# Kinetic Energy of Rotation

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• 0:01 What Is Kinetic Energy…
• 0:51 Equation
• 1:57 Example Calculation
• 3:06 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 kinetic energy of rotation is, provide the equation to calculate it, and solve simple kinetic energy of rotation problems. A short quiz will follow.

## What is Kinetic Energy of Rotation?

Anything that moves has kinetic energy. But what about objects that rotate? The equation for kinetic energy is one-half m v squared (1/2 mv^2). But if the velocity v is zero, what then? A rotating object isn't moving left or right, up or down, forward or backward. So, surely its kinetic energy is zero, too?

But that doesn't make sense. If you push a merry-go-round in the park, it spins faster... and you get increasingly tired. You had to use energy in your body to push it, energy you got from your food. So, that energy has to go somewhere. And it does.

It turns out that there are two types of kinetic energy: translational and rotational. Kinetic energy of rotation is the movement energy an object has due to its spin.

## Equation

The equation for translational kinetic energy was one half mass times the velocity squared. Rotational kinetic energy isn't all that different. In rotational motion, we replace MASS with MOMENT OF INERTIA, and we replace VELOCITY with ANGULAR VELOCITY. So, the rotational kinetic energy equation is just one half, multiplied by the moment of inertia, 'I', measured in kilogram meters squared, multiplied by the angular velocity, omega, squared.

The angular velocity is the number of radians the object rotates by each second. A radian is a measure of angle, pretty similar to degrees, except whereas there are 360 degrees in a circle, there are 2 times pi radians in a circle - 2 pi radians.

And the moment of inertia is the rotational equivalent of mass - it's a quantity that helps an object resist a change in its rotation. Just like more mass makes it harder to accelerate an object linearly, a larger moment of inertia makes it harder to speed up or slow down a rotation.

Moment of inertia depends on the object's shape, its mass, and the way that mass is distributed around the rotation axis.

## Example of Calculation

Okay, let's go through an example. A merry-go-round with uniform mass distribution is rotating around its axis at a rate of two rotations a second. If the moment of inertia of the object is 16 kg m^2, how much rotational kinetic energy does the merry-go-round contain?

First of all, let's write out what we know. We know that the moment of inertia, I, is 16. And we know the rate of rotation. A full rotation contains 2-pi radians, so two rotations a second would be 4-pi radians a second. Which means that the angular velocity is 4-pi radians per second. So, we know the angular velocity, too. All we have to do now is plug the numbers into the equation and solve for the kinetic energy.

One half, multiplied by 16, multiplied by 4-pi squared, gives us 1,263 Joules. And that's it; we're done!

While things can get more complicated with non-uniform objects, the difficult part in such situations is calculating the moment of inertia itself. Once you have it, figuring out the rotational kinetic energy is usually pretty easy.

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