Problem

Let $S$ be a bar $AB$ as on figure $1.$ The bar is homogeneous with the mass $m$ and the length

$2l$ so that the center of mass $G$ of $S$ is also the geometric center of the bar. The motion is

observed with respect to an inertial frame identified with the orthonormal coordinate system

$R=(O,$vec$(i),$vec$(j),$vec$(k))$ as on figure $1.S$ is subjected to a driving force vec$(F{)}_A(t)=F_A(t)vec(i{)}_A(t)$ acting at $A.$

The angle $=((widehat(i_S,$vec$(i{)}_A)))$ is a control parameter of the system so that the set $u=(F_A,)$ is the

control vector.

$(x,y,)$ as depicted on figure $1$ is chosen as a parametrization of the motion, where $(x,y)$ are

the coordinates of $G$ the center of inertia of $S$ and $$ is the angle $(widehat((vec(i)),AB))=((widehat((vec(i)),$vec$(i_S)))).$

An external action is acting on the body and modeled a force vec$(F)=-$Fvec$(j)$ passing throught the

center of mass $G$ with $F>0$ a constant value.

C$.$ Galculate the coordinates of A and $B$ in $R$ as functions of $x,y,$ and of the parameter $l.$

Calculate the velocity vector field $V$ of $S.$

Show that the inertia matrix $J_G(B_S)$ of $S$ at $G$ in $B_S=(vec(i_S),(vec(j)),(vec(k)))$ reads

$J_G(B_S)=\left[\begin{array}{ccc}0& 0& 0\\ 0& \text{I}& 0\\ 0& 0& \text{I}\end{array}\right]$

Calculate I. Use if necessary, $=\frac{m}{22}$ the linear mass density of $S.$

Calculate the kinetic wrench $HI$ of $S.$

Calculate the dynamic wrench $K$ of $S.$

Calculate the equivalent moment field $T_{F_A}$ of the force vec$(F{)}_A.$ The reduction elements of this

moment field will be calculated at $G.$

Calculate the equivalent moment field $T_F$ of the force vec$(F).$ The reduction elements of this

moment field will be calculated at $G.$

Write the dynamic equations of $S.$

Is the attitude $$ controllable by the driving vector $u=(F_A,)?$