Education Technology

Harmonic motion

Published on 06/09/2008

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