Definitions and Calculations — Page 58, Lesson 74

Let’s pick up right where the CG repositioning story gets practical. We’ve already seen how moving load between holds shifts the centre of gravity. Now I want to walk you through the second tool in the box: adjusting the CG by adding or subtracting mass. And the professional name for mass added purely to reposition the CG is ballast. Here’s the setup. In Figure 2.15, the CG has been found to be out of limits, sitting at a distance ‘X’ inches aft of the datum. The forward CG limit is at a distance ‘Y’ inches aft of the datum. To bring the CG back into limits, we’re going to put ballast into compartment B, which is at a distance ‘Z’ inches aft of the datum. Let me give you the notation before the algebra, because everything hangs on it. The total mass of the aircraft is M lb. The total moment is M × X lb in — that’s mass times arm, so pound-inches. If we place ballast of m lb in compartment B to move the CG to its forward limit, the total mass becomes M + m, and the new total moment becomes (M + m) × Y. Now the key physical principle: assuming equilibrium is maintained, the original total moment plus the moment of the added mass must equal the new total moment. That’s the conservation of moment — the sum of the old moment and the cargo moment has to balance the new total moment. Algebraically, that gives us: (M + m) × Y = (M × X) + (m × Z) New Total Moment = Old Total Moment + Cargo Moment. And here’s a nice symmetry: the same formula works for removing mass, you just change the plus sign to a minus. So for any calculation involving adding or subtracting mass, remember the master formula: New Total Moment = Old Total Moment ± Cargo Moment. One critical note before we calculate: when you use this formula, the distances X, Y and Z are always measured from the datum itself. Not from the CG, not from the limit — from the datum. That’s a trap that catches people, so hold onto it. Let’s do a worked example together — Example 6. The CG limits of an aircraft are from 84 inches to 96 inches aft of datum at all masses. The load sheet is: - Basic Mass: 1250 lb, arm 80 in, moment 100,000 lb in - Crew: 340 lb, arm 82 in, moment 27,880 lb in - Fuel: 300 lb, arm 72 in, moment 21,600 lb in - Baggage: 0 lb, arm 140 in, moment 0 lb in Total mass is 1890 lb, total moment is 149,480 lb in. So the CG is 149,480 divided by 1890, which gives 79.1 inches aft of datum. Now compare that to the limits. The forward limit is 84 inches, and our CG is at 79.1 inches. So the CG is out of limits by 4.9 inches — too far forward. To fix it, we put ballast in the baggage compartment, which is at arm 140 inches. The minimum ballast is the amount that brings the CG exactly to the forward limit of 84 inches. Plug into our formula. New Total Moment = Old Total Moment + Cargo Moment: (1890 + m) × 84 = (1890 × 79.1) + (m × 140) Expand the left: 158,760 + 84m. The right: 149,499 + 140m. (Note: 1890 × 79.1 gives 149,499, not the rounded 149,480 from the load sheet — that’s the precise product.) Now rearrange: 158,760 − 149,499 = 140m − 84m. That gives 9261 = 56m. Divide both sides by 56: m = 9261 ÷ 56 = 165.4 lb. So the mass of ballast required is 165.4 lb. Now, two checks before you sign that off. First, you must verify that loading that ballast doesn’t push the total mass above the Maximum Take-off Mass. Second, as always, confirm the aircraft is within limits for landing. Those are the operational gates. I know this looks long-winded, but here’s the payoff: this method always gives the correct answer. And the examiner can ask you three different things with it — the mass to add or remove, the resulting change to the CG, or the position where you must place the ballast. The same formula handles all three.