# Activities

• • • ##### Subject Area

• Math: Calculus: Applications of the Derivative

• ##### Author 9-12

60 Minutes

• TI-86
• ##### Other Materials
This is Activity 5 from the EXPLORATIONS Book:
Differential Equations With The TI-86

## Harmonic motion

#### Activity Overview

In this activity, students learn about the differential equations used to model the simple harmonic motion of a block attached to a spring. They study the effect of air resistance on the motion and create an animation to describe its motion.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 45 - 55 from the attached PDF file for your class
• #### During the Activity

Distribute the pages to the class.

Fitting an Equation to a Graph of Spring Motion:

• Convert the differential equation of displacement, velocity, and acceleration by substitution to a system of 1st order equations
• Graph the solutions to the differential equations ( x = time, y = displacement)
• Use the Sinusoidal regression function to find the x- and y-coordinates of the points on the graph
• Find the sine function that fits the graph
• Plot the sine function graph on the same screen with the earlier graph and observe the two graphs coincide for values of x greater than zero

• Phase Trajectories:
• Use the same initial conditions as example 1, and change axes to x = displacement and y = velocity
• View the direction field
• Notice the data graphed gives a circle, where the plane is the phase plane and the orbit is the phase trajectory
• Use different initial conditions and view the resulting orbit - smaller circle corresponds to a mass oscillating with a smaller amplitude

• Damped Motion:
• Enter the system of differential equations that includes air resistance
• Graph the displacement versus time data and observe that the graph shows damped oscillations
• Select Directional field, change axes x = displacement and y = velocity, graph the data
• Observe the graph shows a spiral orbit, spiraling into the origin showing the spring reaches equilibrium as the amplitude of the oscillations approaches zero

• Animation:
• Select the animation graphing style
• Set x = 1 and y = displacement
• Create a model for actual motion of the mass and spring (including air resistance)
• #### After the Activity

Review student results:

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