Rotational Inertia

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Students will be able to define the moment of inertia and calculate the rotational inertia for standard and rigid bodies.

Big Idea

Today, students learn how to calculate rotational inertia for hoops, spheres, cylinders, and more.


In yesterday's lesson, students completed a lab on center of mass, and they already have a working knowledge of torque. Today, the goal is to expand our knowledge of rotational motion by qualitatively and quantitatively defining rotational inertia ( and it's relationship with Newton's Second Law HS-PS2-1). I try to offer a variety of strategies to accomplish our goal, so the lesson starts with activation of prior knowledge about inertia in a first word activity. Then, students take some brief notes on rotational inertia (SP5). Finally, students get to apply their new knowledge towards the end of class with collaborative problem solving (SP6).

First Word: Inertia

10 minutes

As our introduction to rotational inertia, I have the word 'inertia' written onto the board when students enter the room. Today, class starts with a first word activity that is meant to activate students' prior knowledge of inertia. We covered inertia in a previous lesson, so my students are familiar with the concept.

Students start the activity when they write the word 'inertia' vertically down the side of a blank sheet of paper. Then, in small groups of two or three, students create a word, short sentence, or phrase that begins with each letter of the vertical word. The word, short sentence, or phrase should also be relevant to the word inertia or rotational motion. I give them about 5 full minutes to work on their creations.

After the students have completed the activity, I ask students to share their best phrase or sentence. This quick sharing gives me an opportunity to see how much students remember about inertia and how much review of the concept I need to do in today's notes. Also, when students are sharing they might activate prior knowledge in another student with their contributions.

Notes on Rotational Inertia

25 minutes

It's time for students to take out their notebooks to copy down the definition of moment of inertia, along with some sample problems. My students are operating under the expectation that they must write down key points from the presentation. This expectation of how to take notes has been outlined and ingrained in their learning since freshman year. Because these are AP students, how they organize their notes (notebook, binder, etc.) is a decision the individual student gets to make. I assume at this point in their high school careers they have an established system to stay organized. 

I display the Moment of Inertia presentation (also available in pdf format) to help the students understand what they need to write down. I have a hard copy of the slides in my hands as I circulate throughout the room, which includes notes (viewable when the file is downloaded). These notes help me to stay focused and ensure I mention the highlights as we progress through each slide. I usually have a lot of interaction with my students throughout the presentation. The students ask questions, participate in solving any examples, and connect to real-world situations to stay engaged the entire time.

The specific goal of these notes is to show students that Newton's First and Second Laws still apply to rotating objects. So, I start by showing students how substitutions of rotational quantities can transform Newton's Second Law to produce the moment of inertia equation. Then, I give students a conceptual definition before asking them to apply their knowledge in a quantitative example. Before I provide the solutions to the first example, I give students several minutes to read and think about the problem (individually or with those seated near them). Then, I ask students to contribute information as I walk them through the solution. Finally, I include the different expressions for rotational inertia of standard shape objects before giving students a final example.

Collaborative Rotational Inertia Problem Solving

15 minutes

Students get themselves into groups of 6, and since they are mature and able, I let students choose their own groupings. Usually we have groups of 2, 3, or 4, so today is definitely an opportunity for students to work with someone new. Once students are in their groups, they are instructed to pick one person for each homework problem assigned. As they are deciding who is which number, I go around and pass out today's practice problems and encourage students to write their name and number at the top. 

Once students are organized, I direct them to solve the problem that corresponds to the number they've chosen. For example, if Sonya is number 5, she's in charge of doing only number 5 on her problem set. If students have done this properly, each group has 1 student doing each problem. The purpose of this strategy is to get students practicing what they've just learned before they share their answers.

The students get about 10 minutes to solve their individual problems, although they should use their peers as a resource if needed. I am also walking around the room with the answer key to answer questions and provide students hints as needed. At the end of the work time, I ask all of the students who had the first problem to share their answers. Each person from each group shares, and then I reveal the correct answer. I share the correct answer verbally, unless everyone missed the question. In the event that nobody got the question correct, I work the solution on the front board. The students that did not work on the first problem writes down the correct answer. The process continues with groups 2 through 6 so everyone has a chance to share and record the correct answers.

With any remaining time before the end of class, students debrief in their groups and discuss their work and the solutions. I walk around and encourage students to ask each other questions, so students might ask a peer "Why did you use that distance?" or "Did you round that part of your answer to get the final solution?" This type of closure to the lesson is very informal, but it gives me an opportunity to talk with a variety of students in small groups while checking for understanding. The collaborative conversations between students, with or without me present, continue until the end of the class.