**AP/Calculus-Based Physics:** Periodic and Simple Harmonic Motion Units

Simple Harmonic Motion (SHM). The motion that occurs when an object is accelerated towards a midpoint or equilibruim position. The size of the acceleration is dependent upon the distance of the object from the mid-point. Examples of this type of motion are sea waves, pendulums and springs. Units are not listed in a prescribed order.

### Simple Harmonic Motion (8)

#### Lesson Plans:

This high-quality lesson plan was created to be used with the PhET simulation "Wave on a String". Students will apply the concepts introduced in the computer simulation to explore frequency, amplitude, and tension on an oscillating string. *Editor's Note: this lesson gives students the chance to model the movement of energy on a string by applying properties of sinusoidal function. A great integration of trigonometry and physics.....and it's fun!*

**Item Type:**Simulation-Based Activity

**Level:**High School Physics

**Duration:**One Class Period

#### Activities:

This simple Java simulation will help students see that the period of a pendulum depends largely on its length and the gravitational constant (__not__ the mass of the pendulum bob). They will see that the pendulum swings in simple harmonic motion as long as the initial angle is under 20 degrees.

**Item Type:**Interactive Simulation

**Level:**Grades 8-12

This simple, yet creative simulation offers a way for students to explore the connection between uniform circular motion and SHM. It will help build understanding of the basic equation for objects undergoing simple harmonic motion. *The editors suggest using this resource with the interactive homework problem "Block and Spring" directly below.*

**Item Type:**Interactive Simulation

**Level:**Grades 9-12

This interactive problem takes learners step-by-step through the components of simple harmonic motion. It will help students recognize the connection between the oscillation of a mass on a spring and the sinusoidal nature of SHM. It provides help with the related free-body diagram, graphs depicting SHM, and support in using the Work-Kinetic Energy Theorem to do the calculations.

**Item Type:**Interactive Problem

**Level:**High School Physics

This Java model explores the motion of a block attached horizontally to an ideal spring. You can change the mass of the block, spring constant, and initial position. The model will display energy bar graphs and graphs of position, speed, and acceleration as a function of time. *Try teaming this simulation with the interactive homework problem (directly below) to promote deep understanding of the sinusoidal nature of SHM.*

**Item Type:**Interactive Simulation

**Level:**Grades 9-12

**Duration:**30 minutes

#### Student Tutorials:

This module integrates a Java simulation, related lessons, physics background information, and explicit help in using every tool available in the simulation. It features a block suspended on a frictionless spring. Students can change the spring constant, mass of the block, and amplitude. Tools include a vector display and real-time graphs of potential and kinetic energy, position, velocity, and mechanical energy.

**Item Type:**Simulation-Based Module

**Level:**Grades 9-12

This robust, yet easy-to-use interactive model can be adapted for learners ranging from middle school through AP physics. You can change the string length, mass, and initial angle. Change the gravitational constant to see how the pendulum moves on different planets. View real-time bar graphs to see how energy is converted from kinetic-to-potential and back as the pendulum swings. Advanced learners can view graphs of angular acceleration/velocity.

**Item Type:**Interactive Simulation

**Level:**Grades 6-12

Explore SHM in a multimedia format that integrates video clips, still images, graphs, and diagrams with informative text and lecture presentations. Great tool for students to set their own pace and self-gauge understanding. Covers the basics of periodic motion, Chladni patterns, Lissajous figures, and more.

**Item Type:**Multimedia Tutorial

**Level:**Grades 10-12

### Conservation of Energy and Forces on a Spring (7)

#### Lesson Plans:

This two-hour activity was created to accompany the PhET simulation *Masses & Springs*. In the first lesson, students use the simulation to explore how displacement of a spring is mathematically related to the load applied to it. On Day 2, learners analyze the energy interactions by observing distribution and transfer of kinetic, elastic potential, and gravitational potential energy. Includes explicit directions for use of the simulation, homework problems, and answer key.

**Item Type:**Lesson Plan

**Level:**High School Physics

**Duration:**Two Class Periods

#### Activities:

In this realistic virtual mass-and-spring laboratory, students explore spring motion by hanging weights of different masses on springs. Students can adjust the spring stiffness and damping. (Damping is a force, often friction, that reduces amplitude.) Real-time charts show changing kinetic, potential, and thermal energy levels as the springs oscillate. *Can be adapted for a variety of levels and capabilities. *

**Item Type:**Interactive Simulation

**Level:**Grades 7-12

This item combines a simulation of a simple linear oscillator with detailed content support explaining the physics. Both numerical and analytical solutions are given, as well as puzzles for student interaction.

**Item Type:**Interactive Simulation

**Level:**High School Physics

Student-generated computer models are becoming much more widespread in physics education because of the opportunity for kids to test and apply their own prototypes. This resource lets them develop a model for a mass on a spring. It opens as a simple simulation for them to explore, record, and analyze data. Afterward, they use the Intro SpringLab computer modeling tool to built their own model and compare results with the real-life experimental results.

**Item Type:**Computer Model

**Level:**Grades 9-12

#### Content Support For Teachers:

This lab activity, designed to explore the potential energy of springs, is offered in both an introductory and an advanced version. Using equipment available in most high school science labs, it includes reproducible data tables, "elicit" questions, and graphing activities.

**Item Type:**Instructor's Guide

**Level:**Grades 9-12

#### Student Tutorials:

This applet simulates a harmonic-coupled line of masses moving in one dimension. The load, damping, and stiffness of the springs are adjustable. It includes more complex concepts, such as phases and magnitudes of normal modes.

**Item Type:**Interactive Simulation

**Level:**AP Physics

We highly recommend this resource to promote deeper understanding of the interactions that influence the motion of a mass hanging from a spring. The author anticipates student conceptual roadblocks and helps them recognize how to use alternative problem-solving methods when the kinematic equations would become overly tedious. *Meets numerous Math Common Core standards and AAAS Benchmarks and would be a good supplement for the PhET simulation "Masses & Springs" (above under Activities).*

**Item Type:**Interactive Problem

**Level:**High School Physics

**Duration:**20 minutes

### Teaching About Oscillation (2)

#### Activities:

Looking for a scaffolded simulation on SHM that has a lesson plan, background information, and explicit help for students? This module, developed by the University of Calgary Physics Department, brings it all together in one convenient location. It features a mass on a frictionless spring. You can adjust the spring constant, add/subtract mass, and change amplitude. The tools include a vector display, kinetic/potential energy bar graphs, and real-time graphs of position, velocity, and mechanical energy.

**Item Type:**Simulation-Based Module

**Level:**Grades 9-12

#### Content Support For Teachers:

Teachers who want a refresher on the topic of oscillating motion will appreciate these illustrated lecture notes on pendulum action and damped driven oscillators. Be sure not to miss the animated spreadsheets in Excel format, which can be customized and used in the classroom. Also included are sets of printable student study sheets and problem sets on oscillations.

**Item Type:**Instructor's Guide

**Level:**High School Physics

### Special Topics: The Physics of Music (2)

#### Activities:

Pluck the virtual string on this simulation, watch, and listen. You can adjust the damping, tension on the string, and speed of the simulation. Set your own force frequency or click to see the resonant frequency. Click on "Shape String" to create your own harmonic. *Editor's Note: This resource was designed to help students visualize the different modes of oscillation that happen when a string is plucked. The frequency of the harmonics is a multiple of the fundamental frequency. Teachers: see Reference Materials below for a free textbook written to introduce high school students to the physics of music.*

**Item Type:**Interactive Simulation

**Level:**High School Physics

#### References and Collections:

This well-written free textbook integrates the topics of waves, sound, music, and musical instruments. It would be ideal content support for secondary teachers wishing to teach a unit on harmonics or the physics of music. The mathematics involves only algebraic applications, so the book is also appropriate for high school physics students.

**Item Type:**Digital Textbook

**Level:**Algebra-Based Physics