This interactive simulation offers a way for students to explore the connection between uniform circular motion and simple harmonic motion. The display shows two blocks on springs oscillating horizontally, and two balls traveling in uniform motion in a circular path. The user sets initial values for the blocks: amplitude, mass, and spring constant. The two balls are automatically set to the same values. Students are able to see that the circular motion of each ball corresponds to the motion of the blocks, thus promoting understanding of the basic equation for objects undergoing simple harmonic motion.

To extend the learning, users can set values for the phase angles of each block. Also included by the author is a set of suggested activities to accompany the simulation. See Related Materials for an extensive online multimedia tutorial on the topic of simple harmonic motion from Physclips.

This applet was created with EJS, Easy Java Simulations, a modeling tool that allows users without formal programming experience to generate computer models and simulations.

9-12: 2A/H1. Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.

2B. Mathematics, Science, and Technology

9-12: 2B/H3. Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments.

4. The Physical Setting

4F. Motion

6-8: 4F/M7. Wave behavior can be described in terms of how fast the disturbance spreads, and in terms of the distance between successive peaks of the disturbance (the wavelength).

9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.

9-12: 4F/H6c. The energy of waves (like any form of energy) can be changed into other forms of energy.

9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.

11. Common Themes

11B. Models

6-8: 11B/M5. The usefulness of a model depends on how closely its behavior matches key aspects of what is being modeled. The only way to determine the usefulness of a model is to compare its behavior to the behavior of the real-world object, event, or process being modeled.

9-12: 11B/H1a. A mathematical model uses rules and relationships to describe and predict objects and events in the real world.

Common Core State Standards for Mathematics Alignments

Standards for Mathematical Practice (K-12)

MP.4 Model with mathematics.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.^{?}

F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.^{?}

F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Trigonometric Functions (9-12)

F-TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F-TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^{?}

High School — Geometry (9-12)

Circles (9-12)

G-C.4 (+) Construct a tangent line from a point outside a given circle to the circle.

<a href="http://www.compadre.org/portal/items/detail.cfm?ID=9407">Duffy, Andrew. Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion. August 27, 2009.</a>

A. Duffy, Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion, (2008), <http://physics.bu.edu/~duffy/Ejs/EP_chapter12/reference_circle_v1.html>.

Duffy, A. (2009, August 27). Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion. Retrieved June 28, 2017, from http://physics.bu.edu/~duffy/Ejs/EP_chapter12/reference_circle_v1.html

Duffy, Andrew. Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion. August 27, 2009. http://physics.bu.edu/~duffy/Ejs/EP_chapter12/reference_circle_v1.html (accessed 28 June 2017).

Duffy, Andrew. Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion. 2008. 27 Aug. 2009. 28 June 2017 <http://physics.bu.edu/~duffy/Ejs/EP_chapter12/reference_circle_v1.html>.

%A Andrew Duffy %T Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion %D August 27, 2009 %U http://physics.bu.edu/~duffy/Ejs/EP_chapter12/reference_circle_v1.html %O application/java

%0 Electronic Source %A Duffy, Andrew %D August 27, 2009 %T Boston University Physics Easy Java Simulation: Uniform Circular Motion and Simple Harmonic Motion %V 2017 %N 28 June 2017 %8 August 27, 2009 %9 application/java %U http://physics.bu.edu/~duffy/Ejs/EP_chapter12/reference_circle_v1.html

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This is the full index of Easy Java Simulations (EJS) by author Andrew Duffy, created for first-semester introductory physics. EJS is a modeling tool developed by the Open Source Physics project.

Multimedia tutorial provides extensive background support on simple harmonic motion through video, Flash animations, informative text, diagrams, and graphs.