Problem $1(15$ points$)$: The figure shows a bar OA fixed at point $O,$ and at point $A,$ it is connected to a cantilever

AB $.$ The spring rate of bar OA in torsion is $k_T,\frac{(N)*(m)}{\text{r}\text{a}\text{d}\text{i}\text{a}\text{n}},$ and in bending is $k_L\frac{(N)}{(m)}.$ The spring rate of the cantilever AB is

$k_l,\frac{N}{m}.$ What is the overall spring rate based on the deflection y at point B $?$

Problem $2(25$ points$)$: Derive the deflection equations for the following beam using statics and the double$-$

integration method.