6. A cylindrical rod with mass M has a ring of mass m slid over it, which stays in contact with the rod via a spring during the rod's motion about the fixed-point O. Using the Lagrangian method, derive the equations of motion for the system during a simple harmonic motion of the rod. Here, m is the mass of the bracelet, M is the mass of the homogeneous rod, k is the spring constant, R is the total length of the rod, r is the distance of the bracelet from the fixed- point O at time t, and r0 is the natural length of the spring. ////// G