The Arizona STEM Acceleration Project

An Inquiry Approach to Circular Motion

An Inquiry Approach to Circular Motion

A 9th-12th Grade STEM Lesson

Jeff Steinert

March 2023

Notes for Teachers

Context: This lesson takes place in a classroom for two to three hours. It extensively employs elements of Modeling Instruction.

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Students should work in small groups of 2-4.

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An emphasis on experimental design, precise measurement, and careful analysis of graphical results, with an end goal of development of a simple mathematical model.

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Facilitate student reflection on their analysis of the graphical results as they work in small groups.

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Engage the whole class in coming to consensus on a mathematical model via post-analysis “board meeting”.

List of Materials

• Centripetal Force Apparatus (or similar) for demonstration.
• Student access to a device with a graphing program (Vernier Graphical Analysis is free)
• Internet connection to The Physics Aviary’s Classic Circular Force Lab for students to collect group data.
• 24” by 32” dry erase whiteboards and markers for student groups to display their data and analysis of the results.
• Readings: Johannes Kepler - Before Newton and An Appreciation of Earth

Science Standards:

PlusHS+Phy.P3U1.3: Develop a mathematical model, using Newton’s Laws, to predict the motion of an object or system in two dimensions (projectile and circular motion).

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Science and Engineering Practices:

• ask questions and define problems
• develop and use models
• plan and carry out investigations
• analyze and interpret data
• use mathematical and computational thinking
• construct explanations and design solutions
• engage in argument from evidence
• obtain, evaluate and communicate information

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ELA:

9-10.RI.1 Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

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9-10.RI.2 Determine a central idea of a text and analyze its development over the course of the text, including how it emerges and is shaped and refined by specific details; provide an objective summary of the text.

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11-12.RI.4 Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze how an author uses and refines the meaning of a key term or terms over the course of a text.

9-10.RI.8 Delineate and evaluate the argument and specific claims in a text, assessing whether the reasoning is valid and the evidence is relevant and sufficient; identify false statements and fallacious reasoning.

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Mathematical Practices:

MP.4 Model with mathematics.

MP.6 Attend to precision.

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Arizona Standards

Ed Technology

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Computational Thinker

9-12.5.b. Students collect data or identify relevant data sets, use digital tools to analyze them, and represent data in various ways to facilitate problem-solving and decision-making.

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9-12.5.c. Students break problems into component parts, extract key information, and develop descriptive models to understand complex systems or facilitate problem-solving.

Arizona Standards

Next Generation Science Standards (NGSS)

• HS-PS2-1: Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
• Connection: Students collect data using the "Classic Circular Force Lab" simulation and analyze graphical results (e.g., velocity vs. radius) to derive the mathematical model for centripetal force (Fc = mv^2/r), which is an application of Newton's Second Law.
• HS-PS2-4: Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.
• Connection: Through the reading "Johannes Kepler - Before Newton" and discussions on Earth's orbit, students connect the principles of circular motion to planetary orbits and gravitational forces.

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Common Core Mathematics (CCSS.MATH)

• HSS.ID.B.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
• Connection: Students use graphing software (Vernier Graphical Analysis) to plot their experimental data, analyzing relationships by linearizing graphs (e.g., plotting $v^2$ vs. $R$) to determine the function. * HSA.CED.A.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
• Connection: The specific goal of the lesson is the "development of a simple mathematical model," requiring students to construct the equation that describes the circular motion based on their data analysis.
• HSN.Q.A.1: Use units as a way to understand problems and to guide the solution of multi-step problems.
• Connection: In the differentiation/analysis phase, students are guided to help "simplify the units for the slope" of their graphs, using dimensional analysis to verify their mathematical model.

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National Standards

Common Core English Language Arts (CCSS.ELA)

• RST.11-12.7: Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem.
• Connection: Students synthesize information from the physical demonstration, the computer simulation, and the data analysis software to construct an understanding of circular motion.
• RST.11-12.1: Cite specific textual evidence to support analysis of science and technical texts.
• Connection: Students read the provided texts on Johannes Kepler and Earth ("An Appreciation of Earth") and answer inquiry questions about historical scientific concepts like "motive power".
• SL.11-12.1: Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 11–12 topics, texts, and issues.
• Connection: The lesson utilizes a "board meeting" format where students display their whiteboards and engage in a whole-class discussion to come to a consensus on the mathematical model.

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National Standards

Objectives:

Today we will design an experiment to investigate circular motion.

Today we will collaborate to collect and analyze data to determine the physical laws of circular motion.

Today we will share our understanding of circular motion with others using graphical and mathematical representations.

Today we will explore the application of circular motions to the Earth and other planets through readings about orbits and how Earth would be different if the gravitational field strength on our planet were larger or smaller.

Agenda (60 minutes per day)

Day 1:

Circular motion experiment design

Data collection using online simulation.

Use Graphical Analysis to plot data and determine mathematical relationships.

Prepare small whiteboards for sharing.

Day 2:

Hold “board meeting” to reach preliminary consensus and discuss next steps.

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Complete analysis in small groups and share with class.

Reach final consensus on physical laws of circular motion.

Day 3:

Discuss readings in context of circular motion and central forces.

Complete examples demonstrating the application of mathematical relationships.

Why do some objects follow circular paths?

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Is there something special about curved paths?

What can we measure to understand?

Let’s investigate further!

Hands-on Activity Instructions

• In a group of 2-4, collect data using the Classic Circular Force Lab simulation from the Physics Aviary.
• Your teacher will assign you both the moving mass and the number of washers (10 g each) to use in your experiment.
• Choose eight fairly equally spaced values for the Radius of your circle between 0.25 and 2.00 m. Carefully measure and record the Radius before beginning each trial.
• For each Radius, measure the time it takes for the moving mass to make ten complete trips around the circle, divide by ten, then record it as the Period (T) of the motion.
• Calculate the Speed (v) of your moving mass using v=2πR/T and record the value to two decimal places.
• Open Vernier Graphical Analysis and enter your Radius (R) values in the x column and your Speed (v) values in the y column. Inspect your v vs. R graph and make any necessary adjustments to linearize the plot.
• Find the slope and intercept of your linearized plot using a linear fit. Write the equation you find relating Speed (v) and Radius (R), including units for all numerical quantities.
• On your group whiteboard, sketch your graph and write your equation.
• What do you think the slope of your linearized graph tells you about the motion of your moving mass? Is your intercept significant?

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Assessment

Data Collection and Graphing: What does your v vs R graph tell you? Does it need to be modified to create a linear relationship, so you can write a mathematical relationship? What is your slope?

Compare and Contrast: How do your results compare to your classmates? What is the same and what is different? What can we conclude about all objects traveling along circular paths?

Complete Whiteboard, PostLab, and/or Lab Report: Share what you have learned with others.

Readings: Kepler thought the sun had a “motive power” that caused the planets to move. What do we now understand about that idea? How does the strength of that “motive” power affect our lives on Earth?

Differentiation

• Circulating as groups work on their data collection and analysis of their results will allow the teacher to provide additional assistance to groups that may need it. This might take the form of suggesting possible methods of linearizing the v vs R graph or help in simplifying the units for the slope of the v² vs R graph.
• For issues that perplex multiple groups, a whole-class discussion can help, especially if suggestions for solutions come from the other groups in the classroom instead of the teacher.

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Remediation

Extension/Enrichment

• Groups that complete their analysis quickly can be enlisted to assist groups that might be struggling, especially with the nuts and bolts of using the Classic Circular Force Lab simulation or plotting data in Graphical Analysis. Teaching others is always the best way to become expert in something new.
• The Readings (Johannes Kepler - Before Newton and An Appreciation of Earth) provided can also be used as an extension for students who have completed their analysis quickly. While some of Kepler’s ideas about the Music of the Spheres seem odd to us today, the discussion of how small changes in Earth’s inclination or mass or orbit would affect life here are fascinating for students to contemplate.