Vernier Software and Technology
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 1

Option 2

Additional Requirements

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

Standards Correlations

See all standards correlations for Real-World Math Made Easy »

Real-World Math Made Easy

See other experiments from the lab book.

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

Experiment 27 from Real-World Math Made Easy Lab Book

<i>Real-World Math Made Easy</i> book cover

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.

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