Consider a simple pendulum consisting of mass m attached to a string of length l. The equation of motion for the mass is 6\" = 'g sine, l where positive 9 is counterclockwise. For small angles 6, sin 9 a 9 and the linearized equation of motion is 9\" = :59. I m The acceleration due to gravity is g = 9.81 m/secz, and I = 0.6 m. Assume that the pendulum starts from rest with 6(t = 0) = 10. (a) Solve the linearized equation for 0 5 t S 6 using the following numerical methods: (i) Euler (ii) Backward Euler (iii) Second-order RungeKutta (iv) Fourth-order RungeKutta (v) Trapezoidal method Try time steps, At = 0.15, 0.5, 1. Discuss your results in terms of what you know about the accuracy and stability of these schemes. For each case, and on separate plots, compare your results with the exact solution. (b) Suppose mass m is placed in a viscous uid. The linearized equation of motion now becomes a\" +c9'+ $6 = 0. Let c = 4 sec\". Repeat part (a) with methods (i) and (iii) for this problem. Discuss quantitatively and in detail the stability of your computations as compared to part (a). (c) Solve the non-linear undamped problem with 6(1' 2 O) = 60 with a method of your choice, and compare your results with the correspond- ing exact linear solution. What steps have you taken to be certain of the accuracy of your reSults'? That is, why should your results be believable? How does the maximum time step for the non-linear problem compare with the prediction of the linear stability analysis