Intent

Students perform experiments to test whether each of three variables affects the period of a pendulum.

Mathematics

Students will be conducting a series of controlled experiments in which they hold all of the pendulum variables equal to the values in the standard pendulum exceptfor the one they wish to test. Then they examine the collected period data to see whether they fall within what one would expect due to measurement variation alone.

In physics, a simple pendulum is, in theory, one with a weightless string and a weighted bob that is a single point that oscillates through a small angle. With these simplifying assumptions, the period is a function of only the length of the pendulum. Practically speaking, students’ pendulums only approximate a simple pendulum, so students might observe some additional variability.

Progression

Students work in groups to test each of the three variables in turn. Then the class draws conclusions about each variable.

Approximate Time

35 minutes for activity

30 minutes for discussion

Classroom Organization

Groups, followed by whole-class discussion

Materials

Additional washers and fishing line or dental floss

Class graph from The Standard Pendulum

Doing the Activity

Introduce the activity by referring to the list of variables that might affect the period of a pendulum. Then tell students that their work will now focus on these three variables:

Then have the class read the activity.

Students are told to take each measurement twice but not to average their measurements. The reason for this is that the standard deviation for a set of averages of data is different from the standard deviation for the data set itself. Because the sample standard deviation found in A Standard Pendulum was from individual experiments, students need to look here at results from individual experiments. [link to math maps]

Discussing and Debriefing the Activity

Focusing on one variable at a time, have groups share their measurements. Mark each measurement on the graph of the normal distribution of measurements of the period of the standard pendulum. Use a different color for each variable.

As each variable is discussed, students should examine their collective results and determine if that variable had a significant effect on the period based on whether there are data points that are more than two standard deviations from the mean.

Does weight matter? Students should see that their results for the pendulum with a change in weight seem to fit well within the results that would be expected for the standard pendulum.

Does length matter? Students should see that this variable change has dramatically affected the period; the results fulfill the criterion for “mattering.” Bring out that the decision that “length matters” is based on the fact that the results would be extremely unlikely with the standard pendulum. Circle “length” on the list of possible factors.

Does amplitude matter? The variable of amplitude poses the most difficult “Does it matter?” question. An analysis based on the relevant laws of physics shows that changes in amplitude do affect the period of a pendulum, but the effect is very slight unless the angle exceeds 30° or so. For an increase from 20° to 30°, the change will probably be small enough to produce results within the measurement variation found for the standard pendulum. However, an increase from the standard 20° angle to a 60° angle might appear significant.

If the class experiments seem to show that amplitude does make a difference, students will have to make an assumption about amplitude in order to solve the original unit question.

If the class does not find a difference in pendulum period based on a change in amplitude, they will conclude that the period is determined only by a pendulum’s length. You can mention that, in fact, very large changes in amplitude do affect the period, but that students can make the simplifying assumption described above.

It should be clear that the weight of the bob does not affect the period, that the length of the string does affect the period, and that the amplitude could, but probably doesn’t, have a significant effect.

Tell students that for the rest of the unit, they will assume that a pendulum’s length determines its period. You may want to return to the story to note that Poe’s pendulum is “some thirty or forty feet overhead” and tell students that they will use 30 feet as the length of Poe’s pendulum.

Ask students where they are on the outline of steps for solving the unit problem. They should see that they have finally finished step 1 and are now ready for step 2.

Key Questions

Does weight matter?

Does length matter?

Does amplitude matter?

What’s next in solving the unit problem?

Supplemental Activity

Are You Ambidextrous? (reinforcement or extension) asks students to conduct a data-collecting experiment and then use standard deviation to decide whether a difference in results is significant.