Pitching moment
= Pitching moment
{title2=$M$}
A \b[pitching moment] is a <torque> about an <aircraft pitch axis>, associated with rotation of the nose upward or downward. It requires a reference point, an axis and a sign convention; the numerical value is not an intrinsic property of a <force> alone.<Aircraft Rotations by NASA Glenn Research Center>[Pitch axis and control-surface torques]{ref}
For a planar chord-line example, take $x$ positive aft along the <airfoil chord>, $z$ positive toward the upper surface, and a nose-up moment as positive. An upward <chord-normal aerodynamic force> $N$ applied at $x_f$ has moment about a chord-line reference $x_0$,
$$
M(x_0)=-(x_f-x_0)N.
$$
Thus a stipulated upward $N=100\,\mathrm N$ acting $0.050\,\mathrm m$ behind the <leading edge> gives \b[$M(0)=-5.0\,\mathrm{N\,m}$], a nose-down moment. Moving the reference aft to $x_0=0.020\,\mathrm m$ gives $-3.0\,\mathrm{N\,m}$ without changing the <force>.
For a distributed aerodynamic load, the same translation law gives
$$
M(x_B)=M(x_A)+(x_B-x_A)N.
$$
Dividing by the same $qSc$ yields the <reference-point transformation of a pitching moment coefficient>. Replacing $N$ by <lift> is an approximation unless the body and flow axes coincide; with a stated <angle of attack>, the full <vector> transformation accounts for <drag> as well.
Net <force> can vanish while a <pitching moment> remains. Two stipulated chord-normal forces, $+100\,\mathrm N$ at $x=0.20\,\mathrm m$ and $-100\,\mathrm N$ at $x=0$, have $N=0$ and $M=-20\,\mathrm{N\,m}$. They form a <couple (mechanics)>; no finite location of a single zero resultant represents that load.
When $N\ne0$ and a chord-line resultant is meaningful, its <center of pressure (fluid mechanics)> satisfies $x_{\rm cp}=x_0-M(x_0)/N$. This zero-moment location differs from an <aerodynamic center>, where moment is approximately insensitive to <angle of attack> in a specified regime. The latter can have a nonzero moment, especially for a cambered <airfoil>.<Aerodynamic Center by NASA Glenn Research Center>[Aerodynamic Force and Airfoil]{ref}