Angular Velocity

by Professor Rich Born


In the previous lesson the "intelino® smart train" was introduced, and an activity on speed for 4th grade through middle school students was presented. In that lesson Voyager was "on board" the intelino train and collected data for measuring the speed of the train. With students at the 4th grade level learning angle measurements in degrees and also having a solid foundation in multiplication and long division, there is no reason why they couldn't also learn the concept of angular velocity. Attach Voyager to the top of the intelino smart train engine and you have a perfect and enjoyable way to introduce your students to angular velocity!

Activity 1: Determine the Angle Arc in Degrees of a Single Section of Curved intelino Track

For upcoming activities, your students will need to know the number of degrees in the arc of a single section of curved intelino track. Ask them to construct a perfect circle using only curved pieces of intelino track. From their knowledge of the number of degrees in a circle and long division, they should be able to determine the answer. See Figure 1.

Figure 1 - Angle measure of an intelino curved track

Activity 2: Understanding Angular Velocity

Have the students mount Voyager to the top of the intelino smart engine as shown in Figure 2. It works well to use a 3M damage-free poster hanging strip. The +z-axis is perpendicular to the orange surface of Voyager, pointing straight up. The -z-axis would be pointing straight down. As the train moves around the track, it is experiencing a rotation about the z-axis. As the students will see in this activity, counterclockwise (CCW) motion on the track will represent positive angular velocity while clockwise (CW) rotation will represent negative angular velocity. The magnitude of the angular velocity is the number of degrees that the train rotates is one second.

Figure 2- PocketLab mounted to the intelino smart engine

Kids of all ages love sports, and lots of sports involve movement around a track of some sort:

  • A track and field athlete runner in a 1600 meter race

  • A baseball player traversing the bases in a home run

  • Cars participating in the Indianapolis 500

  • Horses running in the Kentucky Derby

  • A participant in short track speed skating in the Winter Olympics

Ask the students for examples or provide the above examples for them. What can be said about the sign of the angular velocity of participants in every one of these sports?

Activity 3: Making the intelino Smart Engine Go Back-and-forth Around the Circular Track

Now that your students have constructed a perfectly circular track and have learned the concept of angular velocity, it's time for them to get the engine to move around the track. There is no need to use the intelino app, as the engine can simply be started by turning it on and then pressing the button on the engine. The engine will simply keep going around and around the track over and over again. Not too exciting!

Have your students place a color snap command on the track that will cause the engine to reverse itself each time it goes around the track. This time have them also turn on Voyager, connect it via Bluetooth to a device running the PocketLab app, have it display a graph of angular velocity (z only, x and y turned off), and set the data rate to 10 points/sec.

  • What happens to the sign of the angular velocity each time that the engine encounters the color snap command?

  • What is the sign of the angular velocity when the engine is moving CCW around the track? CW around the track?

  • Do the signs agree with what you learned in Activity 2?

See the movie below for a typical run of this activity.

Activity 4: Analyzing Angular Velocity Data from the intelino Smart Engine Going Back-and-forth on the Track

In this activity your students will analyze the angular velocity data collected by the PocketLab app. Figure 3 shows a screen print of angular velocity vs. time for a typical run. The resulting graph resembles a sort of "square wave", in which the angular velocity quickly changes direction each time the engine reverses itself. Ask the students to pick any one of the sections that they want. They need to compute the time for that section as shown in Figure 3. Since there is some random variation of the angular velocity in each section, they also need to "eyeball" the average angular velocity as shown by the red line. Once they have these two items, they can compute the angle traversed by the engine. This is accomplished by multiplying the angular velocity in (⁰/s) by the time in seconds. Lo and behold, they will get an answer fairly close to the number of degrees (360) in a circle! It will be negative when the engine is moving CW and positive when it is moving CCW.

You may wish to point out the similarity of angular velocity to linear velocity. For linear velocity, distance can be computed by multiplying the velocity by time. For angular velocity, the angle can be computed by multiplying the angular velocity by time.

Figure 3 - Data from a typical run with the circular track

Activity 5: The Right Hand Rule for Angular Velocity

There is an easy way to determine the direction of the angular velocity. Hold your right hand as shown in Figure 4, with your four fingers curled in the direction of rotation of the engine on the curve. Your thumb will point in the direction of the angular velocity! Thumb up --positive angular velocity, thumb down - negative angular velocity. Its that easy.

What about when the engine is either at rest or moving on a straight track? In both of these cases, the angular velocity is zero since the engine is not rotating. You can use the author's extension of the right hand rule to handle these cases. Just make a fist--it looks kind of like the digit zero (0).

Figure 4 - The right hand rule for angular velocity

Activity 6: Angular Velocity on a More Elaborate Track Layout

Now that your students have all of the basics, let's consider angular velocity on the more elaborate track layout shown in Figure 5. The figure shows the layout that they need to construct. The direction that the train is moving around the track is shown by the blue arrow. The layout consists of eight segments, each with specific tracks: AB, BC, CD, DE, EF, FG, GH, and HA. Magnets are placed at the center of each segment for help in identifying the segments in the PocketLab app's graph. Students should record data on the PocketLab app using "2 graph" mode: angular velocity and magnetic field magnitude. The data rate should be set to 50 points/second.

Figure 5 - A more elaborate track layout

Here are the challenges for the students in this activity:

  1. Based upon the number of degrees in a single section of curved intelino track (from Activity 1), the students should calculate the number of degrees for each of the following segments: AB, BC, CD, EF, FG, and GH. For which segments would the number of degrees be a positive number, and for which segments a negative number?

  2. In which segments would you expect the angular velocity to be zero?

  3. Analyze the PocketLab graph using the techniques learned in Activity 4. How do the number of degrees in each of the segments obtained from analysis of the graph compare to what you found in question 1?

For question 3, you might want to assign a specific segment to each student group. Here is a video of the intelino train with PocketLab data displayed on an iPad while the train travels around the layout.

The Excel chart of Figure 6 shows typical graphs of angular velocity and magnetic field magnitude versus time. The letters A through H correspond to the points in Figure 5. The colored circles correspond to the magnets shown in Figure 5. The locations of the magnetic field magnitude maxima have been extended upward in Figure 6 to show that they center on each of the track segments of interest. On the straight track segments HA and DE, the visual average of the angular acceleration is zero, as would be expected since there is no rotation of the intelino engine there.

Figure 6 - Angular velocity for the more elaborate track layout

Figure 7 shows a graph of Z angular velocity vs. time zoomed in on segment AB. Calculations from data on the graph show that the angular displacement was about -133⁰. This is in close agreement to the fact that segment AB has three curved tracks of 45⁰ each (3 x 45 = 135).

Figure 7 - Zoomed in on Segment AB

Activity 7: Determining a Track Layout from Its Angular Velocity Graph

The final challenge for your students is to determine the shape of a track layout (curves and straights) from analysis of its angular velocity versus time graph. The graph shown in Figure 8 was obtained from PocketLab Voyager data and represents exactly one complete 10-second cycle of an intelino train moving around a track layout. The train was moving at a constant speed of approximately 40 cm/second. What is the shape of the track, and what is the train's overall direction as it moves around the track?

Figure 8 - Angular velocity graph for an unknown layout