14.2 Moment, CG, and Sample Computations

ACS PA.I.F.S1 is not “know the formula.” It is compute the weight and balance, correct an out-of-limit loading, and decide whether the result is inside the published envelope. FAA-H-8083-1 and PHAK Chapter 10 do that with a table: one row per item, three columns (weight, arm, moment), a total row, and a division. The knowledge test will hand you a table or a loading graph that is the same idea in picture form.

This section uses one generic four-place trainer so the arithmetic is checkable. It is not a substitute for any real AFM. Limits for the examples:

• BEW 1,500 lb at 39.0 in (full oil already in BEW)
• Maximum ramp 2,557 lb; maximum takeoff 2,550 lb; maximum landing 2,450 lb; maximum baggage 120 lb
• Front-seat arm 37.0 in, rear-seat arm 73.0 in, baggage 95.0 in, usable fuel 48.0 in
• Avgas at 6 lb/gal
• Envelope for these weights: forward limit 35.5 in (light) expanding to 39.5 in at 2,550 lb; aft limit 47.3 in at all weights

The table method, every time

1. Start with the current BEW and empty-weight moment from the airplane’s record.
2. Convert fuel gallons to pounds (× 6).
3. Enter each occupant group and each baggage compartment at its published arm. Use actual body weights when you have them.
4. Moment = weight × arm on every line. Add the weight column. Add the moment column.
5. CG = total moment ÷ total weight, then round only at the end (tenths of an inch is the usual training precision).
6. Compare total weight with ramp / takeoff / landing limits, and CG with the envelope at that weight.

Example 1 — two-up local flight

CG = 87,350 ÷ 2,130 = 41.0 in (87,350 ÷ 2,130 = 41.009…).

Takeoff weight 2,130 is 420 lb under 2,550. At this weight the forward limit is well forward of 41.0 and the aft limit is 47.3, so the point is inside. Useful load used is 340 + 50 + 240 = 630 lb of the 1,050 lb available.

Add 10 gallons of fuel (a common “we decided to go farther” move). Ten gallons are 60 lb at arm 48.0, moment +2,880.

• New weight = 2,130 + 60 = 2,190 lb
• New moment = 87,350 + 2,880 = 90,230 lb-in
• New CG = 90,230 ÷ 2,190 = 41.2 in

The fuel arm (48.0) is aft of the old CG (41.0), so adding fuel walked the CG aft 0.2 in — toward the tank. Still inside. Still under 2,550.

Remove the 50 lb baggage from the original 2,130-lb loading, instead of adding fuel. Baggage moment was 4,750.

• New weight = 2,130 − 50 = 2,080 lb
• New moment = 87,350 − 4,750 = 82,600 lb-in
• New CG = 82,600 ÷ 2,080 = 39.7 in

The bag sat at 95.0, far aft of 41.0, so taking it off walked the CG forward 1.3 in. Same airplane, opposite direction from the fuel add. That is PA.I.F.S1: change one line, recompute both totals, read the new inch.

Example 2 — four souls, legal CG, illegal weight

CG = 120,704 ÷ 2,608 = 46.3 in (46.282…).

Aft limit 47.3 — the inch is inside. Maximum takeoff 2,550 — the weight is 58 lb over. A pretty CG does not bless an overweight airplane. You must offload 58 lb before brake release; the examples below take 60 lb so the gallons stay even.

Fix A — drain 10 gal (60 lb) of fuel at arm 48.0, moment −2,880:

• Weight 2,548 lb, moment 117,824, CG 46.2 in (117,824 ÷ 2,548 = 46.242)

Weight is now 2 lb under 2,550. CG barely moved, because the fuel arm (48.0) was close to the already-aft 46.3-in CG.

Fix B — remove 60 lb of baggage at arm 95.0, moment −5,700, leave the fuel alone:

• Weight 2,548 lb, moment 115,004, CG 45.1 in (115,004 ÷ 2,548 = 45.135)

Same 60 lb off the airplane, 1.1 in more forward than Fix A, because 95.0 is much farther from the old CG than 48.0 is. Choose the offload that also keeps you inside baggage and fuel-reserve rules — but computationally, where the pounds come off matters as much as how many.

Landing check after Fix A, 30 gal burned en route (180 lb at 48.0, moment −8,640):

• Landing weight 2,368 lb, moment 109,184, CG 46.1 in
• 2,368 is under the 2,450-lb landing limit; 46.1 is still forward of 47.3

Fuel burn at a tank near the current CG hardly walks the inch. It still has to be computed if the envelope is tight.

Graph method versus table method

The computational table above is the source of truth. The graph method in most light-airplane POHs is the same math drawn as two pictures:

1. A loading graph: enter an item’s weight, read its moment index (moment ÷ 100 or ÷ 1,000).
2. A CG envelope (moment envelope): plot total weight against total index. If the point is inside the polygon, both weight and CG are legal.

For Example 1 with a ÷1,000 index: empty 58.50 + front 12.58 + baggage 4.75 + fuel 11.52 = 87.35. Then CG = (87.35 × 1,000) ÷ 2,130 = 41.0 in — identical. Use the reduction factor printed on that chart. Do not mix a ÷100 index with a ÷1,000 index. Do not read the loading graph in gallons.

How to decide if you are inside

Ask two questions, in either order, and require both yes:

• Is total ramp / takeoff / landing weight at or below the published maximum for that condition?
• At that weight, is CG at or between the forward and aft limits (and inside any side-of-envelope cutoffs, utility-category cutouts, or zero-fuel-weight limits the AFM draws)?

If either answer is no, change the load — offload, shift bags, move a person, or leave fuel in the truck — and recompute. Guessing that “the rear seats will pull it back enough” is not a method.

Scenario: Maya’s full-tank family trip

Maya loads Example 2, sees 46.3 in, and says “CG is fine.” It is. The airplane is still 58 lb over maximum takeoff. She drains 10 gallons, recomputes 2,548 lb and 46.2 in, and only then opens the takeoff-distance chart at that weight. The chart from Chapter 13 does not apply to the 2,608-lb airplane she almost taxied out.