Problem $3(20$pts$)$

Assume that a reference flight condition is given such that none of the variables

$(u_0{v}_0{w}_0{p}_0{q}_0{r}_0{}_0{}_0{}_0)$

is zero. But assume that $(u_0^{}{v}_0^{}{w}_0^{}{p}_0^{}{q}_0^{}{r}_0^{})$ are zero.

$($a$)$ Determine what are the conditions the angular rates ${}_0^{},{}_0^{}$ and ${}_0^{}$ must satisfy in terms of the

above given reference values.

$($b$)$ Find the corresponding reference forces and moments $x_0,Y_0,Z_0,L_0,M_0$ and $N_0$ in terms of the

reference motion variables.

$($c$)$ Obtain the linearized translational equations in body $z$ direction and linearized pitch moment

equations. Show clearly your steps on how you arrive at them.