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