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## Potential Energy of an Elastic System: Instructor's Guide

### Main Ideas

• Calculating changes in Potential Energy from Work
• Build understanding of state variables using a familiar system
• Build understanding of variable dependence and degrees of freedom

Estimated Time: 50 minutes

### Prerequisite Knowledge

• Familiarity with Potential Energy
• Familiarity with Work
• Familiarity with the Partial Derivative Machine

### Props/Equipment

• A handout for each student

### Activity: Student Conversations

As a preface to a major activity associated with the Partial Derivative Machines, students were given a review lecture on:

• Calculating changes in potential energy, $\Delta U$, as the work, $W$, done on the system
• Finding potential energy of stretched springs
• Work as the integral of force, $W=\int F dx$

After this review they proceeded to conduct a laboratory experiment. The primary task was to measure the potential energy stored in the spring system of the Partial Derivative Machine, however a process to determine this function was not explicitly given. The review of work, potential energy, and springs prior to data collection was designed to help students make the connection that the potential energy could be obtained from the work done on the system. Since the system was now two dimensional, using $W=\int \vec{F} \cdot d\vec{r}$ required finding the work done on the system in both the X and Y directions.

One possible solution method that determines all necessary information is:

1. Starting at a particular $x_1=x_{1,o}$, where $\Delta x_1=0$, take measurements of $x_2$ while changing $F_2$ in uniform steps, e.g., $0.05kg \times 9.81 m/s^2$.
2. Set subsequent $x_1$ values by loosening knob D, incrementing $F_1$ by small uniform steps, and then tightening knob D.
3. Repeat step 1 for each new fixed $x_1$ value.
4. Using the data and numerical integration of $F_1\, dx_1$ and $F_2\, dx_2$, approximate the value of $U(x_1,x_2)$.

This process gave students the data needed to get from any state $(x_{1,1},x_{2,1})$ to a different state $(x_{1,2},x_{2,2})$, provided each corresponded to a state generated during the steps outlined above. To verify path independence one would need to conduct a similar process, now measuring $x_1$ for fixed $x_2$ values while varying $F_1$ (changing $F_1$ and $F_2$ by the same increments used above). This lab also provided students practice distinguishing between fixed $x_2$ and fixed $F_2$ processes and the relevance of each to particular measurements.

### Extensions

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

• Preceding activities:
• 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.
• Isowidth and Isoforce Stretchability: In this small group activity, students are challenged to measure a given partial derivative with the PDM.
• 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.

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