Problem 6a. (Lagrangian mechanics ) A pendulum is made from a massless spring, with force constant k and unstretched length lo, that is suspended at one end from a xed pivot O and has a mass m attached to its other end. The spring can stretch and compress, but cannot bend, and the whole system is conned to a single vertical plane. (a) Write down the Lagrangian for the pendulum, using as generalized coordinates the usual angle (f) and the length r of the spring. (b) Write down the two Lagrange equations of the system. (c) The equations of part (b) cannot be solved analytically in general. However, they can be solved for small oscillations. Follow these steps to uncouple and simplify your Lagrange equations: (i) let 1 denote the equilibrium length of the spring with the mass hanging from it, and rewrite your equations using 3*\" = l + 6. Note that l does not equal lg (this would only be true if m = 0.) (ii) \"Small oscillations\" involve only small values of e and Q5, so you can use the small-angle approximation for trigonometric terms. (iii) Next drop from your equations all terms that involve powers of e or q (or their derivatives) higher than the rst power (also drop products of e and (:5 or their derivatives). (iv) Finally, let my = k(l l0). These steps should dramatically simplify and uncouple the equations. Describe the form of the solutions to your simplied radial and angular equations (with the radial equation now in terms of e)