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## Pendulum Lab: Instructor's Guide

### Main Ideas

1. Pendulum
2. Allowing the students to test and refine, and or to define the important features that any proposed model should have.

Estimated Time: 50 minutes

Students work in groups of three to measure the period of a plane pendulum as a function of the amplitude of the oscillation. They record the data using Logger Pro software. They analyze the data either on-screen and tabulate their results, or they export pairs of (time, angle) data to an Excel spreadsheet and analyze the data in Excel (most draw graphs and read period and amplitude from those). Students then plot the period as a function of amplitude. In an in-class follow-up, they discuss their results and compare with a model.

### Prerequisite Knowledge

No prior physics knowledge is necessary to perform the lab. General familiarity with equipment from introductory classes is sufficient to understand the set up. Students should have very basic spreadsheet skills (enter and plot data), but even these can be learned on the fly.

### Activity: Introduction

A general description of the system, equipment at hand, and the task has been assigned as reading before the lab. Students are simply encouraged to explore the system at hand and to investigate the period as a function of the angular displacement. The only pointers that are given are any instructions necessary for safety (the steel pendulum should not be dropped when someone's face is in the way!) or for the protection of equipment (the potentiometers should not be forced beyond their range or they will break). Students are encouraged to “play” before recording data.

### Activity: Student Conversations

Students work in groups of 2-4, and often consult other groups. They should be encouraged to do so.

Many students try to take detailed data before they know the general behavior of the system and the expected extremes. They need to be reminded to “play” for a few minutes, and to recognize that such “play” can save time later. For example, many do not think to start the pendulum from angles larger than 90$^{\circ}$ to the vertical.

Students understand the need to relate the angle of displacement, measured with a large protractor, to the voltage recorded on the computer. They are interested in the conversion process (described in the equipment document), but are happy to treat it as a black box. Ask the students whether two points are sufficient to establish a general relation. Most students need to be prompted to take calibration points over the whole range of expected displacements. Most expect the calibration to be a linear relation (it is), and few entertain the notion that it could be nonlinear (which it well could be!).

Prior classroom discussion will have most often focused on the simple harmonic oscillator for which the period is amplitude independent. The increase in period of the plane pendulum with amplitude becomes evident at angles close to 90$^{\circ}$, and students realize that it must not be a simple harmonic oscillator, but they are not sure what to expect for the large-amplitude period. The debate usually becomes lively.

Most students are rightly concerned about the effect of damping, which has not been considered at this point. Many consider that damping is directly responsible for the change in period. Encourage the students to think of an experiment that would test the validity of this claim.

### Activity: Wrap-up

The lab activity itself is “wrapped up” on the day following the measurements when the students come to class with their data (period, amplitude) analyzed in tabular and graphical form. This must be given as an explicit (minimally graded) assignment due the following day, or else the class participation is insufficient. Students compare their results and notice that the period increases by at least a factor of 2 as the amplitude increase to about 140$^{\circ}$ from vertical. The pendulum is not simple harmonic, and the reasons for this can be discussed. The form of the potential energy ($\cos \theta$ rather than $\theta^2$) can be plotted to show why the period increases rather than decreases and this ties into the discussion of energy diagrams.

The “wrap-up” is really a beginning, because the students now need to take their knowledge and translate it to a proper quantitative model where either by numerical evaluation of the integral expression for the period or by successive approximation (series expansion) of that integral, they can determine that the model is also quantitatively good. This activity follows in the classroom.

Finally, the results should be written up in a coherent fashion. This activity is assigned as homework, but drafts are encouraged. Students anonymously critique their peers' graphical presentation of data in the wrap-up session; this begins the reporting process. Here are some typical results.

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