$(15$ points$).$ The beam shown has roller a support at node $2$ and a fixed support at node $1$ and is subjected to a linearly distributed load and a concentrated moment as shown. Its modulus of elasticity and moment of inertia about the neutral axis is $30,000$ksi and $100i{n}^4.$ The origin for x axis$-$is at node $1.$

$($a$)$ Find the energy equivalent nodal forces and moments for the beam.

$($b$)$ Use the stiffness matrix of the beam and find the rotation at node $2.$

$($c$)$ Use the stiffness matrix of the beam and find the reactions at nodes $1$ and $2,$