Hey, guys. In this video, we're going to talk about something called the parallel axis theorem, which is something that we're going to use to find what I call non-standard moments of inertia. And it's going to be critical to solving a lot of rotational dynamics problems. Alright. Let's get to it.

Now, the moment of inertia is really important to know and to be able to find because it's the rotational analog to mass in rotational equations. Most rotational equations look exactly like linear equations. You simply just replace the linear variables with the angular or rotational variables. For instance, Newton's second law looks like f equals ma for a linear system. For a rotational system, it would look like the rotational, analog to force, which would be torque, the rotational analog to mass, which would be the moment of inertia, as I just reminded us of, and the rotational analog to acceleration, which would be angular acceleration. Every single equation looks like this. If you know the linear one, you know the rotational one. You just replace the variables with the rotational versions of those variables.

Now the problem with the moment of inertia is that it's not a fixed quantity like the mass is. If you know the mass, you always know the mass. The moment of inertia is a relative quantity and it depends upon the rotation about which the motion is occurring. Typically, there will be a select few moments of inertia given on things like homework or your exam, which I would call typical or standard moments of inertia. Usually, they're given about the center of mass of the object. Typically, they're given for regular geometric shapes like spheres, discs, rings, and typically they are given for those objects when they are uniformly distributed, meaning homogeneous, as in the density is the same throughout the object. The mass is evenly distributed throughout the whole object.

But how do you solve a problem when you have a non-typical rotation and you need a non-typical moment of inertia? This is where the parallel axis theorem comes in. If, for instance, we want to know the moment of inertia for a disk rotating about its rim, first, we would start with the disc rotating about an axis through its center, which hopefully you guys remember this. It's a very common one. The moment of inertia is
1
2
m
r
2
. But what if now I wanted to choose a parallel axis? So obviously parallel axis theorem, these axes must be parallel to one another, but the second axis is on the rim, which means it's a distance of the radius away. This is where we need to use the parallel axis theorem to find the new moment of inertia. The parallel axis theorem is going to be the moment of inertia through the center of mass, which in this case we know because we know that through the center, which is the center of mass for a uniform disk, the moment of inertia is
1
2
m
r
2
. This is going to be plus md^{2}, where d is the distance from the center of mass axis to the new axis. The moment of inertia is going to be the moment of inertia of the center of mass plus the mass of the disc times the distance squared, where the distance is the distance between the center of mass and the new parallel axis. You can see that these two axes are definitely parallel.

If we have a disk of mass m and radius r, what is the moment of inertia about an axis perpendicular to the surface of the disk at the rim of the disk? All we need to do is apply the parallel axis theorem. It's the center of mass plus md^{2} where the distance to the rim is the radius. The moment of inertia through the center of a disk is
1
2
m
r
2
plus
m
r
2
. This is going to be
3
2
m
r
2
.

What about a parallel axis halfway to the rim? Right? Here's our original axis through the center of mass. And here is our new axis, where this distance is 1_half of the radius. Halfway to the rim means that the distance is half of the radius. So what is that new moment of inertia? The new moment of inertia is still the moment of inertia through the center of mass plus capital md^{2}, where now the distance is one half of the radius. Remember that it's the distance squared, so the one half is squared as well. So this is 1_half
1
2
m
r
2
, which is simply the moment of inertia through the center of mass, plus 1_fourth
m
r
2
. Because this one half also gets squared. One half plus a quarter is 3 quarters, so this is 3 quarters
m
r
2
.

The moment of inertia of a thin rod. So we have a rod of some length, l, and some mass, m. The moment of inertia about an axis perpendicular to the rod at the end of the rod is one third. This should be a capital m, just because I called this a capital m. The moment of inertia is one third M l 2 . What we want to know is, what is the moment of inertia halfway between the center of the rod and the edge of the rod? This is one quarter of the length of the rod because the distance to the center of mass is clearly one half of the length, right? That's clearly true.

Now should we simply say that this moment of inertia is one third M l 2 plus M l 2 1 4 ? No. We shouldn't say that that's true. And the reason is that to use the parallel axis theorem, you need to know the moment of inertia about the center of mass, which we don't know in this case. We know it about the edge. So first we need to sort of use the parallel axis theorem, in reverse, to find the moment of inertia at the center of mass, then we can find it at the point we're interested in. Here is the rod. Here's the center of mass. This moment of inertia we don't know, but we want to find. Here's the moment of inertia at the edge, which we do know is one third M l 2 , and this guy is one half l. So using the parallel axis theorem here, we know that I is I times the center of mass plus m times one half l squared. We know I, this is clearly a constant and we want to solve for this. So this is I minus one quarter M l 2 . Right? One fourth because this one half just gets squared. And I, we know is one third M l 2 minus one fourth M l 2 . So this whole thing is just 1 12 M l 2 . Now that we know the moment of inertia about the center of mass, now we can apply the parallel axis theorem. This distance is what we're interested in which is still one fourth l. But just to be technical, we're moving it from the radius, from the axis through the center of mass. We're doing that from there to this new axis. So this is the center of mass moment of inertia, which we know is one twelfth M l 2 plus M 1 16 l 2 . Now to find the least common denominator, how I did this is 12 is 4 times 3. 16 is 4 times 4. They both share that common factor of 4. So if I multiply the 12 times 4 and the 16 times 3, I should get a common denominator. So this is going to be 4 48 M l 2 + 3 48 M l 2 . And it turns out I do get a common denominator. And 4 + 3 is just 7. So 7 48 M l 2 . Okay? So that is the moment of inertia of an axis halfway between the center of the rod and the edge of the rod. Remember that the parallel axis theorem requires you to know the moment of inertia at the center of mass. So you cannot do this right here and try to move the axis from the edge to halfway to the edge because the moment of inertia at the edge is not the center of mass. You need to use the parallel axis theorem like this to find the moment of inertia at the center of mass, then you can move that axis to the edge. Alright, guys. Thanks so much for watching. I'll see you guys in another video.