Problem $3$: Lockheed P$-38$ Lightning

The Lockheed P$-38$ Lightning was an American fighter aircraft designed by Clarence

'Kelly' Johnson for use in World War II$.($Kelly Johnson also designed the iconic Lock$-$

heed U$-2$ and SR$-71$ Blackbird, among other aircraft.$)$ The P$-38$ uses a NACA $4412$

airfoil for the outboard portion of its wing, which has a mean camber line defined by:

$\frac{z}{c}=\{\begin{array}{ccc}0\mathrm{.}25[0\mathrm{.}8\frac{x}{c}-(\frac{x}{c}{)}^2]& \text{f}\text{o}\text{r}& 0\frac{x}{c}0\mathrm{.}4\\ 0\mathrm{.}111[0\mathrm{.}2+0\mathrm{.}8\frac{x}{c}-(\frac{x}{c}{)}^2]& \text{f}\text{o}\text{r}& 0\mathrm{.}4\frac{x}{c}1\end{array}$

where $c$ is the chord length, and $x$ and $z$ are in units of length. In the analysis below,

ignore the taper and the inboard portion of the wing.

$($a$)$ Using Thin Airfoil Theory, set up integral expressions for the zero$-$lift angle of

attack $({}_L=0)$ in terms of the transformed variable $.$ Ensure that you include the

correct integration limits$,$ but please do not actually integrate anything.

$($b$)$ Given ${}_L=0=-4\mathrm{.}16,$ calculate the lift coefficient $(C_l)$ when $=3.$

$($c$)$ Set up integral expressions for the coefficients $A_1$ and $A_2$ in the series expansion

of the circulation distribution. Ensure that you include the correct integration limits$,$

but again please do not actually integrate anything.

$($d$)$ Given $A_1=0\mathrm{.}1630$ and $A_2=0\mathrm{.}0277,$ calculate $C_{m,\frac{c}{4}}$ and $\frac{x_{cp}}{c}$ when $=3.$