Education Technology

# Activities

• ##### Subject Area

• Math: Calculus: Other Functions

9-12

60 Minutes

• TI-86
• ##### Other Materials
This is Activity 7 from the EXPLORATIONS Book:
Differential Equations With The TI-86

## Nonlinear Pendulum Problems

#### 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.

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