$2)[15$ points$]$ A symmetric airfoil of span $1$ m and cherd $0\mathrm{.}6$ m is hinged at its mid chord and supported by a translational spring of constant $\backslash ($ K$\_\{1\}=500\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$mat$\}\backslash )$ the leading$-$edge and a translational springs of constant $\backslash ($ K$\_\{2\}=1000\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash )$ at $0\mathrm{.}2$ m from the leading edge. The aerodynamic center is located at the quarter chord point.

a$)$ Assume small deflection in the pitch direction $\backslash (\backslash$theta $\backslash ),$ about the hinge point, and draw the free body diagram of the airfoil including the aerodynamic loads.

b$)$ Start from the Newton's second Law and write the airfoil's static aeroelastic equation in pitch direction about the hinge point. Assume small angles and ignore aerodynamic drag effects.

c$)$ Find an equation for the pitch angle $\backslash (\backslash$theta $\backslash )$ as function of dynamic pressure.

d$)$ Calculate the divergence dynamic pressure using the equation obtained in part c$.$

e$)$ Assume a trailing edge flap of size $0\mathrm{.}2$ m for the original model and calculate its aileron reversal dynamic pressure $($airfoil chord is still $0\mathrm{.}6$ m $).$ Use the thin airfoil theory equations for $\backslash ($ C$\_\{$L$\_\{\backslash$beta$\}\}\backslash )$ and $\backslash ($ C$\_\{$M$\_\{\backslash$beta$\}\}\backslash ),$ and assume incompressible flow.