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UExcel Physics: Study Guide & Test Prep18 chapters | 201 lessons | 13 flashcard sets

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Lesson Transcript

Instructor:
*David Wood*

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what rotational inertia is, describe how it makes it harder to change the speed of rotation, and solve rotational inertia problems. A short quiz will follow.

If you've ever tried to push a really heavy merry-go-round, you know that more massive objects are harder to make rotate. But you might have also noticed they're harder to stop rotating. This is all because of rotational inertia.

What is rotational inertia? Well, first of all, what is inertia?

Inertia is a very impressive-sounding concept that is actually extremely simple. When it really comes down to it, inertia is just mass. **Inertia** is the tendency of mass to resist a change in its motion. But it's the mass that DOES the resisting. Objects with mass require forces to make them speed up or slow down because they have inertia.

Okay, so what's ROTATIONAL inertia? Since so many linear (or translational) quantities have rotational versions, so does mass. Instead of mass, we have rotational inertia. **Rotational inertia** (otherwise known as moment of inertia) is a number that represents how much mass a rotating object has and how it's distributed. An object with more rotational inertia is harder to accelerate.

With regular inertia, the equation that tells you how hard it is to accelerate an object is Newton's 2nd Law. That's still true for rotational inertia. We just have to replace the translational quantities with rotational ones. So instead of *F* = *ma*, we get tau = *I*-alpha, where tau is the torque (force at a distance) you apply to the object, measured in newton meters, *I* is the moment of inertia (or rotational inertia) of the object, measured in kilogram meters squared, and alpha is the rotational acceleration of the object, measured in radians per second per second.

So the rotational version of **Newton's 2nd Law** tells us that objects with more rotational inertia, either through mass or the way it's distributed, take more force to increase or decrease their rotation.

But maybe this would be easier if we went through an example.

Let's imagine that you're now riding ON that merry-go-round. You and the merry-go-round together have a moment of inertia of 600 kilogram meters squared, and your 10-year-old daughter pushes the merry-go-round so that it accelerates from 1 radian per second to 5 radians per second, by pushing it for 4 seconds. How much torque (force at a distance) did she have to apply to make that happen?

First of all, let's write down what we know. The moment of inertia (or rotational inertia),*I*, is 600, and we're asked to calculate the torque, tau. We don't yet know the angular acceleration, alpha, but we do know the initial angular velocity, which is 1, and the final angular velocity, which is 5. We also know the time it took for that angular velocity to change, which is 4 seconds.

So how do we solve this? Well, looking at Newton's 2nd Law, the torque will be equal to your moment of inertia, 600, multiplied by your angular acceleration. But this question is made harder by not giving us the acceleration directly. In another lesson, we learned that angular acceleration is equal to the final angular velocity minus the initial angular velocity divided by the time it takes to make that change in seconds. So, we can calculate the acceleration by calculating 5 minus 1 divided by 4. That gives us an angular acceleration of 1 radian per second.

Finally, we can plug that into Newton's 2nd Law, and we get a torque of 600 newton meters. Which, to be fair, might be a bit of a struggle for a 10-year-old!

Inertia is really just mass. **Inertia** is the tendency of mass to resist a change in its motion. But it's the mass that does the resisting. Objects with mass require forces to make them speed up or slow down because they have inertia. **Rotational inertia** (otherwise known as moment of inertia) is a number that represents how much mass a rotating object has and how it is distributed. An object with more rotational inertia is harder to accelerate. Any object with mass will have rotational inertia, and this makes it harder to speed up or slow down the rotation of an object.

To get the main equation that involves inertia - Newton's 2nd Law - for rotational inertia, we just have to replace the translational quantities with rotational ones. So instead of *F* = *ma*, we get tau = *I*-alpha, where tau is the force you apply to the object, measured in newtons, *I* is the moment of inertia (or rotational inertia) of the object, measured in kilogram meters squared, and alpha is the rotational acceleration of the object, measured in radians per second per second.

The rotational version of **Newton's 2nd Law** tells us that objects with more rotational inertia, either through mass or the way it's distributed, take more force to increase or decrease their rotation.

After you've completed this lesson, you should have the ability to:

- Define inertia and rotational inertia
- Explain how to use Newton's 2nd Law to solve rotational inertia problems

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UExcel Physics: Study Guide & Test Prep18 chapters | 201 lessons | 13 flashcard sets

- Go to Vectors

- Go to Kinematics

- Differences Between Translational & Rotational Motion 4:33
- Rotational Kinematics: Definition & Equations 5:03
- Five Kinematics Quantities & the Big 5 Equations 6:02
- Torque: Concept, Equation & Example 4:52
- Rotational Inertia & Change of Speed 4:30
- Kinetic Energy of Rotation 4:14
- Rolling Motion & the Moment of Inertia 4:27
- Work & Power in Rotational Motion 4:46
- Angular Momentum vs. Linear Momentum 5:52
- Conservation of Angular Momentum 7:00
- Go to Rotational Motion

- Go to Relativity

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