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8. Consider the motion of a simple pendulum mgsin(0 mg mgcos(0) The restoring force is mg sin 0 and hence the governing equation is mL dt2 + mg sin (0) = 0 Let the length of the string be g. Hence, the governing equation simplifies to + sin (0) = 0 dt? At the initial time, the pendulum is pulled to an angle of 0 = 30° = before being let loose without any 6. velocity imparted. Write a code to solve for the motion of the pendulum till t = 100 seconds using (a) Forward Euler (b) Backward Euler (c) Trapezoidal Rule • Recall that you need to need to reformulate the second order differential equation as a system of first order differential equation. • Vary your time step At in {0.01,0.02, 0.05, 0.1,0.2, , 0.5, 1,2, 5, 10, 20}. • For each At plot the solution obtained by the three methods on a separate figure till the final time of 100. • Discuss the stability of the schemes. From your plots, at what At do these schemes become unstable

8. Consider the motion of a simple pendulum mgsin(0 mg mgcos(0) The restoring force is mg sin 0 and hence the governing equation is mL dt2 + mg sin (0) = 0 Let the length of the string be g. Hence, the governing equation simplifies to + sin (0) = 0 dt? At the initial time, the pendulum is pulled to an angle of 0 = 30° = before being let loose without any 6. velocity imparted. Write a code to solve for the motion of the pendulum till t = 100 seconds using (a) Forward Euler (b) Backward Euler (c) Trapezoidal Rule • Recall that you need to need to reformulate the second order differential equation as a system of first order differential equation. • Vary your time step At in {0.01,0.02, 0.05, 0.1,0.2, , 0.5, 1,2, 5, 10, 20}. • For each At plot the solution obtained by the three methods on a separate figure till the final time of 100. • Discuss the stability of the schemes. From your plots, at what At do these schemes become unstable

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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8. Consider the motion of a simple pendulum L. mgsin (0) mg mgcos(0) The restoring force is mg sin 0 and hence the governing equation is mL. ´dt2 + mg sin (0) = 0 Let the length of the string be g. Hence, the governing equation simplifies to + sin (0) = 0 dt2 At the initial time, the pendulum is pulled to an angle of 0 = 30° = - before being let loose without any velocity imparted. Write a code to solve for the motion of the pendulum till t = 100 seconds using (a) Forward Euler (b) Backward Euler (c) Trapezoidal Rule • Recall that you need to need to reformulate the second order differential equation as a system of first order differential equation. • Vary your time step At in {0.01,0.02, 0.05, 0.1,0.2, ,0.5, 1,2, 5, 10, 20}. • For each At plot the solution obtained by the three methods on a separate figure till the final time of 100. • Discuss the stability of the schemes. From your plots, at what At do these schemes become unstable (if at all they become unstable)? 4 • Analyse the stability of the three numerical methods to solve the differential equation by approximating sin (0) to be 0. • Make sure each figure has a legend and the axes are clearly marked. • Ensure that the font size for title, axes, legend are readable. • Submit the plots obtained, entire code and the write-up.
Transcribed Image Text:8. Consider the motion of a simple pendulum L. mgsin (0) mg mgcos(0) The restoring force is mg sin 0 and hence the governing equation is mL. ´dt2 + mg sin (0) = 0 Let the length of the string be g. Hence, the governing equation simplifies to + sin (0) = 0 dt2 At the initial time, the pendulum is pulled to an angle of 0 = 30° = - before being let loose without any velocity imparted. Write a code to solve for the motion of the pendulum till t = 100 seconds using (a) Forward Euler (b) Backward Euler (c) Trapezoidal Rule • Recall that you need to need to reformulate the second order differential equation as a system of first order differential equation. • Vary your time step At in {0.01,0.02, 0.05, 0.1,0.2, ,0.5, 1,2, 5, 10, 20}. • For each At plot the solution obtained by the three methods on a separate figure till the final time of 100. • Discuss the stability of the schemes. From your plots, at what At do these schemes become unstable (if at all they become unstable)? 4 • Analyse the stability of the three numerical methods to solve the differential equation by approximating sin (0) to be 0. • Make sure each figure has a legend and the axes are clearly marked. • Ensure that the font size for title, axes, legend are readable. • Submit the plots obtained, entire code and the write-up.
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