9.weight-shift-formulas. Weight Shift Formulas
When the loaded center of gravity (CG) of an airplane falls outside the approved envelope, or when the pilot wants to predict how moving cargo, baggage, or passengers will affect the CG, weight-shift formulas provide a quick mathematical solution. These formulas come directly from the principle of moments (Weight × Arm = Moment) and are presented in Chapter 9 of the Pilot's Handbook of Aeronautical Knowledge.
The Basic Weight-Shift Formula
When weight is moved from one location to another within the airplane, the total weight does not change, but the CG shifts. The relationship is expressed as:
Weight Shifted / Total Weight = Distance CG Moves / Distance Between Arms
Rearranged to solve for the change in CG:
ΔCG = (Weight Shifted × Distance Between Arms) / Total Weight
Where:
- Weight Shifted = the amount of weight moved (lb)
- Distance Between Arms = the distance between the old and new station (in)
- Total Weight = the airplane's total loaded weight (lb)
- ΔCG = the distance the CG moves (in)
The new CG is then found by adding ΔCG to the old CG if the weight is moved aft, or subtracting if moved forward.
Example 1 — Shifting Baggage
An airplane has a loaded weight of 4,000 lb and a CG located at station 80.0 in. Suppose 100 lb of baggage is moved from the forward baggage compartment at station 55.0 to the aft compartment at station 145.0.
- Distance between arms = 145.0 − 55.0 = 90.0 in
- ΔCG = (100 × 90) / 4,000 = 9,000 / 4,000 = 2.25 in aft
- New CG = 80.0 + 2.25 = 82.25 in
Because the weight moved aft, the CG also shifted aft by 2.25 inches.
Weight to Be Shifted to Move CG a Specified Distance
If a known CG correction is required (for example, to bring the CG within limits), the formula can be solved for the weight that must be shifted:
Weight Shifted = (Total Weight × ΔCG) / Distance Between Arms
Example 2 — Bringing CG Into Limits
A 3,200 lb airplane has a CG at 86.0 in, but the aft limit is 85.5 in. The CG must be moved 0.5 in forward. Available stations are the aft baggage compartment at 130.0 in and the forward baggage compartment at 60.0 in (distance = 70.0 in).
- Weight Shifted = (3,200 × 0.5) / 70.0 = 1,600 / 70 = 22.9 lb
Moving approximately 23 lb of baggage from the aft to the forward compartment will bring the CG within limits.
Weight Addition or Removal Formula
When weight is added or removed rather than shifted, total weight changes and a different formulation is used:
ΔCG = (Weight Added or Removed × (Arm − Old CG)) / New Total Weight
For weight removed, treat the weight as negative. The new CG is the old CG ± ΔCG.
Example 3 — Removing Cargo
A 2,800 lb airplane has a CG at 78.0 in. 150 lb of cargo is removed from station 120.0 in.
- New Total Weight = 2,800 − 150 = 2,650 lb
- ΔCG = (−150 × (120.0 − 78.0)) / 2,650 = (−150 × 42) / 2,650 = −6,300 / 2,650 = −2.38 in
- New CG = 78.0 − 2.38 = 75.62 in (CG shifts forward, away from the removed weight)
Procedure Summary
- Determine the airplane's current weight and CG from the loading graph or computation.
- Identify the weight to be moved, added, or removed and the relevant station arms.
- Apply the appropriate formula.
- Compute the new CG and verify it lies within the published forward and aft limits at that gross weight.
- Recheck after any in-flight fuel burn that significantly changes weight distribution.
Practical Considerations
- Always reference station arms from the airplane's specific weight and balance section in the POH/AFM; arms differ between makes and models.
- A useful check: weight shifted aft moves the CG aft; weight removed from a station moves the CG away from that station.
- Pilots are responsible under 14 CFR 91.9 and 91.103 for ensuring the airplane is operated within its weight and CG limits before every flight.
- Shifting passengers in flight (e.g., asking a rear-seat passenger to move forward) is a legitimate technique, but the new loading must still be within limits and any seat belt/restraint requirements observed.
Mastering these three short formulas — shift, weight-to-shift, and add/remove — gives a pilot the tools to solve virtually any practical loading problem on the ground or in the cockpit.