Definitions and Calculations — Page 50, Lesson 70

We’re starting a fresh topic now: repositioning the centre of gravity by moving mass. This is one of the most practical fixes in mass and balance — when your CG is out of limits, you don’t always have to offload or add weight. Sometimes you can just shift what you already have. Let me set the scene. In Figure 2.12, we have an aircraft whose centre of gravity has been found to be out of limits. It sits at a distance ‘a’ inches aft of the datum. The forward CG limit is ‘b’ inches aft of the datum. So the CG is too far aft — it’s beyond the forward limit, which sounds odd, but think of it this way: the forward limit is the most forward position the CG is allowed to be, and here the CG is further aft than that limit allows. To bring it back into limits, we move some baggage, mass m, from compartment A to compartment B. Here’s the key relationship. If a mass of m pounds is moved from A to B, the change of moment is m multiplied by d. That is, the change of moment equals the mass moved, m, times the distance through which it moves, d. The distance d is the arm difference between the two compartments — how far apart they are. Now, the total mass of the aircraft is M, and the CG is at ‘a’ inches aft of the datum. So the total moment around the datum is M times a. We want to move the CG to ‘b’ inches aft of the datum, so the new total moment will be M times b. The change in moment required is therefore M times b minus M times a, which is M times the quantity (b minus a). And here’s the elegant part: that required change in moment must equal the change we produce by moving the mass. So M(b − a) equals m × d. Now, b minus a is exactly the change in the CG position — we call that ‘cc’. So we can rewrite it as m × d equals M × cc. Let me say that in words, because it’s the whole point of the lesson: the mass you move, multiplied by the distance you move it, is equal to the total aircraft mass multiplied by the distance the CG moves through. That’s the formula we’ll use over and over. Let’s apply it with Example 5. The CG limits of an aircraft are from −4 to +3 inches from the datum. Note the minus sign — that means the forward limit is 4 inches forward of the datum, and the aft limit is 3 inches aft of the datum. The aircraft is loaded as shown. Basic Empty Mass is 2800 pounds at an arm of 2 inches, giving a moment of 5600 pound-inches. Crew is 340 pounds at an arm of −20 inches, moment −6800. Fuel is 600 pounds at arm 10, moment 6000. Forward Hold is 0 pounds at arm −70, moment 0. Aft Hold is 150 pounds at arm 80, moment 12,000. Adding it all up: total mass is 3890 pounds, total moment is 16,800 pound-inches. The CG is total moment divided by total mass — 16,800 divided by 3890, which is 4.32 inches. Now, the aft limit is +3 inches. Our CG is at 4.32, so it’s 1.32 inches out of limits — too far aft. We can correct it by moving some freight or baggage from the rear hold to the forward hold. The distance between them is 150 inches — from 80 inches aft to 70 inches forward, that’s a separation of 150 inches. We use our formula: m × d equals M × cc. So m times 150 equals 3890 times 1.32. Solving for m, we get 3890 times 1.32 divided by 150, which is 34.232 pounds. So we need to move 34.232 pounds of freight and/or baggage from the aft hold to the forward hold. But we don’t stop there. We need to check the aircraft is safe for all fuel states after take-off. So we calculate the CG at the Zero Fuel Mass — that’s the mass of the aircraft with no fuel, which is the critical state because as fuel burns off, the CG shifts. We move 35 pounds of baggage to the forward hold. Now the forward hold has 35 pounds at arm −70, moment −2450. The aft hold is down to 115 pounds at arm 80, moment 9200. Fuel is zero. Basic Mass and Crew stay the same. The Zero Fuel Mass is 3290 pounds, and the ZFM moment is 5550 pound-inches. The ZFM CG is 5550 divided by 3290, which is 1.69 inches — and that’s in limits. So the whole process is: find the CG, compare it to the limits, calculate how much mass to move using m × d = M × cc, then verify the result at the critical fuel state. That’s the complete method for repositioning the CG by moving mass.