Weight, balance, and CG

One formula, three words

Every weight-and-balance problem is the same piece of arithmetic. Pick a starting line, measure how far each item sits from it, multiply, add, divide. The words for the pieces are the whole vocabulary. The datum is the imaginary line all measurements start from; the manufacturer chooses it, and on the beam below it is the nose. The arm is an item’s distance from the datum, in inches. The moment is weight times arm, a measure of how hard that item twists the aircraft about the datum. FAA-H-8083-25C, Ch. 10

The center of gravity (CG) is the point where the loaded aircraft would balance if you hung it from a string. You find it by dividing the total moment by the total weight, and the answer is itself an arm: a distance from the datum. FAA-H-8083-25C, Ch. 10

Now do it with the beam’s own numbers. The airframe weighs 20 lb and sits at an arm of 20 in, so its moment is 20 × 20 = 400. The battery weighs 6 lb at 22 in, so 6 × 22 = 132. Together, with no payload at all, that is 26 lb and 532 lb·in of moment, and the CG is 532 ÷ 26 = 20.46 in. Now add a 4 lb payload sitting at 12 in. It brings 4 × 12 = 48 more moment. The totals become 30 lb and 580 lb·in, and the CG is 580 ÷ 30 = 19.33 in. The payload sat ahead of the old balance point, so it dragged the balance point forward.

The beam's default loading, datum at the nose

Airframe	20 lb × 20 in = 400 lb·in
Battery	6 lb × 22 in = 132 lb·in
Payload	4 lb × 12 in = 48 lb·in
Totals	30 lb and 580 lb·in
CG	580 ÷ 30 = 19.33 in aft of the datum

What the limits protect

The manufacturer publishes a forward and an aft CG limit. Inside them the aircraft behaves as designed; outside, the FAA says plainly, you get control difficulty. FAA-H-8083-25C, Ch. 10 The handbook is written for manned airplanes, but the logic of each end is worth knowing.

Forward of the limit, the aircraft is nose-heavy. It is more stable but harder to control at low speed, with higher stalling speed and higher control forces, and in extreme cases an airplane cannot be flared for landing because there is not enough control deflection left to raise the nose. FAA-H-8083-25C, Ch. 10 Aft of the limit, the aircraft is tail-heavy. Stability drops, so it is slower to right itself after a gust, and the handbook lists extreme control difficulty, violent stall characteristics, and very light control forces that make it easy to overstress the aircraft without noticing. FAA-H-8083-25C, Ch. 10 A stall is the wing exceeding its critical angle of attack and losing lift. FAA UAG Sample Questions, Q21 A tail-heavy airplane that stalls can fall into a spin it cannot recover from.

A multirotor has no wing to stall, and the FAA text is silent on it, so what follows is reasoning, not a quotation. A multirotor hovers by splitting thrust among its motors so that the sum holds up the weight and the turning effects cancel. If the CG sits off-center, some motors must work harder than others just to stay level. That spends control authority on holding still, leaving less to correct for wind, and a motor already near full throttle cannot give the extra thrust the flight controller asks for. It saturates, and the aircraft tips toward that side.

Moving a payload: the proportion rule

Here is the part that makes the beam predictable. Slide a payload by some distance and the CG moves by only a fraction of that distance, and the fraction is the payload’s share of the total weight.

CG shift = payload weight × distance moved ÷ total weight

On the beam, the 4 lb payload sits in a 30 lb aircraft, so every inch you slide it moves the CG by 4 ÷ 30 = 0.13 in. Slide it 10 in aft and the CG moves about 1.33 in. A heavier payload takes a bigger share and moves the CG further per inch; a lighter one, less. That is also why a tiny sensor on a long boom barely moves the CG while a large battery pack moved a few inches can push you out of limits.

You can use it to find the edge of the limits in advance. The aft limit on the beam is 21.5 in, so the total moment must not exceed 21.5 × 30 = 645. The airframe and battery contribute 532, leaving 113 for the 4 lb payload, which makes its arm 113 ÷ 4 = 28.25 in. Past about 28 in, you are aft of the limit. The same sum at the forward limit of 18.5 gives 555 minus 532 = 23, and 23 ÷ 4 = 5.75 in. Ahead of about 6 in, you are forward of it.

The manual is the authority

The beam’s limits are invented for the exercise. Real ones come from the manufacturer, and so does the way to stay inside them: the aircraft loading instructions in the flight manual or pilot’s operating handbook. Do not try to work out limits yourself, and do not borrow them from a similar aircraft. FAA UAG Sample Questions, Q28 · FAA-H-8083-25C, Ch. 10

The regulation puts the duty on you. Before flight, the remote pilot in command must ensure that any object attached or carried by the aircraft is secure and does not adversely affect its flight characteristics or controllability. 14 CFR § 107.49(e) A payload that shifts in flight changes the CG mid-flight, and a payload that pushes the CG out of limits changes how the aircraft flies. Either way that sentence is the rule you broke.

Load factor in a turn

Weight is not the only thing the structure carries. When an aircraft banks and holds its altitude, the wings have to make more lift than the weight, because part of the lift is now pointed sideways to turn the aircraft. The load factor is the ratio of lift to weight, measured in Gs. In a coordinated level turn it is 1 ÷ cos(bank angle). FAA-CT-8080-2H, Fig. 2 · FAA UAS Study Guide, Ch. 4 (Load Factors)

In level flight the load factor is 1.0: the structure carries exactly the weight. At 30° of bank, cos 30° is 0.866 and the load factor is 1.15. At 45° it is 1.41. At 60° it is 2.0, and the study guide notes it climbs “at a terrific rate” after 45° or 50°. FAA UAS Study Guide, Ch. 4 (Load Factors in Steep Turns) Higher load factors raise the stall speed too, in proportion to the square root of the load factor, so a steep turn can stall the wing at a speed that looked safe in straight flight. FAA UAS Study Guide, Ch. 4 (Load Factors and Stalling Speeds)

Load factor in a level turn, 1 ÷ cos(bank)

0° (level)	1.00 G	FAA-CT-8080-2H, Fig. 2
30°	1.15 G	FAA-CT-8080-2H, Fig. 2
45°	1.41 G	FAA-CT-8080-2H, Fig. 2
60°	2.00 G	FAA-CT-8080-2H, Fig. 2
70°	2.92 G	FAA-CT-8080-2H, Fig. 2

The sample question uses the same arithmetic. A 33 lb airplane in a 30° banked turn that holds altitude: 33 × 1.154 = 38.1 lb, so the structure supports about 38 lb. FAA-CT-8080-2H, Fig. 2 · FAA UAG Sample Questions, Q7

Using the beam

Three experiments, each a minute. Start with the defaults and watch the readout as you go.

First, slide the payload aft until the readout flips from within to AFT of limit. With 4 lb you should find the edge between 28 and 28.5 in, matching the 28.25 you computed above. Then do the same moving forward and find it near 6 in. Notice how far the payload moved compared with how far the fulcrum moved. That ratio is the proportion rule.

Second, put the payload at 28 in with its weight at 4 lb, just inside the aft limit. Now raise the weight to 6 lb without touching the position. The spot did not change, but the CG left the limits: more weight at the same arm is more moment, and the payload’s share of the total grew. Check the readout’s moment total against 21.5 times the total weight.

Third, set the bank-angle slider to 60° and read the load. The beam shows the wings carrying twice the weight. Then drag the payload weight to 12 lb and watch the structural load climb with it: the same bank costs more pounds on a heavier aircraft. A bank that is gentle at 30° is not the same animal at 60°.

How the test asks it

What to take away

Weight and balance is one division: total moment over total weight. The limits are the manufacturer’s, found in the manual, and the beam below showed you the shape of the rule: a payload’s pull on the CG grows with its weight and its distance, so a small weight far from the CG matters and a big weight near it can be forgiven. Bank angle is the second half of the story, because the structure carries more than the weight the moment you turn.

Thin air changes the same margins from another direction, so read Density altitude next. For the full chapter on loading and performance, including the numbers the test favours, see Loading and performance.