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