The initial emphasis is on defining a system in such a way that it includes the pendulum and the earth and all the pathways of interaction that connect the two. This serves to provide an opening to discuss the nature of space as an active part of these chains of interaction, because it is the gravitational distortion of space and the stretching, sheering, bending, and compression compression of electromagnetic polarization of space that form the chains of interaction that connect the pendulum ball to the earth. This involves a rather sophisticated and modern conception of space, that clearly fails the "How do we know?" test, but I find it is a conception that is fairly readily accepted by students, and at least addresses the issue, which in my mind makes it actually less "magical" than claiming a gravitational action without any mediator.

 

The next step is observing in order to determine what phenomena are exhibited and which might be most interesting or accessible for further investigation, given an emphasis on using simple qualitative comparisons between nearly identical systems. A major objective is to have the students recognize the rich spectrum of phenomena to which we could pay attention the modeling of all of which together might be required for a full understanding of the system, and then have the students design simple comparative tests to determine which attributes of the system affect the observed motion in which ways. The goal is for students to realize that in order to make progress understanding a complex system scientists have learned to narrow and revise their questions and until their questions are considered "answerable" and then build up their understanding step by step by selectively paying attention to one aspect at a time.

 

The next step after refining our questions, is determining what comparative investigations might legitimately and unambiguously answer the students questions. A key concept that emerges is the difficulty of initially recognizing what is and what isn't a "fair test" of the influence or non-influence of a particular system attribute. So for example when a difference is seen in the motion when mass is added in a way that also slightly shifts the mass distribution, we might not immediately recognize that this was not a "fair test" until a more fair test was designed. Similarly, we might not initially recognize that increasing the mass is not a fair test of the effect of weight on the system, when increasing the mass also increases the inertia. This leads to a challenge to imagine a way that the effect of weight could be investigated independently of the inertia, or inertia independently of the weight (or restoring force).

 

A new theme emerges from our observations and that is the theme of compensation. In two different situations we saw how two attributes might either vary inversely so as to compensate for each other or vary proportionately but have inverse effects. In the first case, the increase in the average speed gained from increasing the angle almost perfectly compensates for the extra distance to be traveled in each swing to keep the time for each full swing nearly independent of angle. In the second case, increasing weight and inertia are seen to have inverse effects on the average speed and thus perfectly compensate for each other in the absence of a change in their proportion determined by the local gravitational field. These two case are developed into two important "touchstones" for how through compensation attributes of the system may remain constant or be conserved even as the system varies. The rule-of-thumb emerges, foreshadowing the development of conservation laws, that existence of a constant attribute of the system implies some form of compensation, while an observed proportional or inverse relation implies the existence some constant in the system.

 

Another important theme emerges around the relations between average speed, time and distance. Since both angle and string length both increase proportionally the distance traveled in one swing, the question is raised how can average speed and time depend upon angle and string length, so that their product increases proportionally. In the first case for angle in which the period is essentially constant, the average speed must increase proportionately. However, if as in the case of string length, both average speed and period increase with angle, what would be the simple guess for the relationship with which both increase. This is the first introduction to the guiding principle of physics that if all else is equal, the simplest possible relation is preferable. The simplest possible relation in this case would be if both increase as a square root of the string length in which they would both be proportional to each other.

 

This led to our hypothesis that both the average speed and the period would vary as the square-root of the string-length. And with a little cooperation of the fire-department we managed to test this by comparing the period of a one meter pendulum with the period of a 1, 4, 9, 16 and 25 meter pendulum, and the results were fairly dramatic and turned out fairly well. The one meter pendulum did in fact swing 1, 2, 3, 4 and 5 times in the time of one swing of the progressively longer pendulum.

 

The final phase was to use CPO timers and LoggerPro, to verify the relations more precisely, that average speed and period are proportional to each other when you vary the length of the string, and that average speed and angle are proportional to each other as you vary the angle. The final conclusion is to try to make sense of these results, i.e. why does average speed increase with angle? Observe that the restoring force increases with angle. Notice how restoring force is independent of string length, as so is the ratio of average speed/time. This requires some discussion of how the average speed is proportional to the change in speed, and so average speed/time is in this case a measure of how fast the pendulum is accelerating.

 

A final extension involves the investigation of how you might change the system so that the average speed remains constant, and all the change happens in the period. So if the speed as the square root of the length of the string the angle would need to decrease as an inverse square-root to compensate and the change in height would remain the same. Alternatively it is possible to have the students just discover this by having them all vary the systems in different way and discover that they all get the same relation when they look at speed vs. change in height, and then make sense of it after the fact.