00:01
Okay, a disc and a hoop of the same mass and radius, or at the same time, at the top of an inclined plane.
00:07
If both are uniform, which one reaches the bottom first, if there is no slipping.
00:13
Okay, so, i've got an incline, and we've got like a ring, and then i'm going to draw them like not side by side.
00:30
Or not at the same height, but i've got a disc as well.
00:33
Okay, let's just pretend that they start with a center of mass height.
00:38
Of h and they travel a distance d down the slope.
00:49
Okay, so the key concept here is that the disc and the hoop have different moments of inertia.
00:59
Okay, so the moment of inertia is the symbol i, tells you how resistant an object is to rotating.
01:28
Okay, so the greater the moment of inertia, the greater the object will resist rotation.
01:35
So let's just see what we have here.
01:37
The moment of inertia of a disk is 1 half m r squared.
01:45
The moment of inertia of a ring, or what do they call it a hoop here? a hoop, yeah, a hoop.
01:53
The moment of inertia of a hoop is m r squared.
01:56
Okay, so which has a greater moment of inertia, the hoop, that means that the hoop is going to be more resistant to rotation, which means it doesn't want to rotate as much as the disk does.
02:11
So the disk, once you let go, the disc is going to want to start rolling.
02:14
It's going to want to...
02:16
But the hoop is going to want to resist that.
02:20
So based off of that alone, we can already tell that the disc is going to win the race.
02:30
But we can also show it.
02:32
So let's look at energy here.
02:34
We're going to look at the energy at the top of the slope and the energy at the bottom of the slope.
02:39
We're going to do it for a general case.
02:41
We're not going to do it for either the disc or the hoop.
02:45
So what energy does the object have at the top of the slope? potential energy, mgh.
02:52
What energy does it have at the bottom of the slope? just kinetic, one -half mv squared, plus one -half moment of inertia, omega -squared.
03:04
Okay, linear velocity equals radius times angular velocity.
03:11
So we can substitute that in...
