For the material element shown below with current forces and dimensions, what is the matrix representation of the $3$D stress tensor $$ given the normal forces $(4$ N $),$ shear forces $(2$ N in both directions$),$ and current dimensions $(1cm2cm2cm).$

$(20$ pts$)$ A person is wearing a weighted sleeve while resting their forearm on a support. This can be modeled by the image on the right. Assume the weighted sleeve has a weight per unit length of $1200\frac{N}{m}.$ The length of the forearm $L$ is $30$ cm $.$ The forearm can be considered to be roughly cylindrical, with a radius of $3$ cm $.$ The bone has an elastic modulus of $10$ GPa $.$

$($a$)$ Determine the reaction forces at the elbow $($left side support$)$ and the support $($right side support$).$

$($b$)$ Determine the internal shear force $(V)$ in the bone as a function of length $x$ along the bone;

$($c$)$ Determine the internal bending moment $(M)$ as a function of length $x$ along the bone.

$($d$)$ What is the normal stress at the bottom surface halfway along the length of forearm $($in kPa $)?$

$R_{A,x}=$

$R_{A,y}=$

$R_{B,y}=$

$22(15$ pts$).$ A fixed rod is subjected to $3$ torques as shown. The rod has length $L=5m,$ radius $c=0\mathrm{.}75m,$ and $To=100Nm.$ The shear modulus of the material is $G=3$GPa. $($For a circular cross$-$section, $J=\frac{c^4}{2})$

Find:

$($a$)$ The total twist angle at the end of the