Answer Happy

Problem 2: Consider a simple pendulum consisting of mass m attached to a string of length 1. The equation of motion for the mass is g 0" sino, --- where positive 0 is counterclockwise. For small angles 0, sin 0 ~ 0 and the linearized equation of motion is 0" = _8. ZI/ т = The acceleration due to gravity is g = 9.81 m/sec², and I = 0.6 m. Assume that the pendulum starts from rest with O(t = 0) = 10°. (a) Solve the linearized equation for 0 <t< 6 using the following numerical methods: (i) Euler (ii) Backward Euler (iii) Second-order Runge-Kutta (iv) Fourth-order Runge-Kutta (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.