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As far as I know, the pitching moment is an aerodynamic moment and is related to the aerodynamic center.
But I am confused with the following figure, which shows the aerodynamic force applied at the aerodynamic center. So I would think, in this case, that the center of pressure is located at the aerodynamic center which is a quarter chord. If so, why is there a moment ?
The definition of the aerodynamic center is the following: "the aerodynamic center is the point along the wing chord where the coefficient of aerodynamic moment is constant with respect to the AOA." It doesn't mean that it is equal to 0.
To make it simple, imagine taking a cambered wing, if you look at the center-line of the profile, it will have a changing AOA along the chord of the profile. Setting the wing at a 0° AOA will result for a positively cambered profile into a negative AOA and a negative lift near the leading edge and a positive AOA and lift near trailing edge. Thus creating an absolute torque pitching down the wing profile. The aerodynamic center is the point where the coefficient related to this torque is not impacted by the AOA.
The point you are referring to is the center of lift ($P$). It's the point where the aerodynamic torque is equal to 0. But this point location is changing with respect to the AOA. You have the following relation:
$$C_{m_F} = C_{m_P}-(x_P-x_F)\cdot C_Z \rightarrow x_P = x_F - \frac{C_{m_F}}{C_Z}$$
Which means that for a classic wing with positive camber and thus negative torque, the center of lift is located after the aerodynamic center in cruise condition with $C_z > 0$.
It is more practical to always express the aerodynamic load at a given fixed point, the aerodynamic center with the couple (lift, torque) than expressing just the lift at the lift center which is moving with respect to flight conditions.
