Problem $2$: Cambered Airfoil

Consider a thin airfoil whose mean camber line is defined by:

$z=4z_m[\frac{x}{c}-(\frac{x}{c}{)}^2]$

where $z_m$ is a positive constant with units of length.

$($a$)$ Differentiate the expression above to obtain the slope of the mean camber line $(d\frac{z}{d}x)$

as a function of $x.$ Determine the location $(x)$ where the camber is at its maximum

on the chord line. Show that $z_m$ is the maximum camber for this airfoil.

$($b$)$ Substitute $x=\frac{c}{2}(1-cos{}_0)$ into your result for $d\frac{z}{d}x$ above $-$in effect making a

change of variables from $x$ to ${}_0.$

$($c$)$ From part $($b$),$ find the zero$-$lift angle of attack $({}_L=0).$ Check that your answer is

dimensionless. Can you tell if it is of the correct sign, and if its magnitude is in a

reasonable range?

$($d$)$ Determine the coefficients $A_1$ and $A_2$ in the vortex strength expression for this cam$-$

bered airfoil. Find the moment coefficient $(C_{\text{m}\text{a}\text{c}})$ about the aerodynamic center.

$($e$)$ Again using $A_1$ and $A_2$ above, find the location of the center of pressure $(x_{cp}).$ Is it

located between the leading edge and trailing edge, and closer to the former?

$($f$)$ Find the value of $z_m$ in terms of the chord length $(c)$ if a lift coefficient of $0\mathrm{.}8$ is

desired at $4$ degrees angle of attack.