, times 0.72, times 10, equals 63 lb. Oil used: the consumption is 1.0 US quart per hour, so in 1.5 hours that's 1.5 quarts. Converting: 1.5 times 0.25, divided by 1.2, times 0.9, times 10, equals 2.8 lb. The 0.25 converts quarts to gallons. So the Landing Mass is the take-off mass minus fuel used minus oil used: 2055 minus 63 minus 2.8 equals 1989.2 lb. That's within the 2100 landing limit. Now the Landing Moment. We take the take-off moment and subtract the moment of the fuel used and the moment of the oil used. The fuel used has an arm of +2, so its moment is 63 times +2, which is 126. The oil used has an arm of -48, so its moment is 2.8 times -48, which is -134. So Landing Moment equals +895 minus 126 minus (-134). Now, minus and minus give plus, so that becomes +895 minus 126 plus 134, which equals +903. Finally, the Landing CG is the landing moment divided by the landing mass: 903 divided by 1989.2. That gives us the CG position for landing. So you can see the full process: convert volumes to masses, build the moment table, sum for take-off, then adjust for the fuel and oil burned to find the landing condition. The key is keeping track of the signs on the arms — negative arms are forward of the datum, positive arms are aft. And remember, when you subtract a negative moment, it adds.