Angular Velocity vs. Speed in Circular Motion
by Professor Rich Born
Have you ever felt like you are going in circles? In this lesson you will have PocketLab Voyager going in circles while riding an intelino smart train engine. If you are not familiar with intelino, here is a quick overview. Designed for all ages, intelino is intuitive with its app, has built-in sensors to provide an interactive experience for the user, and is easily programmed with color snaps that allow the user to control intelino's actions. Most elementary and junior high school students enjoy playing with trains. So why not sneak some physical science of motion into the blend? This can be done by attaching Voyager to the top of an intelino engine. Then place the pair onto a circular track. Record angular velocity with the PocketLab app while varying the linear speed of the intelino engine. Setting the speed is a snap with custom color action snaps (no pun intended, ha! ha!).
The video below shows a typical run in which the speed starts at 30 cm/s, the slowest of three autopilot speeds. It then increases in increments of 10 cm/s to a final speed of 70 cm/s. The color snaps (white-magenta-red) have been programmed using the intelino app to 40 cm/s, yellow to 50 cm/s, green to 60 cm/s, and blue to 70 cm/s. Notice how the angular velocity increases each time the speed is increased.
The goal for your students is to design an experiment using custom color action snaps with the Intelino app and PocketLab Voyager mounted to the intelino engine with a 3M damage-free strip. Specifically, the experiment design should allow the students to:
estimate the average angular velocity from data collected with the PocketLab app with each increment in intelino speed,
prepare a graph of angular velocity in ⁰/s (on the y-axis) versus intelino speed in cm/s (on the x-axis), and
discuss the nature of the relationship between angular velocity and intelino speed (linear, or what?)
Figure 1 shows an Excel chart prepared from PocketLab app angular velocity data for a typical run of this experiment. Although there is a fair amount of "noise" for each of the angular velocities, there is a definite pattern in which the angular velocity increases when the intelino speed is increases. The easiest way to determine the angular velocity is to simply eyeball the average. This is what was done by introducing the red lines shown in the chart. For students with more background in a spreadsheet package, averages could be computed for each of the intelino speeds. However, an average computed in this way will probably not differ significantly from an eyeballed average.
Figure 1 - Excel chart of angular velocity vs. time
The final step in the data analysis is for the students to make a graph of angular velocity vs. intelino speed. The Excel chart of Figure 2 shows such a graph. It was constructed with data from the graph of Figure 1. The point (0,0) was included since the angular velocity was zero when the intelino engine was at rest. The points all lie very close to a straight line through the origin. Therefore, it can be concluded that angular velocity is proportional to speed. For your convenience, a pdf file containing an "empty" graph of angular velocity vs. intelino speed accompanies this lesson. This can be duplicated for you students to use in making their graphs.
Figure 2 - Angular velocity vs. intelino speed
Gedanken Experiment for Student Discussion
Gedanken is a German word for thought. Thus a gedanken experiment is a thought experiment. For all of the data collected in this experiment, the radius of the circle was constant at the natural radius of a circle constructed with eight pieces of curved track. We really cannot make a perfect circle of any other radius with intelino track. But we can think about what would happen if the radius was different than the natural radius. Ask the students what would happen to the straight line of their graph of angular velocity vs. speed if the radius was larger. What if the radius was smaller?