Vernier Software and Technology
Vernier Software & Technology

Swinging Ellipses - Plotting an Ellipse

Figure from experiment 20 from Real-World Math with Vernier

Introduction

Any ellipse centered at the origin can be expressed in the form

\frac{{{x^2}}}  {{{a^2}}} + \frac{{{y^2}}}  {{{b^2}}} = 1

where ± a and ± b represent the x- and y-intercepts of the ellipse.
To graph an ellipse on a calculator, the expression above must first be solved for y to obtain

y =  \pm b\sqrt {1 - \frac{{{x^2}}}  {{{a^2}}}}

This equation is entered into the calculator in two parts, one expression for the positive part (upper half of the ellipse) and one for the negative part (lower half of the ellipse).

In this activity you will use the Motion Detector to record the position and velocity of a swinging pendulum. You will find that the plot of velocity versus position is elliptical, and that you can model it with the standard equation of an ellipse.

Objectives

  • Record position and velocity versus time data for a swinging pendulum.
  • Plot data as a velocity versus position phase plot.
  • Determine an ellipse that fits the phase plot.

Sensors and Equipment

This activity 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 with Vernier »

Real-World Math with Vernier

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
6Mix It Up - Mixing Liquids of Different Temperatures
7Spring Thing - Newton's Second Law
8Stretch It to the Limit - The Linear Force Relation for a Rubber Band
9What Goes Up - Position and Time for a Cart on a Ramp
10That's the Way the Ball Bounces - Height and Time for a Bouncing Ball
11Walk This Way - Definition of Rate
12Velocity Test - Interpreting Graphs
13From Here to There - Applications of the Distance Formula
14Under Pressure - The Inverse Relationship between Pressure and Volume
15Light at A Distance - Distance and Intensity
16Chill Out: How Hot Objects Cool
17Charging Up, Charging Down - Charging a Capacitor
18Bounce Back - The Pattern of Rebound Heights
19Sour Chemistry - The Exponential pH Change
20Swinging Ellipses - Plotting an Ellipse
21Lights Out! - Periodic Phenomena
22Tic, Toc: Pendulum Motion
23Stay Tuned: Sound Waveform Models
24Up And Down: Damped Harmonic Motion
25How Tall? Describing Data with Statistical Plots
26And Now, the Weather - Describing Data with Statistics
27Meet You at the Intersection: Solving a System of Linear Equations
28Titration Curves: An Application of the Logistic Function
29Clock Design: Period and Length of a Simple Pendulum
30Graph It in Pieces: Piecewise Defined Functions
31Stepping to the Greatest Integer: The Greatest Integer Function
32Crawling Around: Parametric Plots

Activity 20 from Real-World Math with Vernier Lab Book

<i>Real-World Math with Vernier</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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