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Internal Energy of the "Derivative Machine": Instructor's Guide

Main Ideas

• Integration as “accumulating pieces”

• Measuring integrals experimentally

Students' Task

Estimated Time:

  • Determine the internal energy, $U$, of the system at several locations

Prerequisite Knowledge

  • Knowledge of integration from previous physics and calculus courses


Activity: Introduction

The following prompts can be used to begin this activity:

  • “Find the relationship between $F$ and $x$.” This encourages students to think about multiple representations in order to express this relationship which likely include a table of data, a plot of the data, and an attempt to find an algebraic expression. Additionally, students can begin to discuss what the dependent and independent variables of the system are and recognize that the system is nonlinear.
  • “For various forces, how much energy is stored in the system?”
  • “Find the potential energy for the system.”

This activity serves as an introduction to experimentally measuring an integral. Students are asked to determine the internal energy, $U$, of the system at several locations. This integration can be done by approximating the internal energy as $U=\int Fdx\approx\sum F_{i}\Delta x_i$. This requires students to choose a starting point, or zero point, for internal energy of the system and then add up, incrementally, the force at each small change in distance. This type of integration is numerical rather than the typical integration students perform with formulas. By doing integration numerically, the idea of integration as a way of adding up small changes is emphasized.

Activity: Student Conversations

  • Energy: Students may discuss what is meant by energy and consider various formulas used in introductory physics courses to calculate energy such as $U=mgh$ and $U=\frac{1}{2}kx^2$. Giving students the more general expression for work in one dimension by $W=\int{F_x dx}$, relating work to energy by $W=\Delta U$, and explicitly stating the measurable position is $x$ and the measurable force is $F_x$ may provide students with the necessary information to get them on track to thinking less about the formulas which describe energy in various contexts.
  • Zero Internal Energy: The derivative machine doesn't have a built-in “zero point” for the internal energy, and therefore students may discuss where the “zero point” of internal energy occurs for the system. The total internal energy of the system at a point in the system can be determined by adding up small changes in the internal energy from the “zero point” to the point of interest. If students can accept that energy can be negative, then it does not matter where they set the “zero point” for internal energy.
  • Independent and Dependent Variables: It may be natural for students to think about $x$ as their independent variable and $F$ as their dependent variable, as would be typical in a mathematics course. However, the derivative machine does not define which variable is dependent or independent. Both force and distance can be controlled when using the derivative machine which means dependent and independent variables are indistinguishable in this activity.
  • “Small enough” changes: Students may choose changes in mass or distance which are too small or too large for an accurate estimate of the integral. The changes students use to control mass or distance must be sufficiently large to be measurable, however, they must be sufficiently small to observe small nonlinear changes.
  • How to “accumulate pieces”: Students must use numerical integration to determine the internal energy of the derivative machine. This may be a difficult step for students because the summation, $U \approx \sum F_{i} \Delta x_i$, requires a change in position to be multiplied by a single measurement of force, not a difference, at each step. Instructors can prompt students to graph their data which will encourage students to interpret their data and summations visually.
  • Comparing $ U = \sum_i F_i \Delta x_i$ and $ \sum_i x_i \Delta F_i$: Because the weights are easier to control on the machine than the distance, students may try to calculate the internal energy by finding small changes in the force at fixed values of position. Encourage students to represent each summation on a graph to compare whether they are calculating the same value. This should draw their attention to these sums being different quantities and the need to choose small changes in $x$ at fixed $F$.
  • Curve fitting: Asking students to determine the relationship between $x$ and $F$ may result in students using Excel or another computer program to curve fit their data without an underlying physical model in order to produce an algebraic relationship between $F$ and $x$. Discourage students from using a curve fit because they do not have a reason to pick a particular fit since the system is unknown.

Activity: Wrap-up

A whole class discussion can follow about measuring integrals experimentally and different representations of integration.

An alternative wrap-up can be in a homework problem where students write a brief report on how to calculate the internal energy of their system using the data that was collected in class.


This is the initial activity within a sequence of activities addressing Scalar Integration in Curvilinear Coordinates. The following activities are included within this sequence:

  • Follow-up activities:
    • Curvilinear Coordinates: This lecture introduces students to curvilinear coordinates and highlights the notation difference of $\theta$ and $\phi$ in physics and mathematics.
    • Scalar Distance, Area, and Volume Elements: In this small group activity students derive expressions for infinitesimal distances in order to find area and volume elements in cylindrical and spherical coordinates and can be done with Pineapples and Pumpkins to give students a three dimensional object to explore the geometry and construction of a volume element.
    • Pineapples and Pumpkins: This activity can be done in small groups or as an instructor led whole class activity where a pineapple (for cylindrical) and/or pumpkin (for spherical) can be cut to demonstrate the geometry of an infinitesimal volume element used in integration.
    • Acting Out Charge Densities: This kinesthetic activity provides students with an embodied understanding of charge density and total charge by using their bodies to represent charges and act out linear, surface, and volume charge densities which prompts a whole class discussion on the meaning of constant charge density, the geometric differences between linear, surface, and volume charge densities, and what is “linear” about linear charge density.
    • Total Charge: In this small group activity, students calculate the total charge within spherically or cylindrically symmetric volumes by using multivariable integration in various coordinate systems in order to find the total charge contained within the volume due to a specific charge density.

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