$3)$ The force, $\backslash ($ F$\_\{$T$\}=1\backslash$mathrm$\{$kN$\}\backslash ),$ and moment, $\backslash ($ M$\_\{$T$\}=0\mathrm{.}5\backslash$mathrm$\{$kN$\}-\backslash$mathrm$\{$m$\}\backslash ),$ at the tip are caused by a wing tip vortex and a winglet, not shown. $\backslash ($ L$=12\backslash$mathrm$\{$~m$\}\backslash )$ and the spar has an elastic modulus of $\backslash ($ E$=70\backslash$mathrm$\{$GPa$\}\backslash )$ and a Poisson's ratio of $\backslash ($ v$=0\mathrm{.}33\backslash ).$ The mass of the wing is $4000$ kg $,$ and the weight of the engine is $107$ kN $.$ Use $\backslash (9\mathrm{.}8\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ for the acceleration due to gravity.$(15$ points$)$

Consider the cross$-$section shown $($you can look these up$).$ Pay attention to the coordinate system given in the drawing.

a$)$ what is the moment of inertia about $\backslash ($ y $\backslash )-$axis in terms of the symbolic dimensions shown?

b$)$ what is the moment of inertia about $\backslash ($ z $\backslash )-$axis in terms of the symbolic dimensions shown?

c$)$ what is the polar moment of inertia in terms of the symbolic dimensions shown $($i$.$e$.$ about the $\backslash ($ x $\backslash )-$axis$)?$

d$)$ where is the centroid?

e$)$ what is the area?

f$)$ Calculate the shear modulus

g$)$ Compute each of these moments of inertia with the values $\backslash ($ b$=0\mathrm{.}4\backslash$mathrm$\{$~m$\},$ d$=0\mathrm{.}5\backslash$mathrm$\{$~m$\},$ t$=\backslash )0\mathrm{.}1$ m $.$

$4)$ The aerodynamic center for problem $3$ is $0\mathrm{.}1$ m from the $\backslash ($ y $\backslash )$ axis $($neutral axis$).$ Calculate the twist angle caused by the lift force. $(10$ points$)$

$5)$ For problem $3,$ calculate the reactions at the fixed end. $(15$ points$)$