Moment Coefficient⚠ unverified
Aerospace / Aerodynamics · Compute the pitching moment coefficient about the centre of gravity
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| CL | CL | — | 1.0 | Lift coefficient (dimensionless) |
| x_cg | xcg | m | 1.0 | Centre-of-gravity location |
| x_ac | xac | m | 1.0 | Aerodynamic-centre location |
| c | c | m | 1.0 | Reference chord length |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | CM | — | Moment coefficient about the centre of gravity (dimensionless) |
The science & history
Understanding the Parameters
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CG–AC offset $(x_{cg} - x_{ac})$ — the static margin in dimensional form. Its sign determines stability: with the CG ahead of the AC ($x_{cg} < x_{ac}$), a lift increase produces a nose-down (restoring) moment → statically stable. CG behind the AC → unstable. This is the single most important relationship in airplane balance.
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Aerodynamic center $x_{ac}$ — the special chordwise point (≈ quarter-chord for subsonic airfoils) about which the pitching moment does not change with angle of attack. Referencing lift there is what makes the transfer term clean.
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Reference chord $c$ — the mean aerodynamic chord, non-dimensionalising the moment arm so $C_M$ is a coefficient. Normalised, $(x_{cg}-x_{ac})/c$ is the static margin (in fractions of chord).
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Sign conventions — a consistent datum and sign convention are essential; the registry's raw $(x_{cg}-x_{ac})/c$ takes positive aft, so watch the sign against your stability convention.
Derivation (Approaching a Proof)
The wing's lift $L$ effectively acts at the aerodynamic center, a distance $(x_{cg} - x_{ac})$ from the CG. The moment of that force about the CG is force times arm:
$$M_{\text{lift}} = L\,(x_{cg} - x_{ac}).$$
Non-dimensionalise by the standard pitching-moment reference $q\,S\,c$ (dynamic pressure × area × chord), and use $L = q S C_L$:
$$C_{M,\text{lift}} = \frac{M_{\text{lift}}}{q S c} = \frac{q S C_L\,(x_{cg}-x_{ac})}{q S c} = C_L\,\frac{x_{cg}-x_{ac}}{c}.$$
Adding the constant airfoil moment about the AC gives the full result $C_{M,cg} = C_{M,ac} + C_L(x_{cg}-x_{ac})/c$. Differentiating with respect to $\alpha$ (through $C_L$) yields the stability derivative $dC_M/d\alpha = a\,(x_{cg}-x_{ac})/c$, negative (stable) when the CG is ahead of the AC.
Dimensional check. $C_M = C_L\,\dfrac{x_{cg}-x_{ac}}{c} = (\text{–})\cdot\dfrac{\text{m}}{\text{m}}$ = dimensionless — a coefficient, as required.
History and Development
The transfer of lift to the CG and the resulting stability criterion (CG ahead of the aerodynamic center) are foundational to aircraft longitudinal static stability, formalised as aeronautics matured in the early 20th century. Expressing it through the aerodynamic center — the point of angle-of-attack-invariant moment — is the elegant simplification that makes balance and tail-sizing tractable, and it connects directly to the Static Margin.
Related Concepts: Pitching Moment, Static Margin, Lift Force, Lift Curve Slope, Downwash Angle, Lift-to-Drag Ratio
Notes: Lift-transfer term only — total $C_{M,cg} = C_{M,ac} + C_L(x_{cg}-x_{ac})/c$ (registry omits the constant $C_{M,ac}$). $(x_{cg}-x_{ac})/c$ = static margin; CG ahead of AC → stable (nose-down restoring). AC ≈ quarter-chord. Watch sign convention.