# Application point of the lift force and pitching moment

#### fog37

Hello

As the angle of attack of an airfoil changes, the lift force $L$ changes both in magnitude and position (not in direction, always upward). The force location is a point called the center of pressure $x_{cp}$. It is possible to transfer the force to any another different point along the chord (even if it is always physically applied to $x_{cp}$). But that transfer implies that we add a pure couple moment $M$. However, as we change the AoA, the pure couple moment $M$ varies, i.e. it is dependent on the AoA even if we keep the reference point about which we calculate the moment fixed.

However, there is a special point called the aerodynamic center $x_{ac}$, located at a quarter chord distance from the leading edge $LE$. If the lift force $L$ is transferred to $x_{ac}$ as its application point, this pure couple moment $M_{ac}$ does NOT vary with AoA. Is that correct? This pure couple moment is always nonzero, even when $L=0$, for any AoA, and always equal to zero for a symmetric airfoil.

I believe, not sure if correct though, that if we leave the lift force $L$ at the center of pressure $x_{cp}$ and calculate the moment of the lift force $L$ about the aerodynamic center, with the lever arm being the distance $|x_{cp} - x_{ac}|$, the resulting moment
$|M|$ = $L |x_{cp} - x_{ac}|$ remains constant and is independent of the angle of attack even if both the distance $|x_{cp} - x_{ac}|$ and $|L|$ change with AoA . Is that correct? Also, this moment happens to be equal to the constant pure couple moment $M=M_{ac}$ described in the first paragraph...

Thanks and any clarification and validation!!

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#### fog37

Hopefully, just to add clarity to my question, here another picture illustrating the lift force $L$ applied at the center of pressure $CP$, which varies with AoA. If the pitching moment due to $L$ is calculated about the aerodynamic center $AC$, the moment remains constant: even if the magnitude of $L$ increases from $L_1$ to $L_2$, the lever arm $d$ decreases from $d_1$ to $d_2$ keeping the moment constant...

The second figure shows how the force can be moved to different points of applications causing the moment $M$ to change depending on the application point. But only when the force is applied at c/4 the moment remains constant with AoA.

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