In physics licensure, students were working through a tutorial about tension. The tutorial guides students through a series of scenarios and questions to generate the reasoning behind the approximation that tension forces exerted on/by both ends of a very light string are equal in magnitude.
Near the end of the tutorial, students apply this knowledge to the Atwood’s machine. The first case is where there are equal masses on both sides , but the masses are at different heights. Based on intuition, some students expect the masses to stay put , but others expect the higher mass to descend in order to match heights .
I nudge students to subject their initial ideas to further analysis. Students conclude correctly that each mass has two forces… Tension up and weight down. The students also reason that the weights are same because the masses are the same. They also reason that the tensions must be the same by the small mass approximation–not by Newton’s third law!
For the student who anticipated that the masses would balance, this looked like proof. Everything is equal and balanced.
For the student who wasn’t sure if they would balance, this was not settled. The student noted that while we knew the tensions were equal and that the masses were equal we knew nothing about how the tensions compared to the masses. So true.
Eventually, with probably too much help from me, we sorted this out using proof by contradiction. Let’s assume that the tension is greater than the weight. If that’s the case , then both masses would accelerate up! Which is impossible, both intuitively, and logically based on the constraint imposed by the rope. You get a similar contradiction if you assume the tension is less, with both masses descending. You are only left with the possibility that the tension is equal to the weight. Both masses sit still perfectly content to be at different heights.
What I loved about this moment was the following:
(1) Being a scientist in this moment was not about knowing the right answer, but rather about pursuing reasoning to help settle a matter.
(2) The person holding us accountable to rigorous reasoning was, in fact, the one with the wrong intuitive prediction. The person who was confident of the right answer was actually briefly convinced further of their answer by incomplete reasoning.
(3) While everyone at the end was convinced of the correctness of final reasoning, the student with the initial wrong prediction wanted to see it to believe it.
That’s a lot of science in there–argumentation, application of newly learned scientific tools to settle disputes, offering and critique of lines of reasoning, and insistence of empirical support for theoretically drawn.
We used to use this tutorial for faculty development. Good stuff.