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.
Follow the Activity procedures:
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