Rolling Motion & the Moment of Inertia

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  • 0:05 What is Moment of Inertia?
  • 0:43 Measuring Moment of Inertia
  • 2:50 Theoretical Value
  • 3:20 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 should be able to figure out the moment of inertia of an object as it rolls down a ramp and compare it to a theoretical value. A short quiz will follow.

What Is Moment of Inertia?

Have you ever opened a really heavy door? Like the kind of door you'd find on an old, medieval castle. The kind with gigantic hinges and solid wood or metal construction.

If you have, you might have noticed that it's much harder to get a heavy door to swing than a light door. It's also much harder to get a door to swing by pushing it nearer the hinge than it is by pushing it further away from the hinge. A heavy door has more mass, and a large door has more mass away from the hinge. Because of both of these things, we can say that it has a greater moment of 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.

Measuring Moment of Inertia

So how can we measure moment of inertia? There are many ways, but today we're going to look at one particular scenario. We're going to imagine a ball rolling down a hill without friction. Obviously in real life, there's always friction, but we're just going to keep things simple.

Thanks to conservation of energy, we know that whatever energy the ball started with must be equal to whatever energy it finishes with. A ball at the top of a hill, released from rest, only has gravitational potential energy, which as we learned in another lesson is equal to mgh. Once you release it, it starts moving, getting faster and faster as it rolls down the hill. But it moves in two ways: translationally and rotationally. The ball as a whole translates down the hill, but the ball also turns - it rolls. So that gravitational potential energy turns into a mixture of translational kinetic energy and rotational kinetic energy. Again, we learned the equations for both of these in other lessons. Translational kinetic energy is 1/2-m-v-squared, and rotational kinetic energy is 1/2-I-omega-squared. Just as a reminder, m is the mass of the ball, v is the translational velocity of the ball at the bottom of the hill, g is the acceleration due to gravity (which is 9.8 on Earth), h is the height of the hill, and I is the moment of inertia of the ball.

So if we can measure everything except for I in this equation, we can figure out the moment of inertia. We get started and find out that the ball has a mass of 2 kilograms, a radius of 0.1 meters, and the hill has a height of 2 meters, after measuring multiple times, like all good scientists. We then take video of the ball rolling down the hill, and use some computer software to calculate its translational and rotational velocities at the bottom. After doing that, let's say it comes out to be a velocity of 6 meters per second and an angular (or rotational) velocity of 28 radians per second. What is the moment of inertia of the ball?

Well, all we have to do is plug numbers into the equation and solve for I. After doing a lot of algebra to make I the subject, we end up with this equation:

equation to solve for I

We plug in our numbers and solve, and we get a moment of inertia of 0.00816 kilograms meters squared.

Theoretical Value

How good was our experimental value? To do this we'll have to look at what the theory says SHOULD be the moment of inertia of a sphere. Assuming it's a solid sphere, a bit of calculus tells us that the moment of inertia of a solid sphere should be equal to two fifths of the mass multiplied by the radius squared. The mass of our sphere was 2 kilograms, and the radius 0.1 meters. So if we plug those in, we find that the moment of inertia of our solid sphere should have been 0.008. So we were pretty close!

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