Rotational Inertia & Change of Speed

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  • 0:00 What Is Rotational Inertia?
  • 1:02 Newton's 2nd Law
  • 1:51 Example Calculation
  • 3:19 Lesson Summary
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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.

What Is Rotational Inertia?

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.

Equation: Newton's 2nd Law

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.

Example Calculation

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.

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