N 21 k 1 relle o m 9 Let the position vector for m have the form A particle of mass m is suspended by a spring-pendulum system, as shown on the left. The spring has stiffness k and natural length 1, while the length of the pendulum is also 1, such that the distance between floor and ceiling is 21. The origin is taken to be the fixed point of the pendulum, denoted O, such that OQ=213. The system evolves under the influence of gravity, and is constrained to move in the two dimensions of the page. r = lsin (0) 1 + 1 cos(0). (i) (3 marks) Determine expressions for the kinetic and potential energies. (ii) (3 marks) Using a small angle approximation for 0, retain terms up to order 04 to give an approximation for the potential. (iii) (3 marks) Using the approximation for the potential, write down the Lagrangian and compute the equation of motion.