Vernier Software & Technology

# Up And Down: Damped Harmonic Motion

## Introduction

An object hanging from a spring can bounce up and down in a simple way. The vertical position of the object can be described mathematically in terms of a simple sinusoidal equation. In the real world, however, resistive forces such as friction are always present and cause the object to slow down. This effect is called damping.

Most oscillating objects experience damping and move in a modified periodic manner so that the amplitude gets smaller and smaller with each cycle. Common examples of damped oscillators include an empty rocking chair as it comes to rest after being pushed and a vibrating diving board after a swimmer leaves it. At first, the problem of modeling this type of motion with a mathematical equation may seem extraordinarily complex. Surprisingly, it can be analyzed rather thoroughly using basic math concepts with which you are already familiar.

In this activity, you will collect motion data as a paper plate attached to a light spring oscillates up and down above a Motion Detector. Then, you will find an appropriate mathematical model for the resulting data set.

## Objectives

• Record the motion data for a plate bouncing at the end of a light spring.
• Analyze the motion data to determine frequency, period and amplitude information.
• Model the oscillatory part of the data using trigonometric functions.
• Model the damping using an exponential function.
• Create a composite model of damping and oscillation.
• Compare the composite model to experimental data.

## Sensors and Equipment

This experiment features the following Vernier sensors and equipment.

### Option 2

You may also need an interface and software for data collection. What do I need for data collection?

## Real-World Math Made Easy

See other experiments from the lab book.

 1 Walk the Line - Straight Line Distance Graphs 2 Making Cents of Math: Linear Relationship between Weight and Quantity 3 Pool Plunge - Linear Relationship between Water Depth and Pressure 4 Funnel Volumes - Volume and Weight 5 Keep It Bottled Up - Rates of Pressure Increase 6 Graph It in Pieces: Piecewise Defined Functions 7 Mix It Up - Mixing Liquids of Different Temperatures 8 Spring Thing - Newton's Second Law 9 Stretch It to the Limit - The Linear Force Relation for a Rubber Band 10 What Goes Up - Position and Time for a Cart on a Ramp 11 That's the Way the Ball Bounces - Height and Time for a Bouncing Ball 12 Walk This Way - Definition of Rate 13 Velocity Test - Interpreting Graphs 14 From Here to There - Applications of the Distance Formula 15 Under Pressure - The Inverse Relationship between Pressure and Volume 16 Light at A Distance - Distance and Intensity 17 Chill Out: How Hot Objects Cool 18 Charging Up, Charging Down - Charging a Capacitor 19 Bounce Back - The Pattern of Rebound Heights 20 Sour Chemistry - The Exponential pH Change 21 Stepping to the Greatest Integer: The Greatest Integer Function 22 Swinging Ellipses - Plotting an Ellipse 23 Crawling Around: Parametric Plots 24 Lights Out! - Periodic Phenomena 25 Tic, Toc: Pendulum Motion 26 Stay Tuned: Sound Waveform Models 27 Up And Down: Damped Harmonic Motion 28 How Tall? Describing Data with Statistical Plots 29 And Now, the Weather - Describing Data with Statistics 30 Meet You at the Intersection: Solving a System of Linear Equations 31 Titration Curves: An Application of the Logistic Function

### Experiment 27 from Real-World Math Made Easy Lab Book

#### Included in the Lab Book

Vernier lab books include word-processing files of the student instructions, essential teacher information, suggested answers, sample data and graphs, and more.