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## Isowidth and Isoforce Stretchability: Instructor's Guide

Estimated Time:

### Prerequisite Knowledge

• Familiarity with partial derivatives and their interpretations

### Activity: Introduction

Once students were familiar with the machine, they were asked in a second exercise to find $\frac{\partial x_1}{\partial F_1}$ and had to consider that there were two possible options: $\left(\frac{\partial x_1}{\partial F_1}\right)_{x_2}$ and $\left(\frac{\partial x_1}{\partial F_1}\right)_{F_2}$. As an introduction, the instructor defined the concept of stretchability as it relates to the system and distinguished between the “isowidth” (constant $x_2$) and “isoforce” (constant $F_2$) stretchabilities.

After collecting data sets, plotting results, and calculating numerical values for both quantities students were asked to present their results to the class. The focus of the presentation was not to provide the class with numerical values, but to explain the techniques used to both measure and calculate the necessary information. The approach of some groups was to take a few measurements of the form $(F_1,x_1)$ and approximate the derivative with the quantity $\frac{\Delta x_1}{\Delta F_1}$. Other groups chose instead to plot $x_1$ as a function of $F_1$.

Due to the different systems under the black boxes, the numerical values for the “isowidth” stretchability and “isoforce” stretchability varied widely from group to group. The relationship between $x_1$ and $F_1$ also varied from system to system — some groups found a linear relationship while others found that the plot was clearly nonlinear.

After these presentations and discussion of the results, students removed the “black box” to see the central systems. Students then walked around the classroom observing other groups' systems to see how each apparatus was different. This allowed for discussion of why particular systems behaved as they did and why particular variables were dependent or independent of each other for each system.

### Activity: Wrap-up

These observations were followed by a whole-class discussion. The instructor asked students to consider if this activity was consistent with or contradicted the idea that one takes a partial derivative while holding “everything else” constant. Next, the class revisited the number of independent variables and which could be set simultaneously.

It was not obvious to some students that $x_2$ and $F_2$ were relevant quantities when changing $x_1$ and $F_1$. To address this concern the instructor conducted a demonstration making use of a piece of spandex. Having one student grab a pair of opposite handles and hold them a fixed distance apart, a second student was instructed to stretch the spandex in the other direction, which simulated measuring $\left(\frac{\partial x_1}{\partial F_1}\right)_{x_2}$. It then became abundantly clear to the first student that in order to maintain a constant $x_2$ it was necessary for $F_2$ to increase as $F_1$ increased. We have repeatedly found that the kinesthetic effect of feeling the force increase in this demonstration helps people notice that the force and displacement in the two directions are coupled.

### Extensions

This activity is the second activity of the Partial Derivative Machine (PDM) Sequence on measuring partial derivatives and potential energy. This sequence uses the Partial Derivative Machine (PDM).

• Preceding activity:
• Quantifying Change: This small group activity introduces students to the PDM by asking them to determine how many measurable quantities exist within the system and how many of these quantities are simultaneously controllable.
• Follow-up activities:
• Easy and Hard Derivatives: This small group activity asks students to write each partial derivative that can be formed from $x_1$, $x_2$, $F_1$, and $F_2$ and then categorize each as “hard” or “easy” to measure on the PDM.
• Legendre Transforms on the PDM: In this small group activity, students get a chance to work with physical analogues of Legendre transforms.
• Potential Energy of an Elastic System: In this integrated laboratory activity, students use the PDM to determine the change in potential energy between two states of a nonlinear system.

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