Exercise 1: Plane pendulum with spring Given is a planar arrangement according to the figure on the right. A point mass m is attached to a thread of length I .A spring is attached between the point mass and the wall on the side of the pendulum. The spring has the spring constant k and satisfies Hooke's law. The distance between the suspension point of the pendulum and the wall and the resting length of the spring are also I, so that the spring is relaxed when the pendulum hangs vertically. The mass is under the influence of the gravitational acceleration 1 Determine a suitable Lagrangian. 2 Determine the equation of motion, 3 linearize the equation of motion for small displacements, and determine the natural frequency of the system