(40pts) 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 7 and it is making an angle 6 with the vertical. a. (Spts) Sketch a neat and clear free body diagram of the particle showing all forces acting on it. b. (10pts) Derive the equations of motion of the particle for the coordinates r and 8, using the Newtonian method. ) Note: Express all vectors using the rotating v 6 unit-vector triad shown. c. (Spts) Derive an expression of the reaction force on the particle from the ring. d. (5pts) Derive the relation between r and 6. e. (15pts) Using @ as the generalized coordinate, derive the Lagrange's equation of motion for this particle