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 activity features the following Vernier sensors and equipment.