= Chord-axial force coefficient
{title2=$C_A=A/(qS)$}
The \b[chord-axial force coefficient] is the dimensionless <chord-axial aerodynamic force>:
$$
C_A=\frac{A}{qS},\qquad q=\frac12\rho U^2>0.
$$
Here $S$ is the stated <aerodynamic reference area>, $\rho$ the reference <mass density> and $U$ the reference <speed>. The chord axis and its positive direction accompany the normalization. A signed coefficient can be negative even when the flow-relative <drag coefficient> is positive.
An original example uses $\rho=1.2$ kg/m$^3$, $U=25$ m/s, $S=0.40$ m$^2$ and aft-directed load $A=9$ N. Then $q=375$ Pa and $C_A=9/(375\cdot0.40)=0.060$. Reporting that load with area 0.80 m$^2$ instead gives 0.030. The <reference-area scaling of aerodynamic coefficients> changes the reported coefficient without changing the balance reading.
For a section, axial load per unit span $A'$ is instead normalized as $c_a=A'/(qc)$ using <airfoil chord> $c$. The denominator $qc$ has units N/m, whereas $qS$ has units N. A per-span load divided by a finite area does not produce the same dimensionless quantity.
With common references, the inverse <airfoil force-axis transformation> is
$$
C_A=C_D\cos\alpha-C_L\sin\alpha.
$$
For stipulated $C_D=0.040$, $C_L=0.60$ and $\alpha=5\,{}^\circ$, this gives $C_A\simeq-0.01245$. The subtraction shows why labeling a body-axis coefficient as drag can change even the sign of the apparent result. NASA defines lift and drag relative to the flow, which is the physical basis for keeping these axes distinct.<Aerodynamic Forces by NASA Glenn Research Center>[Definitions of Lift and Drag]{ref}
At fixed area, density and measured force, a small fractional speed error contributes approximately $-2\,\delta U/U$ to the coefficient's fractional error. Normalization consequently does not replace <measurement calibration> or an uncertainty assessment of the reference flow.
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