Video: Moment of Inertia

In this video, we will learn how to calculate the angular mass, called the moment of inertia, of rotating objects of various regular shapes.

14:25

Video Transcript

In this video, we’re talking about moment of inertia. This is a term that describes how different objects respond to rotational force. For example, if we release the three masses at the top of this inclined plane at the same moment, then depending on each one’s moment of inertia, it will start to roll more or less quickly. In this lesson, we’ll come to a better understanding of what moment of inertia means, and we’ll study what the moments of inertia are for some common shapes rotating about common axes.

The first thing we can say about this term is that it describes the capacity of a given mass to rotate about a particular axis. Now, if we have some massive object, say this solid sphere right here, we can see that there are lots of different ways this mass could be rotating. For example, it could be rotating about an axis that goes right through its center, like this. But really, that axis could be anywhere. It could be, say, off center for the shape or it could even be outside the shape completely, with our mass moving around the axis like this, much like the Earth orbits the Sun.

So we see right away that there are many different ways a mass can be rotating. And not only that, but given a certain axis of rotation, there are many different masses that could be rotating around it. Let’s say that for each one of these four spheres, the density of the material making up the solid sphere is the same. And therefore, the largest sphere has the most mass, the next largest sphere has the second most, and so on. We can imagine all four of these spheres starting out still, not rotating at all, but requiring a push from us to start spinning around the axis through their centers. Thinking this way, we can see that it would take a smaller push, less force, to get the smaller sphere rotating, while the biggest sphere would take a lot of effort and a lot of pushing.

We could say then that the term, moment of inertia, has to do with how difficult it is to get an object rotating or stop it once it is. The larger an object’s moment of inertia is, the more likely it is to continue on with its rotational motion unchanged. In this way, moment of inertia is like another physics term we’ve learned before. Say that next to our rotating spheres, we have these masses of different sizes, but again with the same density, sitting on a frictionless surface, say it’s a surface of ice. We know intuitively that the force we need to apply to this smallest box to give it some acceleration is less than the force we would need to apply to the largest one to give it that same acceleration. So the larger the mass, the harder it is to get this box moving, just as we saw that the larger the moment of inertia, the harder it is to get an object rotating. So moment of inertia, which applies to rotating objects, is like mass when we talk about those masses moving in a straight line.

Knowing this, let’s now consider the simplest moment of inertia of some mass that’s rotating about some axis. In this example, we’ll say that our mass is a point mass that it literally takes up no space, but it does have some mass value. And then we’ll say that this mass is rotating about an axis that’s a distance 𝑟 away. So then the path our rotating mass would follow might look like this. If we want to calculate the moment of inertia of this point mass as it rotates about this axis, first off, the way we symbolize that moment is using the letter capital 𝐼. And in our case, we’ll give it this subscript pm because we’re talking about the moment of inertia of a point mass.

Mathematically, this is equal to the mass of our object multiplied by its distance from the axis of rotation squared. So if we consider again that moment of inertia applies to a mass that’s rotating, we can see those two elements in this mathematical definition. We have our mass 𝑚, and then the fact that it is rotating means that there’s a radius 𝑟 that we can include in this equation. And we notice that this radius is squared. For this reason, moment of inertia is sometimes also called the second moment of mass for an object about some axis. But anyway, this moment of inertia, for the simplest possible object we could have, a point mass, shows us how moment of inertia can be calculated even for larger objects like our sphere or a cube or some other shape.

What we do is we take each little bit of mass in that shape, and we multiply it by the distance of that bit of mass from the axis of rotation squared. And then we add all of those values together. In just a minute, we’ll see more about what moments of inertia for common shapes rotating about common axes are. But for now, let’s note that the units of the moment of inertia of a point mass and indeed the units of the moment of inertia for any object are the units of mass. The SI base unit of mass is the kilogram multiplied by distance squared where the SI base unit of distance is the meter. Whenever we calculate a moment of inertia then, we expect to have these units, kilograms meters squared. Knowing this, let’s now look at moments of inertia for shapes that are different from a point mass.

The first shape we’ll consider is a solid sphere, once again rotating about an axis through its center. The moment of inertia for this object, rotating about an axis like this, is two-fifths its mass times its radius squared. So then if we know the sphere’s radius and we know its mass, then we can calculate its moment of inertia as it would rotate about an axis through its center. Next, let’s consider a sphere that’s similar, but this one is hollow. We can think of it as an empty spherical shell, again rotating about an axis through its center. The moment of inertia of a hollow sphere rotating this way is equal to two-thirds of the sphere’s mass times its radius squared. Comparing these two equations, we see that if we had two spheres, one solid and one hollow, but they both had the same mass and the same radius, then the hollow sphere would actually have a greater moment of inertia than the solid one.

For our next shape and axis, say that we have a solid cylinder rotating about a line along its central axis. A mass of this shape rotating this way has a moment of inertia of one-half its overall mass times its radius squared, where the radius of the cylinder is this distance right here. It’s the radius of the circle that makes up the end of the cylinder. Now so far, we’ve considered the moments of inertia of different shapes, but shapes that rotate about the same axis we could say, an axis through their center. But what if we had an object, say a rod of length 𝐿, that rotated about an axis in contact with one of the ends of the rod? It turns out that the moment of inertia of such an object is equal to one-third its overall mass times its length squared.

Note, though, that if we had the same rod of length 𝐿 and mass 𝑚 rotating now about an axis through its center, then its moment of inertia is different. It’s one twelfth the mass of the rod times its length squared. The difference between this relationship and this one underscores the fact that the axis about which our shape is rotating affects its moment of inertia. Note, though, that regardless of the axis of rotation, for all of our shapes so far, the units of the moment of inertia are kilograms meters squared. That stays constant, no matter what kind of shape we’re working with or how it rotates. Another shape we can consider is a ring with an axis through its center. This ring is defined by two radii, an inner one here, we’ll call 𝑟 one, and an outer one here, we’ll call 𝑟 two.

Set up this way, the moment of inertia of our ring is one-half its overall mass times the quantity, its interior radius 𝑟 one squared plus its exterior radius 𝑟 two squared. Considering this moment of inertia for our ring and also recalling the one we wrote down for our cylinder, notice something interesting. In both cases, this moment of inertia does not depend on what we could call the height of our object. It only depends on the masses and radii of these objects. Having seen these, let’s now consider just a few more moments of inertia of common shapes as they rotate.

Say, we have a shape, we can call a hoop, rotating about an axis like this. If the hoop has a radius 𝑟, then its moment of inertia about this axis is the mass of the hoop times that radius squared. Note that, once again, this moment of inertia does not depend on what we could call the height of the hoop. And lastly, let’s imagine a shape called a cuboid. If we say that this shape has a width 𝑤 and a depth 𝑑, then its moment of inertia as it rotates about an axis that goes through the center of what we could call its top is one twelfth its mass times the quantity 𝑤 squared plus 𝑑 squared, where 𝑤 is its width and 𝑑 is its depth. Knowing all this about moment of inertia, let’s now get some practice with these ideas through an example.

Which of the following correctly shows the SI unit for the moment of inertia? (A) Kilograms per meter squared, (B) kilograms squared times meters, (C) kilograms squared times meter squared, (D) kilograms times meter squared, (E) the quantity kilograms times meters squared.

Considering this question, we can see that we’re not talking about moment of inertia for a particular shape, rotating about a particular axis. But instead we’re speaking of this term in general; we want our answer to give the units that apply to all moments of inertia. Along these lines, we can recall that this term, moment of inertia, applies to a mass that’s rotating. In general, the equation for the moment of inertia of a particular mass rotating about a particular axis depends on those two values. But nonetheless, it is true that all moments of inertia share the same SI base units. One way then to identify which of our five options is correct is to recall the equation for the moment of inertia of any shape rotating about any axis.

Perhaps the simplest moment of inertia we can recall is that of a point mass, where that mass is in rotation about an axis a distance 𝑟 away. The moment of inertia of this point mass is its mass 𝑚 times that distance 𝑟 squared. Now, like we said, not all moments of inertia have the same form. In fact, most are different. But they do all have the same units that this one has. And considering the units of this expression, we know that the SI base units of mass are kilograms and that the SI base unit of distance is the meter. So we have some mass in kilograms multiplied by some distance in meters squared, which means that our units for this expression will be kilograms times meter squared.

And like we said, these units apply not just to the moment of inertia of a point mass, but to all moments of inertia. Looking over our answers, we see that option (D) matches up with what we found. And so we choose this as our answer. Kilograms times meter squared are the correct SI units for moment of inertia.

Let’s look now at a second example exercise.

The objects shown in the diagram both rotate around the same axis and have the same moment of inertia. The radius 𝑟 of both objects is the same. What is the ratio of the mass of the disk to the mass of the hoop?

In our diagram, we see first this solid disk of radius 𝑟 rotating about an axis through its center. And then above it, we see this hoop also with radius 𝑟 rotating about that same axis. Because each of these objects has a mass and is also rotating about some axis, each one has a moment of inertia. And we’re told that those moments are actually the same. So if we call the moment of inertia of the hoop 𝐼 sub h and the moment of inertia of the disk 𝐼 sub d, then we know that these two quantities are equal.

Our question goes on to ask, what is the ratio of the mass of the disk to the mass of the hoop? To answer this question, we’ll need to know what is the moment of inertia of the hoop and of the disk in terms of each object’s mass and size. Often, for a given shape rotating about a certain axis, we’re able to look up that object’s moment of inertia in a table. And if we do this for a hoop rotating about an axis through its center, we find that its moment of inertia is equal to its mass times its radius squared. In a similar way, if we look up the moment of inertia for a disk which notice has the same shape as a cylinder, we find that moment of inertia is equal to one-half the disk’s mass times its radius squared.

Now, one important thing about these two equations is that we don’t know offhand whether this 𝑟 is the same as this 𝑟 and whether this 𝑚 is the same as this 𝑚. For each equation, we’re speaking specifically about that object’s mass and that object’s radius. In general then, we can’t assume that these 𝑟’s are the same or that these 𝑚’s are the same. In our specific scenario, though, we are told that the radius of both objects is the same. So we’ll refer to both of those radii with the same symbol; we’ll just use lowercase 𝑟, while the masses of these objects, which we are not told are the same, we’ll represent using 𝑚 sub h for the mass of the hoop and 𝑚 sub d for the mass of our disk.

All this means that we can write the moment of inertia of our hoop as 𝑚 sub h times 𝑟 squared. And we can write the moment of inertia of our disk as one-half 𝑚 sub d times 𝑟 squared. And as we’ve seen, these two quantities are equal. Now, we want to find the ratio of the mass of the disk to the mass of the hoop. In other words, we want to calculate 𝑚 sub d divided by 𝑚 sub h. That’s the ratio of interest. And we can solve for that ratio by rearranging this equation.

As a first step, we can notice that the same radius, 𝑟, multiplied by itself, 𝑟 squared, appears on both sides. So that factor can cancel out of this expression. And now, knowing that we want to get the fraction 𝑚 sub d divided by 𝑚 sub h in our equation, we can divide both sides by 𝑚 sub h, which means that our left-hand side simplifies to one. And then lastly, if we multiply both sides of this expression by two, we find that the ratio 𝑚 sub d to 𝑚 sub h is equal to two. This then is our answer. The mass of the disk is twice as large as the mass of the hoop.

Let’s summarize now what we’ve learned about moment of inertia. In this lesson, we saw that moment of inertia is a term describing a mass that is rotating. We saw that an object’s moment of inertia depends on how its mass is distributed and the axis that that object rotates around. The SI base units of moment of inertia are kilograms times meter squared. And lastly, we studied the moments of inertia for some common shapes and rotations. This is a summary of moment of inertia.

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