Education Technology

Nonlinear Pendulum Problems

Published on 06/09/2008

Activity Overview

In this activity, students explore the nonlinear model for a simple pendulum. They use the calculator to solve higher order differential equations.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 69 - 75 from the attached PDF file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:
    The Small Angle Assumption:

  • Use the nonlinear model equation for motion of a simple pendulum and assume the angle between the axis and release point is small (5, 10, and 15 degrees)
  • Enter the differential equation on the calculator with angles in radians to find values of t
  • Graph the equation for each angle for a short time span
  • Find numeric solutions for long time span
  • Also, plot data as a phase diagram by using direction field function and observe the circular nature of the solution


  • Angles within 30 to 60 degrees:
  • Enter the nonlinear pendulum equation and the linear pendulum equation
  • Compare the direction fields
  • Set initial angles to be between 30 and 60 degrees and compare the solution for the two models
  • As the initial amplitude changes the period of the solution for the nonlinear model changes but not in the linear model


  • Pendulums with a Push:
  • Change the initial conditions to represent a big push given to the pendulum as it passes the vertical
  • Graph both linear and nonlinear models
  • Note the nonlinear model actually flips over the top as seen in its graph
  • After the Activity

    Students complete the problems on the exercise page.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary