We know from previous posts that lift keeps an airplane aloft, while drag acts against an airplane’s direction of motion, thereby impeding its ability to fly. However, we can deepen our understanding of lift and drag by exploring the following concepts.

The only forces in a fluid are pressure forces and shear forces. In the case of an airplane wing flying through air, pressure forces act normally (perpendicularly) to the wing, and shear forces act tangentially to it. The net force on the wing due to all pressure forces is often called the “resultant force,” denoted by a bold **R**. This **R** force vector points upwards and to the rear of a wing. In turn, engineers can label the upward component of the **R **vector as “lift” and the rearward component of the **R** vector as “drag.” Therefore, lift and drag are components of the “resultant force,” rather than independent forces. We can also write detailed equations for lift and drag:

L = ½ ρ u^{2 }S C_{L} and D = ½ ρ u^{2 }S C_{D}.

In these equations, the variable ρ represents the density of air, u represents the velocity of the plane, S represents the wing’s planform (or two-dimensional) area, and C_{L} and C_{D} represent the coefficients of lift and drag, respectively. These equations tell us that if an airplane flies faster, it experiences both greater lift and greater drag. Meanwhile, the magnitudes of lift and drag can increase if a plane has larger wings, flies at lower altitudes where air is denser, or if C_{L} or C_{D} increase.

One important question engineers can ask at this point is how to increase the value of C_{L}. The answer to this question comes by considering the lift coefficient as a function of the plane’s angle of attack, denoted by α. As it turns out, there is a positive, linear relationship between C_{L} and α, meaning that if an aircraft flies with a higher angle of attack, then its lift coefficient will increase. Moreover, in ideal cases where our airfoil is thin and we can measure our angle of attack in radians, we can specifically write:

C_{L} = 2π (α – α_{0L}),

where α denotes our current angle of attack and α_{0L} denotes the angle of attack that gives our aircraft zero lift. This equation can make determining C_{L} a relatively simple computation.

However, the positive linear relationship between the lift coefficient and angle of attack only holds for small values of α. Indeed, if α becomes too large, C_{L} can start decreasing precipitously. This breakdown in our model happens because, at high angles of attack, airflow separates from the surface of our wings, preventing them from producing lift and causing our plane to stall. There is thus an optimal angle of attack that an airplane can fly at, one where C_{L} is large but the danger of stalling is low.

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