Vernier Software & Technology

# Swinging Ellipses - Plotting an Ellipse

## 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 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 22 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.