$(40$pts$)$ Consider the system shown below. The semicircular ring of radius $R$ is fixed and the collar of mass $m$ can slide on the ring without friction. The mass is connected to the point $O$ by a spring of spring constant $k.$ The unstressed length of the spring is $R.$ An arbitrary configuration is shown where the length of the spring is $r$ and it is making an angle $$ with the vertical.

a$.(5$pts$)$ Sketch a neat and clear free body diagram of the particle showing all forces acting on it$.$

b$.(10$pts$)$ Derive the equations of motion of the particle for the coordinates $r$ and $,$ using the Newtonian method.

Note: Express all vectors using the rotating $r-$ unit$-$vector triad shown.

c$.(5$pts$)$ Derive an expression of the reaction force on the particle from the ring.

d$.(5$pts$)$ Derive the relation between $r$ and $.$

e$.(15$pts$)$ Using $$ as the generalized coordinate, derive the Lagrange's equation of motion for this particle.

need urgent detailed solution with clear drawing.

please do it fast