## Internal Energy of a One Dimensional System

This sequence of activities introduces integration experimentally by quickly building to have students find the Internal Energy of the "Derivative Machine" of a nonlinear spring system.

This sequence can be used with a background of introductory physics and calculus courses because it requires only a brief introduction to energy, work, and integration. This sequence can precede or follow Representations of Ordinary Derivatives which does a similar sequence for experimental measurement of derivatives.

### Activities

• Recall Energy (Estimated time: 5 minutes): This small whiteboard question is used for students to recall something about energy. Students may respond with a variety of answers such as units, equations of types of energy, conservation, and work.
• Recall Work (Estimated time: 5 minutes): This small whiteboard question prompts students to recall something about work. If Recall Energy does not prompt students to write that work is $\int{\vec{F}\cdot d\vec{r}}$ this question can be used to prompt that thinking because it is essential prior to Internal Energy of the "Derivative Machine" activity.
• Internal Energy of the "Derivative Machine" (Estimated time: 25 minutes): This small group activity can be used to introduce integration. Students are first asked how they will analyze their data which prompts discussions within groups for how to calculate work experimentally.

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