The Piper Cherokee, a light, single$-$engine general aviation aircraft, has a wing area of $170f{t}^2$ and a wingspan of $32$ ft $.$ Its maximum gross weight is $2450$ lb $.$ The wing uses an NACA $65-415$ airfoil, which has a lift slope of $0\mathrm{.}1033$ degree ${?}^{-1}$ and ${}_{F_L}=0=-3.$ Assuming $=0\mathrm{.}12,$ if the airplane is cruising at $120m\frac{i}{h}$ at standard sea level at its maximum gross weight in straight$-$and$-$level flight, calculate the geometric angle of attack of the wing.

Equations for Chap $5$

Effective angle of attack

${}_{\text{e}\text{f}\text{f}}=-{}_i$

where ${}_i$ is the induced angle of attack.

At a spanwise location $y=y_0,$ if the downwash velocity is $-w(y_0),$ then

${}_i=\frac{-w(y_0)}{V_{}}$

Prandtl's lift$-$line theory: for a given spanwise location $y=y_0$

$(y_0)=\frac{(y_0)}{V_{}c(y_0)}+{}_{F_L}=0+\frac{1}{4V_{}}{\displaystyle \underset{-\frac{b}{2}}{\overset{\frac{b}{2}}{}}}\frac{(d\frac{}{d}y)dy}{y_0-y}$

Prandtl's lift$-$line theory for elliptic wing:

$(y)={}_0\sqrt[\phantom{2}]{1-(\frac{2y}{b}{)}^2}$

$w=-\frac{{}_0}{2b}$

${}_i=\frac{C_L}{AR}$

$C_{D,i}=\frac{C_L^2}{AR}$

$a=\frac{a_0}{1+\frac{a_0}{AR}}$

Prandtl's lift$-$line theory for general wing:

$C_{D,i}=\frac{C_L^2}{AR}(1+)$

where $$ is the induced drag factor.

$a=\frac{a_0}{1+[\frac{a_0}{AR}](1+)}$

where $$ is the lift efficiency factor.