Exercise 1 Consider a spring pendulum system. It consists of a mass on the end of a spring which is free to move in a plane. The small angle approximation cannot be used. A schematic diagram is shown in the figure below. Here, I is the length the spring stretches to when the weighted system is in equilibrium, lo is the length the spring is when the unweighted spring is at equilibrium. Finally, g and m are the gravity and mass respectively. A coordinate system for this setup can be defined a few ways. In this assignment we will use an x - y coordinate system centred at the equilibrium position of the mass. y 19 M x Show that the equations of motion for this system are dx k ==x+ k lox and dt m m x + (l y) - d y k k lo (e-y) =-g+ - dt (e y) - m m x + ( l y) ' Hint: It is useful to get expressions for sine and cose in terms of the length of the spring (note this is not, in general) and use as an intermediary variable.