Consider the mechanical system given. They are all constrained to move horizontal plane. Choose the coordinates necessary first $,$then solve the problem.

a$)$ Draw necessary free body diagrams, derive the differential equations of motion using Newtonian mechanics

b$)$ Derive the differential equations of motion using Lagrange$$s equations

c$)$ Show that they are the same.

d$)$ Finally put the equations in the second order matrix form

e$)$ Put the system equations in vector$-$matrix form. State $=$Ax$+$Bu$,$ y$=$Cx$+$Du