Definitions and Calculations — Page 58, Lesson 74

We're moving into a new part of the mass and balance work now — repositioning the centre of gravity by adding or subtracting mass. This is a really practical skill, so let's build it up properly. First, the key idea. We've already seen how to move the CG by shifting load between compartments. But there's another way to adjust the CG position: you can add mass, or remove mass. Now, when mass is added simply to reposition the CG — not because you need to carry that load for any other reason — that mass is given a specific name. It's called ballast. So ballast is just dead weight you put on board for the sole purpose of moving the centre of gravity back into limits. Let me set up the geometry, because the notation matters. Imagine the datum, and the CG has been found to be out of limits at a distance 'X' aft of the datum. The forward CG limit is at a distance 'Y' aft of the datum. So the CG is sitting too far aft — beyond the forward limit — and we need to pull it forward. To do that, we're going to put ballast in compartment B, which is at a distance 'Z' aft of the datum. Now, the total mass of the aircraft is M, in pounds. The total moment is therefore M times X, in pound-inches. That's the old state. If we place ballast of mass m, in pounds, into compartment B to move the CG to its forward limit, then the total mass increases to M plus m, and the new total moment becomes (M + m) times Y. Here's the physical principle that holds it all together: 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 moments all add up. So algebraically, using that notation, we write: (M + m) × Y = (M × X) + (m × Z) Let me read that out in words. The new total moment, (M + m) times Y, equals the old total moment, M times X, plus the cargo moment, m times Z. So the new total moment equals the old total moment plus the cargo moment. Now, 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 this master formula: New Total Moment = Old Total Moment ± Cargo Moment. Plus when you're adding mass, minus when you're removing it. One critical note before we do an example. When you're calculating a change in CG position using this formula — New Moment = Old Moment ± Change in Moment — the distances X, Y, and Z are always measured from the datum itself. Not from the CG, not from the limit — always from the datum. That's a trap people fall into, so keep it firmly in mind. Let's work through Example 6 together, because this is where it all comes alive. The aircraft's CG limits are from 84 inches to 96 inches aft of datum, at all masses. So the forward limit is 84 inches, the aft limit is 96 inches. It's loaded as follows. Basic mass: 1250 pounds, arm 80 inches, moment 100,000 pound-inches. Crew: 340 pounds, arm 82 inches, moment 27,880. Fuel: 300 pounds, arm 72 inches, moment 21,600. Baggage: 0 pounds, arm 140 inches, moment 0. So the total mass is 1890 pounds, and the total moment is 149,480 pound-inches. Now we find the CG. CG equals total moment divided by total mass — 149,480 divided by 1890, which gives 79.1 inches aft of datum. Compare that to the limits. The forward limit is 84 inches, and the CG is at 79.1 inches. So the CG is out of limits — it's 4.9 inches too far forward. 84 minus 79.1 is 4.9. So we need to bring it back into limits, and we'll do that by putting ballast in the baggage compartment. The baggage compartment arm is 140 inches — that's our Z. Now, the minimum ballast required is the amount that brings the CG exactly to the forward limit, 84 inches. That's our Y. So let's plug into the formula. New Total Moment = Old Total Moment + Cargo Moment. (1890 + m) × 84 = (1890 × 79.1) + (m × 140) Let me expand that. On the left: 1890 times 84 is 158,760, plus 84m. On the right: 1890 times 79.1 is 149,499, plus 140m. So we have 158,760 + 84m = 149,499 + 140m. Now we rearrange. Bring the constants together: 158,760 minus 149,499 equals 9261. And bring the m terms together: 140m minus 84m equals 56m. So 9261 = 56m. Divide both sides by 56: 9261 divided by 56 equals 165.4. So m = 165.4 pounds. The mass of ballast required is 165.4 pounds. Now, two checks are essential after this. First, you must verify that loading this ballast doesn't push the total mass over the Maximum Take-off Mass. And second, as always, you must confirm the aircraft is still within limits for landing. Those are the operational safety checks that follow any ballast calculation. Now, I know this method looks long-winded — it's a lot of algebra. But here's the thing: it will always give you the correct answer. And you should remember that you might be asked to calculate any one of three things: the mass to add or remove, the resulting change to the CG, or the position where you need to put the ballast. The same formula handles all three — you just solve for whichever unknown you're given. Let me show you the diagram that goes with this — it's Figure 2.15, "Adding or Removing Mass," which lays out exactly the geometry we just used: the CG at X, the forward limit at Y, and the ballast compartment at Z. So the takeaway for this whole technique: ballast is mass added purely to reposition the CG, the governing equation is New Total Moment = Old Total Moment ± Cargo Moment, all distances are measured from the datum, and you always finish by checking take-off mass and landing limits.