Why is the rotational inertia of a hoop with mass, M, and a...

Name: Why is the rotational inertia of a hoop with mass, M, and a radius, R, greater than the rotational inertia of a disk that has the same mass and radius? Why is the rotational inertia of a spherical shell of mass, M, and a radius, R, greater than the rotational inertia of a solid sphere that has the same mass and radius?
Uploaded: 2022-02-22T21:49:40-08:00
Duration: 7 min 37 s
Channel: Gregory Devenport
Description: Why is the rotational inertia of a hoop with mass, M, and a radius, R, greater than the rotational inertia of a disk that has the same mass and radius? Why is the rotational inertia of a spherical shell of mass, M, and a radius, R, greater than the rotational inertia of a solid sphere that has the same mass and radius?

Why is the rotational inertia of a hoop with mass, M, and a radius, R, greater than the rotational inertia of a disk that has the same mass and radius? Why is the rotational inertia of a spherical shell of mass, M, and a radius, R, greater than the rotational inertia of a solid sphere that has the same mass and radius?

Transcript

00:01 Okay, we want to think about the moments of inertia for different objects that are similar, but one that is solid and one that is hollow.

00:11 So we start with a hoop.

00:17 So a hoop has a moment of inertia of its mass times its radius squared.

00:25 So here's our hoop, and it has radius r and mass m.

00:35 Okay, we're comparing that to a disk, and this disk has a moment of inertia of one half m r squared.

01:09 Okay, so here's our disk.

01:14 Let's see, try to make these the same size here.

01:20 Okay, pretend they're the same size.

01:25 So, same radius, same mass, but this one's just...

01:28 Filled in, so this is all solid or whatever.

01:33 Okay, and we want to think about why these are different.

01:42 So why is the disk one half m r squared? and the hoop is just mr squared.

01:48 Okay, the reason for this is that the moment of inertia describes how an object is resistant to being rotated, or resistant to being accelerated.

02:04 Angular observation, how it resists standing still, not rotating.

02:12 So if you have an object with a large moment of inertia, then it's harder to rotate.

02:18 It doesn't like to be rotated.

02:19 And if you have an object with a small moment of inertia, it's easier to rotate.

02:23 So a disk is easier to spin than a hoop.

02:28 That's what this is telling us.

02:30 And this is because the formula for the moment of inertia is what you do is you take your point that you're rotating about and you sum up all the little pieces of mass and you take so some tiny bit of mass multiply it by the perpendicular distance from the rotation axis to that point square that multiply by the mass and you add all those up so so what happens is on the disk, you have some bits of mass that are close to the axis of rotation.

03:28 And actually, let me just draw that over here.

03:32 So here's our, here's this red, is our rotation axis.

03:36 So we're rotating about that center point, right? so for the hoop, you have all of the mass is here on the outside.

03:45 So it's all the same distance away.

03:46 It's a distance r away.

03:49 Now for the disk, you have some mass that's really close.

03:52 Close.

03:54 So that distance r is really small.

03:57 You have some that's like a little bit farther out, so the distance r is a little bigger...