Check out this trailer video from OpenStax about the Moment of Inertia.

https://www.youtube.com/watch?v=IpZL-gxTKy4

Pre-lecture Study Resources

Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.

BoxSand Videos

Learning Objectives

Summary

We have previously studied Newton's laws of motion while only considering what happens to the center-of-mass (COM) of an object, otherwise known as a point particle model.  In statics and dynamics, we still work under the framework of Newton's laws of motion, but we extend our consideration from the point particle model to also include the shape and size of the object. The sum of the forces on a object still determine the linear motion of it's COM, but where those forces are applied will tell us something about how and if the object rotates.

Atomistic Goals

Students will be able to...

Statics and Dynamics

1. Understand the application of Newton's laws of motion to rigid bodies, while also identifying the similarities when compared to the point particle model.

2. Demonstrate the ability to analyze a system in rotational equilibrium using the rotational version of Newton's second law.

3. Define moment of inertia.

4. Understand moment of inertia and its dependence on mass distribution.  Also recognize its analogy to the previous working definition of inertia as applied to the point particle model.

5. Connect quantities found from a rotational dynamics analysis to both rotational and linear kinematics.



Stability

6. Be able to calculate the center of mass of an object or a system of objects.

7. Be able to determine if an object or system of objects will be stable based on center of mass location relative to normal forces.

BoxSand Introduction

Statics and Dynamics  |  Dyamics and Moment of Inertia

Dynamics involves applying Newton's 2nd Law for rotation.

A representation with the words newton's second law on the top. There is an equation that shows that the net torque from forces external to the system acting about axis naught, is equal to the product of the moment of inertia about axis naught, and the angular acceleration about axis naught. this is also written in words below.

In a system that is not in rotational equilibrium, the angular acceleration is not zero. This results in the RHS (right-hand side) of the 2nd law to not be zero. You will need to know the moment of inertia for a mass distribution. Below are the equations for a the moment of inertia for some common mass distributions.

A representation with the words moment of inertia on the top. There is an equation that shows that the moment of inertia is found by adding up all the pieces of mass multiplied by the square of the distance from each piece of mass to the axis of rotation. This is also written in words below.

A representation with the words moment of inertia point particle on the top. There is an equation that shows that the moment of inertia of a point particle is equal to the product of the mass of the point particle and the square of the distance to the rotation axis. This is also written in words below.

A representation with the words moment of inertia point particle on the top. There is an equation that shows that the moment of inertia of a disk is equal to one half the product of the mass of the disk and the square of the distance to the rotation axis. This is also written in words below.

Here are some examples of moments of inertia for various mass distributions.

Key Equations and Infographics

Now, take a look at the pre-lecture reading and videos below.

OpenStax Reading


OpenStax Section 10.3  |  Dynamics of Rotational Motion: Rotational Inertia.

Openstax College Textbook Icon

 

Equations, definitions, and notation icon Concept Map Icon
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Additional Study Resources

Use the supplemental resources below to support your post-lecture study.

YouTube Videos

Moment of inertia explanation.  At 12:25 doc begins to derive moments of inertia for various objects, you can ignore this. 

https://www.youtube.com/watch?v=PBo7llsc6LA

Doc Schuster talks about rotational dynamics.

https://www.youtube.com/watch?v=B_yYrgRs8Sk

Simulations


PhET simulation on dynamic torque. Note that when you let the wheel rotate without acceleration, it is actually in torsional equilibrium despite the fact that it is in motion.

Phet Interactive Simulations Icon

For additional simulations on this subject, visit the simulations repository.

Simulation Icon

You need to have Java installed and updated. Download and run the file.

 

Demos


A demonstration of the intermediate axis theorem, an application of moment of inertia.

https://www.youtube.com/watch?v=-Si6iRL5Fj8

For additional demos involving this subject, visit the demo repository

Demos Icon

History


Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

Physics Fun

Other Resources


Read pages 1-10 for static equilibrium.

Other Content Icon

This links to a set of slides detailing static equilibrium. 

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Resource Repository

This link will take you to the repository of other content related resources.

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Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

Assumptions

 

Checklist

 

Misconceptions & Mistakes

 

Pro Tips

 

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

Physical

Physical Representations describes the physical phenomena of the situation in a visual way.

 

Mathematical

Mathematical Representation uses equation(s) to describe and analyze the situation.

A representation with the words newton's second law on the top. There is an equation that shows that the net torque from forces external to the system acting about axis naught, is equal to the product of the moment of inertia about axis naught, and the angular acceleration about axis naught. this is also written in words below.

A representation with the words moment of inertia on the top. There is an equation that shows that the moment of inertia is found by adding up all the pieces of mass multiplied by the square of the distance from each piece of mass to the axis of rotation. This is also written in words below.

A representation with the words moment of inertia point particle on the top. There is an equation that shows that the moment of inertia of a point particle is equal to the product of the mass of the point particle and the square of the distance to the rotation axis. This is also written in words below.

A representation with the words moment of inertia point particle on the top. There is an equation that shows that the moment of inertia of a disk is equal to one half the product of the mass of the disk and the square of the distance to the rotation axis. This is also written in words below.

 

Graphical

Graphical Representation describes the situation through use of plots and graphs.

 

Descriptive

Descriptive Representation describes the physical phenomena with words and annotations.

 

Experimental

Experimental Representation examines a physical phenomena through observations and data measurement.

 

Practice

Use the practice problem sets below to strengthen your knowledge of this topic.

Fundamental examples

Practice Problems

    2. There are additional practice problems you can work for credit | Calendar